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Micron Document

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<title>Horner-Schema</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Horner-Schema</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Das <b>Horner-Schema</b> (nach <a href="William_George_Horner" title="William George Horner">William George Horner</a>) ist ein Umformungsverfahren für <a href="Polynom" title="Polynom">Polynome</a>, um die Berechnung von Funktionswerten zu erleichtern. Es kann genutzt werden, um die <a href="Polynomdivision" title="Polynomdivision">Polynomdivision</a> sowie die Berechnung von <a href="Nullstelle" title="Nullstelle">Nullstellen</a> und <a href="Differentialrechnung#Ableitungsberechnung" title="Differentialrechnung">Ableitungen</a> zu vereinfachen.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Zu einem <a href="Polynom" title="Polynom">Polynom</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)=b_{0}+b_{1}x+b_{2}x^{2}+\dotsb +b_{n}x^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle p(x)=b_{0}+b_{1}x+b_{2}x^{2}+\dotsb +b_{n}x^{n}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fdb42d248a306323f22bf69562ea55a9c402c8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:36.214ex; height:3.176ex;" alt="{\displaystyle p(x)=b_{0}+b_{1}x+b_{2}x^{2}+\dotsb +b_{n}x^{n}}" loading="lazy"></span> vom Grade <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> aus einem beliebigen <a href="Polynomring" title="Polynomring">Polynomring</a> ist das <i>Horner-Schema</i> definiert als:

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<annotation encoding="application/x-tex">{\displaystyle p(x)=(\dotso (b_{n}x+b_{n-1})x+\dotsb )x+b_{0}.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b1b15c59eb0b80b574fb8accde21905093d3c86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:38.69ex; height:2.843ex;" alt="{\displaystyle p(x)=(\dotso (b_{n}x+b_{n-1})x+\dotsb )x+b_{0}.}" loading="lazy"></span><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Funktion_des_Hornerschemas">Funktion des Hornerschemas</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Rechenvorteile">Rechenvorteile</h3></div>

<p>Bei <a href="Polynom" title="Polynom">Polynomen</a> in der klassischen Schreibweise müssen die <a href="Potenz_(Mathematik)" title="Potenz (Mathematik)">Potenzen</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle a^{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f564e5dc0b6e68af32ca8614e972f5b36e944a24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.284ex; height:2.676ex;" alt="{\displaystyle a^{2}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle a^{3}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/abd83c98f7301a720f69dd6d4043461e4cc83daa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.284ex; height:2.676ex;" alt="{\displaystyle a^{3}}" loading="lazy"></span> usw. errechnet werden, wenn der Funktionswert an einer Stelle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a}">

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<annotation encoding="application/x-tex">{\displaystyle x=a}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaae23950e96a955ab5b07015a168fd931d4d82b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\displaystyle x=a}" loading="lazy"></span> errechnet werden soll. Im umgeformten Polynom nach dem Horner-Schema kommen keine Potenzen, sondern nur noch <a href="Multiplikation" title="Multiplikation">Multiplikation</a> und <a href="Addition" title="Addition">Addition</a> vor. Die Berechnung wird beschleunigt, weil weniger Multiplikationen nötig sind: Deren Anzahl wird durch die Anwendung des Horner-Schemas auf fast die Hälfte reduziert.

</p><p>In der klassischen Schreibweise sind <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n-1}">

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<annotation encoding="application/x-tex">{\displaystyle 2n-1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb6ce8a02613283b1e60305814a1457335b44437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.56ex; height:2.343ex;" alt="{\displaystyle 2n-1}" loading="lazy"></span> Multiplikationen bei einem Polynom vom Grad <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle n-1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbd0b0f32b28f51962943ee9ede4fb34198a2521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" loading="lazy"></span> Multiplikationen zur Bildung der Potenzen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2},\,x^{3},\dotsc ,x^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle x^{2},\,x^{3},\dotsc ,x^{n}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b8c27d45af1acaa91bb349483cc174d9976c802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.915ex; height:3.009ex;" alt="{\displaystyle x^{2},\,x^{3},\dotsc ,x^{n}}" loading="lazy"></span>;<sup id="cite_ref-AnzMult_2-0" class="reference"><a href="#cite_note-AnzMult-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li>

<li>weitere <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> Multiplikationen zur Multiplikation der Potenzen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,\,x^{2},\,x^{3},\dotsc ,x^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle x,\,x^{2},\,x^{3},\dotsc ,x^{n}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d61e45895bf01deef6ec9d235a90b1f93b9a809.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.666ex; height:3.009ex;" alt="{\displaystyle x,\,x^{2},\,x^{3},\dotsc ,x^{n}}" loading="lazy"></span> mit ihren Koeffizienten.</li></ul>

<p>Insgesamt benötigt man deshalb <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n-1}">

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</p><p>Im Hornerschema hingegen kommt man mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</p><p>Die Zahl der – rechnerisch weniger aufwändigen – Additionen ist in beiden Fällen gleich, nämlich <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<div class="mw-heading mw-heading3"><h3 id="Verfahren">Verfahren</h3></div>

<p>Durch fortgesetztes Ausklammern der freien Polynomvariablen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<div class="mw-heading mw-heading3"><h3 id="Beispiel">Beispiel</h3></div>

<p>Das folgende Beispiel illustriert den geringeren Rechenaufwand beim Hornerschema:

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<annotation encoding="application/x-tex">{\displaystyle 2x^{4}-4x^{3}-5x^{2}+7x+11\;=\;(((2\cdot x-4)\cdot x-5)\cdot x+7)\cdot x+11}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/466c92f91fa6073ba8b487180aa99648c0ab40cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:67.006ex; height:3.176ex;" alt="{\displaystyle 2x^{4}-4x^{3}-5x^{2}+7x+11\;=\;(((2\cdot x-4)\cdot x-5)\cdot x+7)\cdot x+11}" loading="lazy"></span></dd></dl>

<p>In der klassischen Darstellung (linke Seite) werden zusätzlich zu den Additionen, Subtraktionen und Multiplikationen noch drei Potenzen gebildet, die durch die Verwendung des Horner-Schemas (rechte Seite) von den Multiplikationen erfasst werden und somit wegfallen. Bei Wiederverwendung der Zwischenergebnisse spart man sich dadurch drei Multiplikationen.

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<div class="mw-heading mw-heading3"><h3 id="Anwendung">Anwendung</h3></div>

<p>In der <a href="Analysis" title="Analysis">Analysis</a> müssen häufig die Werte eines Polynoms und seiner Ableitung berechnet werden: Sei es, um eine <a href="Nullstelle" title="Nullstelle">Nullstelle</a> zu bestimmen, eine <a href="Kurvendiskussion" title="Kurvendiskussion">Kurvendiskussion</a> durchzuführen oder um einen Graphen zu skizzieren.

</p><p>Die hier dargestellte Form eignet sich besonders gut für die Berechnung in der <a href="Umgekehrte_polnische_Notation" title="Umgekehrte polnische Notation">umgekehrten polnischen Notation (UPN)</a>.

</p><p>Zwischen 1975 und 2003 wurde die Einkommensteuer in der BRD nach dem Horner-Schema berechnet, um <a href="Rundungsfehler" title="Rundungsfehler">Rundungsfehler</a> bei der Kalkulation mit elektronischen Taschenrechnern oder Computern zu vermeiden und damit die Rechtssicherheit zu gewährleisten.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Tabellarische_Schreibweise_des_Hornerschemas">Tabellarische Schreibweise des Hornerschemas</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Herleitung">Herleitung</h3></div>

<p>Betrachten wir nochmals obiges Beispiel und setzen:

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<annotation encoding="application/x-tex">{\displaystyle \beta \;:=}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69936a69e4f3ddab8f3e292aa18eacff9edc4e07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.077ex; height:2.509ex;" alt="{\displaystyle \beta \;:=}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle \;\,\;2\cdot x-4=\alpha \cdot x-4}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac31d029acb4397b48f7361d21ce440ae44164ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.449ex; height:2.343ex;" alt="{\displaystyle \;\,\;2\cdot x-4=\alpha \cdot x-4}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle \;\;(2\cdot x-4)\cdot x-5=\beta \cdot x-5}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4445f8e2027f4d754a8ae769d1803fb0b9385ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.727ex; height:2.843ex;" alt="{\displaystyle \;\;(2\cdot x-4)\cdot x-5=\beta \cdot x-5}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle \delta \;:=}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \;((2\cdot x-4)\cdot x-5)\cdot x+7=\gamma \cdot x+7}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b159fefa15b9edacfc27dabd6722a5d173f3e05c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.834ex; height:2.843ex;" alt="{\displaystyle \;((2\cdot x-4)\cdot x-5)\cdot x+7=\gamma \cdot x+7}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle \epsilon \;:=}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle (((2\cdot x-4)\cdot x-5)\cdot x+7)\cdot x+11=\delta \cdot x+11}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6a6b5b66ddabf18259f69d356e722210aec872c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.121ex; height:2.843ex;" alt="{\displaystyle (((2\cdot x-4)\cdot x-5)\cdot x+7)\cdot x+11=\delta \cdot x+11}" loading="lazy"></span>

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<p>Nun überträgt man die Koeffizienten, die Zwischenprodukte und Teilsummen in eine dreizeilige Tabelle, wobei in die

erste Zeile die Koeffizienten eingetragen werden. In die dritte Zeile kommen die Teilsummen. Dabei wird der erste <a href="Koeffizient" title="Koeffizient">Koeffizient</a> des Polynoms direkt übernommen. Die zuvor berechnete Teilsumme multipliziert mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> ergibt dann den nächsten Summanden, den man dann in die zweite Zeile unter den folgenden Koeffizienten einträgt.

</p><p>So erhält man nach und nach das folgende Rechenschema:

</p>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6775feb220c5fa4bdd9a16a476df8a808bb2e778.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle {\,2}}" loading="lazy"></span>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,-4}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a21a162ca867dfed8480371e637493654a22d7b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.358ex; height:2.343ex;" alt="{\displaystyle {\,-4}}" loading="lazy"></span>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,-5}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c6fd9579a823a15b490ab547b5e02eeae861ead.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.358ex; height:2.343ex;" alt="{\displaystyle {\,-5}}" loading="lazy"></span>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,7}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af9ed9287aabbc387b4a6046b963be88d8f3bb20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle {\,7}}" loading="lazy"></span>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,11}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f828de069fb02b39c88619f7998b72c3119019e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.712ex; height:2.176ex;" alt="{\displaystyle {\,11}}" loading="lazy"></span>

</td></tr>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\Bigg \downarrow }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d7ae3ac32f2fa39ad9fc890dd8dcabbadf3220a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:1.55ex; height:6.176ex;" alt="{\displaystyle {\Bigg \downarrow }}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +}</annotation>

</semantics>

</td></tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\alpha x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3b9b7fe41546210983e5f6dae66d7edf26b8dbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.204ex; height:1.676ex;" alt="{\displaystyle \,\alpha x}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\beta x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ecc9c515db6d5c893683bdd666f934cfa6964f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.049ex; height:2.509ex;" alt="{\displaystyle \,\beta x}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\gamma x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66d8ee83c85fbc1216d4a848e38b503f5438ec20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.979ex; height:2.176ex;" alt="{\displaystyle \,\gamma x}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\delta x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/664c83f7c55eec5a0157acbb50ea436cb8093596.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.765ex; height:2.343ex;" alt="{\displaystyle \,\delta x}" loading="lazy"></span>

</td></tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13726ca48b64be8035bbf69dedc5de51b6c59b62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.324ex; height:2.676ex;" alt="{\displaystyle \nearrow }" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mid }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f7b2136e276c4aec285a6c40b91180c16432b9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:0.647ex; height:2.843ex;" alt="{\displaystyle \mid }" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mid }</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mid }</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mid }</annotation>

</semantics>

</td></tr>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdot {x}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43e8f823fe3847a1a0b1dce4c9d794e2f78ebe01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle \cdot {x}}" loading="lazy"></span></td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {=}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc16000e5b6bb4b073f9f2a34ab4e3e2ef3eb5c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle {=}}" loading="lazy"></span>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdot {x}}</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {=}}</annotation>

</semantics>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdot {x}}</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {=}}</annotation>

</semantics>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdot {x}}</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {=}}</annotation>

</semantics>

</td></tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4618f22b0f780805eb94bb407578d9bc9487947a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \downarrow }" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation>

</semantics>

</td></tr>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,2=\alpha }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f64d907300e31ea0eb7bce568c5908247c86c217.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.136ex; height:2.176ex;" alt="{\displaystyle {\,2=\alpha }}" loading="lazy"></span>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,\alpha x-4=\beta }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d35ad8c599faa504fc7ab20fcfaba0067cb3cd9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.638ex; height:2.509ex;" alt="{\displaystyle {\,\alpha x-4=\beta }}" loading="lazy"></span>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,\beta x-5=\gamma }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8539d0d3dc9dddb0348159e3cb2bac94e64c2406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.412ex; height:2.676ex;" alt="{\displaystyle {\,\beta x-5=\gamma }}" loading="lazy"></span>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,\gamma x+7=\delta }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b69713f579053d557e7bd120a634a086e00049a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.129ex; height:2.843ex;" alt="{\displaystyle {\,\gamma x+7=\delta }}" loading="lazy"></span>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,\delta x+11=\epsilon }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b67548c4d0ec21ce2d8bbe61b9cb4d498afa98ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.973ex; height:2.509ex;" alt="{\displaystyle {\,\delta x+11=\epsilon }}" loading="lazy"></span>

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Beispiel_2">Beispiel</h3></div>

<p>Die Berechnung des obigen Polynoms für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=2}</annotation>

</semantics>

</p>

</td>

<td style="width:3ex;">2

</td>

<td style="width:3ex;">−4

</td>

<td style="width:3ex;">−5

</td>

<td style="width:3ex;">7

</td>

<td style="width:3ex;">11

</td></tr>

<tr>

<td style="background:#FFFF99;">2

</td>

<td>

</td>

<td>4

</td>

<td>0

</td>

<td>−10

</td>

<td>−6

</td></tr>

<tr>

<td>

</td>

<td>2

</td>

<td>0

</td>

<td>−5

</td>

<td>−3

</td>

<td style="background:#99FF99;">5

</td></tr></tbody></table>

<p>Den Wert, für den man das Polynom berechnen möchte, schreibt man dabei zur Erinnerung üblicherweise in die mittlere Zeile vor das Schema, die erste Zahl der oberen Zeile schreibt man auch in die untere Zeile. Anschließend multipliziert man diese Zahl mit dem Wert, für den man das Polynom berechnen möchte, schreibt das Ergebnis in die mittlere Zeile der zweiten Spalte, addiert die beiden Werte der zweiten Spalte und schreibt das Ergebnis in die untere Zeile. Anschließend wird wiederholt die Spaltensumme aus der unteren Zeile mit dem Wert, für den man das Polynom berechnen möchte, multipliziert und das Ergebnis in die mittlere Zeile der nächsten Spalte geschrieben, die Spalte addiert usw. Die letzte Zahl (hier fünf) ist das Endergebnis.

</p><p>Für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=5}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=5}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0f8441cf157c3f0ed6b88edd716956517c9d66c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=5}" loading="lazy"></span> ergeben sich jedoch wesentlich höhere Zwischenergebnisse:

</p>

</td>

<td style="width:3ex;">2

</td>

<td style="width:3ex;">−4

</td>

<td style="width:3ex;">−5

</td>

<td style="width:3ex;">7

</td>

<td style="width:3ex;">11

</td></tr>

<tr>

<td style="background:#FFFF99;">5

</td>

<td>

</td>

<td>10

</td>

<td>30

</td>

<td>125

</td>

<td>660

</td></tr>

<tr>

<td>

</td>

<td>2

</td>

<td>6

</td>

<td>25

</td>

<td>132

</td>

<td style="background:#99FF99;">671

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Anwendungsmöglichkeiten_des_Hornerschemas"><span id="Anwendungsm.C3.B6glichkeiten_des_Hornerschemas"></span>Anwendungsmöglichkeiten des Hornerschemas</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Umwandlung_zwischen_verschiedenen_Zahlensystemen">Umwandlung zwischen verschiedenen Zahlensystemen</h3></div>

<p>Unsere vertraute Darstellung von Zahlen im dezimalen <a href="Stellenwertsystem" title="Stellenwertsystem">Stellenwertsystem</a> ist nichts anderes als eine verkürzte Schreibweise für besondere Polynome, nämlich Polynome mit der Basis <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=10}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=10}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9dc68280f58d5939fb1cc2d63e7587bf5da1053a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.753ex; height:2.176ex;" alt="{\displaystyle x=10}" loading="lazy"></span>. Das Gleiche gilt für alle anderen Stellenwertsysteme, beispielsweise das <a href="Dualsystem" title="Dualsystem">Binärsystem</a>. Dort ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=2}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f39b6e42e5ffb81ac7b051b9e48b9a91d0713c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=2}" loading="lazy"></span>. Wir können uns das Horner-Schema zunutze machen, um Zahlen aus jedem anderen Stellenwertsystem in das <a href="Dezimalsystem" title="Dezimalsystem">Dezimalsystem</a> umzuwandeln, und umgekehrt.

</p>

<div class="mw-heading mw-heading4"><h4 id="Umwandlung_ins_Dezimalsystem">Umwandlung ins Dezimalsystem</h4></div>

<p>Beispiel: Die Binärzahl 110101 soll in das Dezimalsystem umgewandelt werden. Wie lautet die sich ergebende Dezimalzahl <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"></span>?

</p><p>Wir schreiben 110101<sub>binär</sub> als Polynom:

</p>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{110101}(x)=1\cdot x^{5}+1\cdot x^{4}+0\cdot x^{3}+1\cdot x^{2}+0\cdot x^{1}+1\cdot x^{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0a5b26d00181e1b22d14184c60e7cbc62ad9c84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.449ex; height:3.176ex;" alt="{\displaystyle P_{110101}(x)=1\cdot x^{5}+1\cdot x^{4}+0\cdot x^{3}+1\cdot x^{2}+0\cdot x^{1}+1\cdot x^{0}}" loading="lazy"></span></dd></dl>

<p>so ist

</p>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} =P_{110101}(2)=1\cdot 2^{5}+1\cdot 2^{4}+0\cdot 2^{3}+1\cdot 2^{2}+0\cdot 2^{1}+1\cdot 2^{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/974c7630b7ba2a25de95cec4d34f6b7e12b083db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.669ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} =P_{110101}(2)=1\cdot 2^{5}+1\cdot 2^{4}+0\cdot 2^{3}+1\cdot 2^{2}+0\cdot 2^{1}+1\cdot 2^{0}}" loading="lazy"></span></dd></dl>

<p>Nach dem Horner-Schema:

</p>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} =((((1\cdot 2+1)\cdot 2+0)\cdot 2+1)\cdot 2+0)\cdot 2+1}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad7a2ee312dc5a0074dab804ddf14cec9ed09093.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.013ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} =((((1\cdot 2+1)\cdot 2+0)\cdot 2+1)\cdot 2+0)\cdot 2+1}" loading="lazy"></span></dd></dl>

<p>Wir brauchen das nun nicht in einem Zuge auszurechnen, sondern können schrittweise vorgehen. Jeder Schritt besteht aus einer Multiplikation mit 2 und einer Addition. Der Übersicht halber schreiben wir die Schritte untereinander und notieren die Zwischenergebnisse:

</p>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<mtext>(1. Ziffer)</mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<mtext>(2. Ziffer)</mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<mtext>(3. Ziffer)</mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<mtext>(4. Ziffer)</mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<mtext>(5. Ziffer)</mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<mtext>(6. Ziffer)</mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}d_{0}&=&&&&&1&&{\text{(1. Ziffer)}}\\d_{1}&=&d_{0}\cdot 2+1&=&1\cdot 2+1&=&3&&{\text{(2. Ziffer)}}\\d_{2}&=&d_{1}\cdot 2+0&=&3\cdot 2+0&=&6&&{\text{(3. Ziffer)}}\\d_{3}&=&d_{2}\cdot 2+1&=&6\cdot 2+1&=&13&&{\text{(4. Ziffer)}}\\d_{4}&=&d_{3}\cdot 2+0&=&13\cdot 2+0&=&26&&{\text{(5. Ziffer)}}\\d_{5}&=&d_{4}\cdot 2+1&=&26\cdot 2+1&=&53&&{\text{(6. Ziffer)}}\\\mathbf {d} &=&d_{5}&&&=&\mathbf {53} \\\end{matrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/514d4c9982bf1b44750e0389930ff9dd4d031445.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.005ex; width:57.604ex; height:23.176ex;" alt="{\displaystyle {\begin{matrix}d_{0}&=&&&&&1&&{\text{(1. Ziffer)}}\\d_{1}&=&d_{0}\cdot 2+1&=&1\cdot 2+1&=&3&&{\text{(2. Ziffer)}}\\d_{2}&=&d_{1}\cdot 2+0&=&3\cdot 2+0&=&6&&{\text{(3. Ziffer)}}\\d_{3}&=&d_{2}\cdot 2+1&=&6\cdot 2+1&=&13&&{\text{(4. Ziffer)}}\\d_{4}&=&d_{3}\cdot 2+0&=&13\cdot 2+0&=&26&&{\text{(5. Ziffer)}}\\d_{5}&=&d_{4}\cdot 2+1&=&26\cdot 2+1&=&53&&{\text{(6. Ziffer)}}\\\mathbf {d} &=&d_{5}&&&=&\mathbf {53} \\\end{matrix}}}" loading="lazy"></span></dd></dl>

<p>Wir haben unsere gesuchte Dezimaldarstellung gefunden.

</p><p>Verallgemeinert lautet das Verfahren: Eine Zahl aus einem Stellenwertsystem zur Basis <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</p>

<ul><li>der Wert der ersten Ziffer als Anfangswert genommen wird</li>

<li>danach schrittweise das Ergebnis aus dem vorigen Schritt mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

<li>bis alle Ziffern aufgebraucht sind.</li></ul>

<p>Am einfachsten schreibt man die Rechnung wieder in tabellarischer Form auf:

</p>

<td>

</td>

<td width="25">1

</td>

<td width="25">1

</td>

<td width="25">0

</td>

<td width="25">1

</td>

<td width="25">0

</td>

<td width="25">1

</td></tr>

<tr>

<td>2)

</td>

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">2

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">6

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">12

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">26

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">52

</td></tr>

<tr>

<td>

</td>

<td>1

</td>

<td>3

</td>

<td>6

</td>

<td>13

</td>

<td>26

</td>

</td></tr></tbody></table>

<div class="mw-heading mw-heading4"><h4 id="Kaskadiertes_Horner-Schema">Kaskadiertes Horner-Schema</h4></div>

<p>Der Nachteil des einstufigen Horner-Schemas besteht darin, dass Multiplikationen mit großen Faktoren nötig werden können (im obigen Beispiel 2 · 26 = 52). Um innerhalb des kleinen <a href="Einmaleins" title="Einmaleins">Einmaleins</a> zu bleiben, wendet man das kaskadierte oder mehrstufige Horner-Schema an. Hier werden jedoch mehr Zwischenschritte benötigt.

</p><p>Dabei wird nur der Einer für die Multiplikation herangezogen. Der Zehner wird wie ein Übertrag in die nächste Zeile unter den Einer geschrieben. Bei der 13 aus dem obigen Beispiel wird also die 3 unter die 12 geschrieben und die 1 unter die 3. Im nächsten Schritt wird nur 3 · 2 + 0 = 6 gerechnet (statt 13 · 2 + 0 = 26). Dieses Ergebnis wird ebenso behandelt; der Zehner ist hier 0. Die letzte Rechnung (6 · 2 + 1), ergibt wieder 13. Der Einer dieses Ergebnisses ist die letzte Ziffer des Endergebnisses.

</p>

<td>

</td>

<td width="25">1

</td>

<td width="25">1

</td>

<td width="25">0

</td>

<td width="25">1

</td>

<td width="25">0

</td>

<td width="25">1

</td></tr>

<tr>

<td>2)

</td>

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">2

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">6

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">12

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">6

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">12

</td></tr>

<tr>

<td>

</td>

<td style="border-bottom-width:medium;border-bottom-style:solid">1

</td>

<td style="border-bottom-width:medium;border-bottom-style:solid">3

</td>

<td style="border-bottom-width:medium;border-bottom-style:solid">6

</td>

<td style="border-bottom-width:medium;border-bottom-style:solid">3

</td>

<td style="border-bottom-width:medium;border-bottom-style:solid">6

</td>

</td></tr>

<tr>

<td>

</td>

<td>

</td>

<td width="25">0

</td>

<td width="25">0

</td>

<td width="25">1

</td>

<td width="25">0

</td>

<td width="25">1

</td></tr></tbody></table>

<p>Um die weiteren Ziffern zu berechnen, wird auf die in der letzten Zeile stehenden Zehner (00101) dasselbe Schema erneut angewandt. Dabei kann man die führenden Nullen vernachlässigen:

</p>

<td>

</td>

<td width="25">1

</td>

<td width="25">1

</td>

<td width="25">0

</td>

<td width="25">1

</td>

<td width="25">0

</td>

<td width="25">1

</td></tr>

<tr>

<td>2)

</td>

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">2

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">6

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">12

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">6

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">12

</td></tr>

<tr>

<td>

</td>

<td style="border-bottom-width:medium;border-bottom-style:solid">1

</td>

<td style="border-bottom-width:medium;border-bottom-style:solid">3

</td>

<td style="border-bottom-width:medium;border-bottom-style:solid">6

</td>

<td style="border-bottom-width:medium;border-bottom-style:solid">3

</td>

<td style="border-bottom-width:medium;border-bottom-style:solid">6

</td>

</td></tr>

<tr>

<td>

</td>

<td>

</td>

<td width="25">0

</td>

<td width="25">0

</td>

<td width="25">1

</td>

<td width="25">0

</td>

<td width="25">1

</td></tr>

<tr>

<td>2)

</td>

</td>

</td>

</td>

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">2

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">4

</td></tr>

<tr>

<td>

</td>

</td>

</td>

</td>

<td style="border-bottom-width:medium;border-bottom-style:solid">1

</td>

<td style="border-bottom-width:medium;border-bottom-style:solid">2

</td>

</td></tr>

<tr>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>0

</td>

<td>0

</td></tr></tbody></table>

<p>Da jetzt nur noch Nullen in der Übertragszeile stehen, ist das Verfahren beendet. Das Gesamtergebnis (53) liest man in der letzten Spalte der Einer von unten nach oben.

</p>

<div class="mw-heading mw-heading4"><h4 id="Senkrecht_kaskadierende_Schreibweise">Senkrecht kaskadierende Schreibweise</h4></div>

<p>Die Ziffern der Ausgangszahl werden zunächst senkrecht untereinander geschrieben. Links daneben wird eine senkrechte Linie gezogen. Unterhalb der letzten Ziffer eine waagerechte Linie, unter der am Ende das Ergebnis steht.

</p><p>Zuerst wird die höchstwertige Ziffer (die erste 1) eine Zeile tiefer in die vorhergehende Spalte übertragen. Diese steht jetzt links neben der zweiten Ziffer (ebenfalls eine 1). Die linke Zahl wird mit der Zahlenbasis (hier 2) multipliziert, die rechte Zahl addiert (1 · 2 + 1). Vom Ergebnis (3) wird der Zehner eine Spalte weiter links geschrieben, der Einer eine Zeile tiefer.

</p><p>Das gleiche Verfahren wird mit dem Einer des Ergebnisses (3) und der nächsten Ziffer (0) durchgeführt. Das Ergebnis (3 · 2 + 0 = 6) wird ebenso notiert wie das vorige Ergebnis.

</p><p>Die dritte Rechnung lautet 6 · 2 + 1 = 13, danach ist 3 · 2 + 0 = 6 und schließlich wieder 6 · 2 + 1 = 13 zu berechnen. Wie in den vorherigen Schritten werden Einer und Zehner des Ergebnisses diagonal untereinander geschrieben.

</p><p>Unter der waagerechten Linie steht jetzt die letzte Ziffer des Endergebnisses (3).

</p>

<table>

<td>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

<td width="8" style="text-align:right">1

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8">0

</td>

<td width="8" style="text-align:right">1

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8">0

</td>

<td width="8" style="text-align:right">1

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8">0

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

</td>

<td width="8" style="text-align:right">6

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8">0

</td>

<td width="8" style="text-align:right">1

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8">0

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

<td width="8">1

</td>

<td width="8" style="text-align:right">6

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8">0

</td>

<td width="8" style="text-align:right">1

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8">0

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

<td width="8">1

</td>

<td width="8" style="text-align:right">6

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8">0

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; text-align:right;">6

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

<p>Zur Berechnung der weiteren Ziffern wird jetzt die führende Spalte mit den bisher unberücksichtigten Zehnern genauso behandelt wie die Ausgangszahl.

</p><p>Die erste gültige Ziffer wird eine Zeile tiefer in die vorherige Spalte übertragen. Diese Zahl (1) wird mit der Basis (2) multipliziert und zum Produkt (2) die nächste Ziffer (0) addiert. Zehner und Einer des Ergebnisses (02) werden diagonal wie oben gezeigt in das Schema eingetragen.

</p><p>Das Ergebnis der letzten Rechnung (2 · 2 + 1 = 05) wird ebenso eingetragen. Die Einer dieses Ergebnisses (5) sind die nächste Ziffer des Endergebnisses.

</p>

<table>

<td>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="8" style="text-align:right">1

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td>

<td width="8" style="text-align:right">6

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">1

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; text-align:right;">6

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="8" style="text-align:right">1

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td>

<td width="8" style="text-align:right">6

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">1

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; text-align:right;">6

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="8" style="text-align:right">1

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td>

<td width="8" style="text-align:right">6

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

<td width="8" style="text-align:right">1

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">1

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; text-align:right;">6

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="8" style="text-align:right">1

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td>

<td width="8" style="text-align:right">6

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

<td width="8">0

</td>

<td width="8" style="text-align:right">1

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid">2

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">1

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; text-align:right;">6

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="8" style="text-align:right">1

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td>

<td width="8" style="text-align:right">6

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

<td width="8">0

</td>

<td width="8" style="text-align:right">1

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="8" style="text-align:right">3

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">0

</td></tr>

<tr>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid">0

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid">2

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">1

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; text-align:right;">6

</td>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">1

</td></tr>

<tr>

</td>

</td>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

<p>Da in der Zehnerspalte nur noch Nullen stehen, ist die Rechnung beendet. Das Endergebnis (53) lässt sich jetzt in der Ergebniszeile ablesen, diesmal sogar in der richtigen Reihenfolge.

</p>

<div class="mw-heading mw-heading4"><h4 id="Verfahren_für_die_umgekehrte_Richtung"><span id="Verfahren_f.C3.BCr_die_umgekehrte_Richtung"></span>Verfahren für die umgekehrte Richtung</h4></div>

<p>Auf die umgekehrte Weise lässt sich eine Dezimalzahl in eine Zahl eines anderen Zahlensystems umrechnen. An Stelle einer fortgesetzten Multiplikation mit der Basis des anderen Zahlensystems tritt eine fortgesetzte Division durch diese Zahl. Die Ziffern der Zahl im anderen Zahlensystem ergeben sich von rechts nach links durch die Divisionsreste.

</p><p>In der Tabellenschreibweise werden die Ziffern der Ausgangszahl untereinander geschrieben und für das Ergebnis wird eine waagerechte Linie gezogen. Die senkrechte Linie wird hier jedoch rechts der Ziffern gezogen. Zur Erinnerung kann die Zahlenbasis rechts unten notiert werden.

</p>

</td>

<td width="16">5

</td>

</td></tr>

<tr>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid">3

</td>

</td></tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

<p>Die erste Ziffer, vermehrt um eine führende Null, (05) wird durch die Zahlenbasis (2) geteilt. Der Quotient (2) wird in die vorangehende Spalte geschrieben. Der Rest (1) in die Zeile darunter.

</p>

<td>

</td>

</td>

<td>

</td>

</td>

</td></tr>

<tr>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">3

</td>

</td></tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid"><b>1</b>3

</td>

</td></tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

<p>Dieser Rest bildet mit der nächsten Ziffer (3) eine neue zweistellige Zahl (13). Diese Zahl wird wiederum durch die Basis geteilt, das Ergebnis (6 Rest 1) wie oben diagonal in das Schema eingetragen.

</p>

<td>

</td>

</td>

<td>

<td width="8">2

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

</td>

</td>

</td></tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

<td width="8">2

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

</td></tr>

</td>

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

<p>Da jetzt alle Ziffern abgearbeitet sind, ist der Rest der letzten Rechnung (1) die letzte Ziffer des Endergebnisses.

</p><p>Die nicht bearbeiteten Quotienten werden wie eine neue Dezimalzahl behandelt (26), auf die dasselbe Verfahren angewandt wird.

</p>

<td>

</td>

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">6

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

</td></tr>

</td>

</td>

<td width="16">1

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">02

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid"><b>0</b>6

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

</td></tr>

</td>

</td>

<td width="16">1

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

<td width="8">1

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">02

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

</td>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

</td></tr>

</td>

</td>

<td width="16">1

</td>

<td width="8" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

<td width="8">1

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">02

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

</td></tr>

</td>

</td>

<td width="16">1

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

<p>Die gewonnene Ziffer ist eine 0. In der Spalte der unbearbeiteten Quotienten steht jetzt eine 13.

</p>

<td>

</td>

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">02

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">3

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

</td></tr>

</td>

</td>

<td width="16">0

</td>

<td width="16">1

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">01

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">02

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid"><b>1</b>3

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

</td></tr>

</td>

</td>

<td width="16">0

</td>

<td width="16">1

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

<td width="8">0

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">01

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">02

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

</td>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

</td></tr>

</td>

</td>

<td width="16">0

</td>

<td width="16">1

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

<td width="8">0

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">01

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">02

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

</td></tr>

</td>

</td>

<td width="16">0

</td>

<td width="16">1

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

<p>Nach diesem Schritt steht in der Quotientenspalte eine 06. Die führende Null wird ignoriert, das Verfahren startet mit der 6.

</p>

<td>

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">01

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">02

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

</td>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

</td></tr>

</td>

</td>

<td width="16">1

</td>

<td width="16">0

</td>

<td width="16">1

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">01

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">02

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

</td></tr>

</td>

</td>

<td width="16">1

</td>

<td width="16">0

</td>

<td width="16">1

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

<p>Die jetzt noch zu behandelnde Zahl ist 3.

</p>

<td>

</td>

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">01

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">02

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

</td>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

</td></tr>

</td>

</td>

<td width="16">0

</td>

<td width="16">1

</td>

<td width="16">0

</td>

<td width="16">1

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

</td>

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">01

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">02

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">03

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

</td></tr>

</td>

</td>

<td width="16">0

</td>

<td width="16">1

</td>

<td width="16">0

</td>

<td width="16">1

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

<p>Jetzt ist nur noch eine 1 übrig.

</p>

<td>

</td>

</td>

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">01

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">02

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

</td>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">03

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

</td></tr>

</td>

</td>

<td width="16">1

</td>

<td width="16">0

</td>

<td width="16">1

</td>

<td width="16">0

</td>

<td width="16">1

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

</td>

</td>

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">01

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">02

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">01

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">03

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

</td></tr>

</td>

</td>

<td width="16">1

</td>

<td width="16">0

</td>

<td width="16">1

</td>

<td width="16">0

</td>

<td width="16">1

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

</td>

</td>

<td>

</td>

</td>

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">0

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">01

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">02

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">05

</td>

</td></tr>

<tr>

<td width="8" style="border-bottom-width:thin;border-bottom-style:solid">0

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">01

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">03

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">06

</td>

<td width="16" style="border-bottom-width:thin;border-bottom-style:solid; border-left-width:thin;border-left-style:solid">13

</td>

</td></tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

<td width="16" style="border-left-width:thin;border-left-style:solid">(2)

</td></tr></tbody></table>

<p>Nach der letzten Rechnung steht in der Quotientenspalte eine 0. Das Verfahren ist damit abgeschlossen. In der Ergebniszeile steht die gesuchte Zahl in richtiger Reihenfolge.

</p>

<div class="mw-heading mw-heading3"><h3 id="Polynomdivision">Polynomdivision</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Polynomdivision_mit_linearem_Divisor">Polynomdivision mit linearem Divisor</h4></div>

<p>Am folgenden Beispiel

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x^{5}-4x^{4}+4x^{3}+3x^{2}-8x+4)\,:\,({x-2})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2d69c79ba61be25c4c4fc677407d9608b059e90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.542ex; height:3.176ex;" alt="{\displaystyle (x^{5}-4x^{4}+4x^{3}+3x^{2}-8x+4)\,:\,({x-2})}" loading="lazy"></span></dd></dl>

<p>wird zunächst die <a href="Polynomdivision" title="Polynomdivision">Polynomdivision</a> mit einem linearen Divisor im Horner-Schema dargestellt.

</p><p>Die Polynomdivision wird üblicherweise in einer schriftlichen Form durchgeführt.

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{}&(1x^{5}&-4x^{4}&+4x^{3}&+3x^{2}&-8x&+4)&:&({1x-2})={1x^{4}-2x^{3}+0x^{2}+3x-2}\\-(&{\underline {1x^{5}}}&{\underline {-2x^{4}}})\\&&-2x^{4}\\&-(&{\underline {-2x^{4}}}&{\underline {+4x^{3}}})\\&&&0x^{3}\\&&-(&{\underline {0x^{3}}}&{\underline {-0x^{2}}})\\&&&&3x^{2}\\&&&-(&{\underline {3x^{2}}}&{\underline {-6x}})\\&&&&&-2x\\&&&&-(&{\underline {-2x}}&{\underline {+4}})\\&&&&&&0\\\end{matrix}}}">

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

</mtd>

<mtd>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

</mtd>

<mtd>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

</mtd>

<mtd>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

</mtd>

<mtd>

<mrow>

</mrow>

</munder>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mrow>

</mrow>

</munder>

</mrow>

</mtd>

<mtd>

<mrow>

</mrow>

</munder>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{}&(1x^{5}&-4x^{4}&+4x^{3}&+3x^{2}&-8x&+4)&:&({1x-2})={1x^{4}-2x^{3}+0x^{2}+3x-2}\\-(&{\underline {1x^{5}}}&{\underline {-2x^{4}}})\\&&-2x^{4}\\&-(&{\underline {-2x^{4}}}&{\underline {+4x^{3}}})\\&&&0x^{3}\\&&-(&{\underline {0x^{3}}}&{\underline {-0x^{2}}})\\&&&&3x^{2}\\&&&-(&{\underline {3x^{2}}}&{\underline {-6x}})\\&&&&&-2x\\&&&&-(&{\underline {-2x}}&{\underline {+4}})\\&&&&&&0\\\end{matrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c7fd502d3b5547409eb0fa6b4856f108944b3b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -18.171ex; width:92.07ex; height:37.509ex;" alt="{\displaystyle {\begin{matrix}{}&(1x^{5}&-4x^{4}&+4x^{3}&+3x^{2}&-8x&+4)&:&({1x-2})={1x^{4}-2x^{3}+0x^{2}+3x-2}\\-(&{\underline {1x^{5}}}&{\underline {-2x^{4}}})\\&&-2x^{4}\\&-(&{\underline {-2x^{4}}}&{\underline {+4x^{3}}})\\&&&0x^{3}\\&&-(&{\underline {0x^{3}}}&{\underline {-0x^{2}}})\\&&&&3x^{2}\\&&&-(&{\underline {3x^{2}}}&{\underline {-6x}})\\&&&&&-2x\\&&&&-(&{\underline {-2x}}&{\underline {+4}})\\&&&&&&0\\\end{matrix}}}" loading="lazy"></span>

</p><p>Lässt man nun die Potenzen von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</p>

<td style="width:25px;">4

</td>

<td style="width:25px;">=

</td>

<td style="width:25px;">−2

</td></tr>

<tr>

<td>−(

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">1

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">−2

</td>

<td style="text-align:left">)

</td></tr>

<tr>

<td>−2

</td></tr>

<tr>

<td>−(

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">−2

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">4

</td>

<td style="text-align:left">)

</td></tr>

<tr>

<td>0

</td></tr>

<tr>

<td>−(

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">0

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">0

</td>

<td style="text-align:left">)

</td></tr>

<tr>

<td>3

</td></tr>

<tr>

<td>−(

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">3

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">−6

</td>

<td style="text-align:left">)

</td></tr>

<tr>

<td>−2

</td></tr>

<tr>

<td>−(

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">−2

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">4

</td>

<td style="text-align:left">)

</td></tr>

<tr>

<td>0

</td></tr></tbody></table>

<p>Verdichtet man nun dieses Schema auf drei Zeilen und übernimmt den ersten Koeffizienten des Dividenden in die dritte Zeile, so erhält man:

</p>

<td>(

</td>

<td style="width:16px;">1

</td>

<td style="width:25px;">−4

</td>

<td style="width:25px;">4

</td>

<td style="width:25px;">3

</td>

<td style="width:25px;">−8

</td>

<td style="width:25px;">4

</td>

<td style="text-align:left">)     

</td>

<td style="text-align:center; width:30px;">:

</td>

<td>( 1

</td>

<td style="width:25px;">−2

</td>

<td>)

</td></tr>

<tr>

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">−2

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">4

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">0

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">−6

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">4

</td>

<td style="text-align:left">)

</td></tr>

<td>

</td>

<td style="border-bottom-style:double">1

</td>

<td style="border-bottom-style:double">−2

</td>

<td style="border-bottom-style:double">0

</td>

<td style="border-bottom-style:double">3

</td>

<td style="border-bottom-style:double">−2

</td>

<td>0

</td></tr></tbody></table>

<p>Wie man nun sieht, sind die doppelt unterstrichenen Werte der letzten Zeile die Koeffizienten des Ergebnispolynoms und der letzte Wert dahinter ist der Divisionsrest (hier Null).

</p><p>Multipliziert man nun das <a href="Vorzeichen_(Zahl)" title="Vorzeichen (Zahl)">Vorzeichen</a> in die zweite Zeile, so erfolgt die Berechnung nach folgendem Ablauf:

</p>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0cd14c533889231abcd6c5686d2b3f9f5ceae286.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle {\,1}}" loading="lazy"></span>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,-4}}</annotation>

</semantics>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,4}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2612d2074f13f85f934198d226c41d1e23b16d25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle {\,4}}" loading="lazy"></span>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,3}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9afa798964ff972cac027c4fe94a83773c262e68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle {\,3}}" loading="lazy"></span>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,-8}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76aaa93527f3e63062b123fecf711353bb8e0d5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.358ex; height:2.343ex;" alt="{\displaystyle {\,-8}}" loading="lazy"></span>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,4}}</annotation>

</semantics>

</td></tr>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\Bigg \downarrow }}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +}</annotation>

</semantics>

</td></tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,2}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0978327a55a2bff305d3e977c99896084a1f7447.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \,2}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,-4}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/354cf0adb419f7c965f3fea853b8b0fb53d917af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.358ex; height:2.343ex;" alt="{\displaystyle \,-4}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d8f9502c23a3583421ba7a0735330be129b2447.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \,0}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,6}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c4b709082597fdd2d3f912fe1110ffe66fa9d4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \,6}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,-4}</annotation>

</semantics>

</td></tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mid }</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mid }</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mid }</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mid }</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mid }</annotation>

</semantics>

</td></tr>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdot {2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ca521edf748d85a33ce3aa0c13603aa79b3e963.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle \cdot {2}}" loading="lazy"></span></td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {=}}</annotation>

</semantics>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdot {2}}</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {=}}</annotation>

</semantics>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdot {2}}</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {=}}</annotation>

</semantics>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdot {2}}</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {=}}</annotation>

</semantics>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdot {2}}</annotation>

</semantics>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {=}}</annotation>

</semantics>

</td></tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nearrow }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation>

</semantics>

</td></tr>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,1}}</annotation>

</semantics>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,-2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/809a6775a68020208f9717b7caf1d91af6ab99dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.358ex; height:2.343ex;" alt="{\displaystyle {\,-2}}" loading="lazy"></span>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c72b6eefc54960ce46c8f17d0ae942d05d3a33dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle {\,0}}" loading="lazy"></span>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,3}}</annotation>

</semantics>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,-2}}</annotation>

</semantics>

</td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\,0}}</annotation>

</semantics>

</td></tr></tbody></table>

<p>Vermerkt man nun noch den vorzeichengedrehten Wert des Absolutglieds des Divisors vor dem Schema, so bekommt man die allgemeine Darstellung des Horner-Schemas:

</p>

<td>

</td>

<td style="width:16px;">1

</td>

<td style="width:25px;">−4

</td>

<td style="width:25px;">4

</td>

<td style="width:25px;">3

</td>

<td style="width:25px;">−8

</td>

<td style="width:25px;">4

</td></tr>

<tr>

<td>2)

</td>

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">2

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">−4

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">0

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">6

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">−4

</td></tr>

<tr>

<td>

</td>

<td style="border-bottom-width:medium;border-bottom-style:double">1

</td>

<td style="border-bottom-width:medium;border-bottom-style:double">−2

</td>

<td style="border-bottom-width:medium;border-bottom-style:double">0

</td>

<td style="border-bottom-width:medium;border-bottom-style:double">3

</td>

<td style="border-bottom-width:medium;border-bottom-style:double">−2

</td>

<td>0

</td></tr></tbody></table>

<p>Das obige Beispiel kann nun in folgender Formel zusammengefasst werden:

</p><p>Hat die Divisionsaufgabe:

</p>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x^{1}+a_{0}):(x+d)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ff991b5b1839fc64daddff25bb3f26558c22368.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.681ex; height:3.176ex;" alt="{\displaystyle (a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x^{1}+a_{0}):(x+d)}" loading="lazy"></span></dd></dl>

<p>als Ergebnis

</p>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e_{n-1}x^{n-1}+e_{n-2}x^{n-2}+\dotsb +e_{2}x^{2}+e_{1}x^{1}+e_{0},{\mbox{ mit dem Rest }}r}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/062150f60edc77df1c36f135d81f79df9e591130.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:62.926ex; height:3.009ex;" alt="{\displaystyle e_{n-1}x^{n-1}+e_{n-2}x^{n-2}+\dotsb +e_{2}x^{2}+e_{1}x^{1}+e_{0},{\mbox{ mit dem Rest }}r}" loading="lazy"></span></dd></dl>

<p>so bestimmen sich die Koeffizienten nach folgender Vorschrift:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}e_{n-1}&=&a_{n}\\e_{k}&=&a_{k+1}-d\cdot e_{k+1},&{\text{für }}k=n-2,n-3,\dotsc ,1,0\\r&=&a_{0}-d\cdot e_{0}\\\end{matrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b19398f021baf5550e8197ede906e81e8f693b1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:57.301ex; height:10.176ex;" alt="{\displaystyle {\begin{matrix}e_{n-1}&=&a_{n}\\e_{k}&=&a_{k+1}-d\cdot e_{k+1},&{\text{für }}k=n-2,n-3,\dotsc ,1,0\\r&=&a_{0}-d\cdot e_{0}\\\end{matrix}}}" loading="lazy"></span>

</p><p>Das Horner-Schema stellt sich dann wie folgt dar:

</p>

<td>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,a_{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c4349c0c2edfd4fed53c0e05b8646b76d68f4e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.835ex; height:2.009ex;" alt="{\displaystyle \,a_{n}}" loading="lazy"></span></td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,a_{n-1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2427e32c79899c052c22565444d730d591a6c63f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.936ex; height:2.009ex;" alt="{\displaystyle \,a_{n-1}}" loading="lazy"></span></td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,a_{n-2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/084537e18dfe6e4b43c66bcf262cf9179d06b33e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.936ex; height:2.009ex;" alt="{\displaystyle \,a_{n-2}}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dotso }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59920ff6acbf42b0989dc14dc975eb6f1a9b2dff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.723ex; height:0.843ex;" alt="{\displaystyle \dotso }" loading="lazy"></span>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,a_{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08694bd86b4876b28b7c27c6b878a46c578e9be1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.671ex; height:2.009ex;" alt="{\displaystyle \,a_{2}}" loading="lazy"></span></td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,a_{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e181e882baf10e8296d6d396343edb48a2d5c66b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.671ex; height:2.009ex;" alt="{\displaystyle \,a_{1}}" loading="lazy"></span></td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,a_{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61b9105ed81e566c43240628e3dbfc3f292fe734.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.671ex; height:2.009ex;" alt="{\displaystyle \,a_{0}}" loading="lazy"></span>

</td></tr>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {-d\,})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a361c7a7da7bf6609eec1941d5bd0c73ff7c3da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.316ex; height:2.843ex;" alt="{\displaystyle {-d\,})}" loading="lazy"></span>

</td>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -d\,e_{n-1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00881ac7dba9e768b8e0cb3c4728c6387ef3633e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.814ex; height:2.509ex;" alt="{\displaystyle -d\,e_{n-1}}" loading="lazy"></span>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -d\,e_{n-2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b18705b266451e7888d5b644aa5dd73ca44761a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.814ex; height:2.509ex;" alt="{\displaystyle -d\,e_{n-2}}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dotso }</annotation>

</semantics>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -d\,e_{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8828288087e141527555267e80f0d0fc5d33dcde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.549ex; height:2.509ex;" alt="{\displaystyle -d\,e_{2}}" loading="lazy"></span>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -d\,e_{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a5200b9e214ae72c52edf9ba89719e66ddc7ef13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.549ex; height:2.509ex;" alt="{\displaystyle -d\,e_{1}}" loading="lazy"></span>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -d\,e_{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c05d14133f33e6e0101bdf7838ed493372debf7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.549ex; height:2.509ex;" alt="{\displaystyle -d\,e_{0}}" loading="lazy"></span>

</td></tr>

<td>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,e_{n-1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e859169ade9c472473a39348a5005d35b735ad3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.79ex; height:2.009ex;" alt="{\displaystyle \,e_{n-1}}" loading="lazy"></span>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,e_{n-2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ed9cff47355948de041aa7d8fca78f6099d2fe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.79ex; height:2.009ex;" alt="{\displaystyle \,e_{n-2}}" loading="lazy"></span>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,e_{n-3}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a12042eec53a9f174cb3bd8a61e0ffc48531ddf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.79ex; height:2.009ex;" alt="{\displaystyle \,e_{n-3}}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dotso }</annotation>

</semantics>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,e_{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed66fb0d400a31bac1850044a077cdff5a02e9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.525ex; height:2.009ex;" alt="{\displaystyle \,e_{1}}" loading="lazy"></span>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,e_{o}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2b31608d8f38a8d76a9d4a30dc27a8d7413adc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \,e_{o}}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,r}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/726f3541766cfc610c6527d1ca6fc2ebb456b490.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.436ex; height:1.676ex;" alt="{\displaystyle \,r}" loading="lazy"></span>

</td></tr></tbody></table>

<td><b>Anmerkung:</b> Mit diesem Ergebnis lässt sich die Berechnung von Funktionswerten eines Polynoms <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> an der Stelle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

<p>Betrachten wir die Division <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89833156eff2c51bfb8750db3306a0544ce34e14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}" loading="lazy"></span> durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {x-a}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {x-a}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9165890584d539bc27f0ce2655c25ba15193a02c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.4ex; height:2.176ex;" alt="{\displaystyle {x-a}}" loading="lazy"></span>

</p>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {P(x)}{(x-a)}}=E(x)+{\frac {r}{x-a}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8e60d0135c16af2e8dc72aa4266a395ae7cfed6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.135ex; height:6.509ex;" alt="{\displaystyle {\frac {P(x)}{(x-a)}}=E(x)+{\frac {r}{x-a}}}" loading="lazy"></span></dd></dl>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Rightarrow P(x)={(x-a)}E(x)+{r}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/810e063eee700d12304922b57c459808779199b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.965ex; height:2.843ex;" alt="{\displaystyle \Rightarrow P(x)={(x-a)}E(x)+{r}}" loading="lazy"></span>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Rightarrow P(a)=0\cdot E(a)+{r}=r}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9c98ba2a7aae499d436358ed64c3880b09576bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.544ex; height:2.843ex;" alt="{\displaystyle \Rightarrow P(a)=0\cdot E(a)+{r}=r}" loading="lazy"></span>

</p>

</td></tr></tbody></table>

<div class="mw-heading mw-heading4"><h4 id="Polynomdivision_mit_einem_Divisor_2._Grades">Polynomdivision mit einem Divisor 2. Grades</h4></div>

<p>Hat die Divisionsaufgabe:

</p>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x^{1}+a_{0}):(x^{2}+d_{1}x+d_{0})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6e5d92ed7f5bedb66fa15b1fcb42c8ea67faa52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.216ex; height:3.176ex;" alt="{\displaystyle (a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x^{1}+a_{0}):(x^{2}+d_{1}x+d_{0})}" loading="lazy"></span></dd></dl>

<p>als Ergebnis

</p>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e_{n-2}x^{n-2}+e_{n-3}x^{n-3}+\dotsb +e_{2}x^{2}+e_{1}x^{1}+e_{0},{\mbox{ mit dem Rest }}r_{1}x+r_{0},}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d7fe4ae8949330a6af1b637da30344ac2d4b340.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:70.9ex; height:3.009ex;" alt="{\displaystyle e_{n-2}x^{n-2}+e_{n-3}x^{n-3}+\dotsb +e_{2}x^{2}+e_{1}x^{1}+e_{0},{\mbox{ mit dem Rest }}r_{1}x+r_{0},}" loading="lazy"></span></dd></dl>

<p>so bestimmen sich die Koeffizienten nach folgender Vorschrift:

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}e_{n-2}&=&a_{n}\\e_{n-3}&=&a_{n-1}-d_{1}\,e_{n-2}\\e_{k}&=&a_{k+2}-d_{1}\,e_{k+1}-d_{0}\,e_{k+2}&{\text{für }}k=n-4,n-5,\dotsc ,1,0\\r_{1}&=&a_{1}-d_{1}\,e_{0}-d_{0}\,e_{1}\\r_{0}&=&a_{0}-d_{0}\,e_{0}\\\end{matrix}}}">

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}e_{n-2}&=&a_{n}\\e_{n-3}&=&a_{n-1}-d_{1}\,e_{n-2}\\e_{k}&=&a_{k+2}-d_{1}\,e_{k+1}-d_{0}\,e_{k+2}&{\text{für }}k=n-4,n-5,\dotsc ,1,0\\r_{1}&=&a_{1}-d_{1}\,e_{0}-d_{0}\,e_{1}\\r_{0}&=&a_{0}-d_{0}\,e_{0}\\\end{matrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f5d0743f6facd04b9a9f639b4a95e92828807b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:66.172ex; height:16.509ex;" alt="{\displaystyle {\begin{matrix}e_{n-2}&=&a_{n}\\e_{n-3}&=&a_{n-1}-d_{1}\,e_{n-2}\\e_{k}&=&a_{k+2}-d_{1}\,e_{k+1}-d_{0}\,e_{k+2}&{\text{für }}k=n-4,n-5,\dotsc ,1,0\\r_{1}&=&a_{1}-d_{1}\,e_{0}-d_{0}\,e_{1}\\r_{0}&=&a_{0}-d_{0}\,e_{0}\\\end{matrix}}}" loading="lazy"></span>

</p><p>Das verallgemeinerte Horner-Schema stellt sich dann wie folgt dar:

</p>

<td>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,a_{n}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,a_{n-1}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,a_{n-2}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,a_{n-3}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99e8daeffbe6415f83fe3ab1869a448d862c0aca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.936ex; height:2.009ex;" alt="{\displaystyle \,a_{n-3}}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dotso }</annotation>

</semantics>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,a_{3}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a294e9edcef67faca87b50e64146eaa156ff70b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.671ex; height:2.009ex;" alt="{\displaystyle \,a_{3}}" loading="lazy"></span></td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,a_{2}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,a_{1}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,a_{0}}</annotation>

</semantics>

</td></tr>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {-d_{0}\,})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b969e3b01e6e3c985bdde5bac392b73b5dbe3a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.363ex; height:2.843ex;" alt="{\displaystyle {-d_{0}\,})}" loading="lazy"></span>

</td>

<td>

</td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -d_{0}\,e_{n-2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07be69f851b4903e95056cc29bed08a166ab388e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.861ex; height:2.509ex;" alt="{\displaystyle -d_{0}\,e_{n-2}}" loading="lazy"></span></td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -d_{0}\,e_{n-3}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c3294b23e5d07e3c936bf4b15bb4d0c4edb92c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.861ex; height:2.509ex;" alt="{\displaystyle -d_{0}\,e_{n-3}}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dotso }</annotation>

</semantics>

</td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -d_{0}\,e_{3}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/504efc3e6a102906e25a7f3372e3d8cf559cda65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.596ex; height:2.509ex;" alt="{\displaystyle -d_{0}\,e_{3}}" loading="lazy"></span></td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -d_{0}\,e_{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a01cc6ca91066845bd280f9da1747b5b3a33b7dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.596ex; height:2.509ex;" alt="{\displaystyle -d_{0}\,e_{2}}" loading="lazy"></span></td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -d_{0}\,e_{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ce41c16ed2e198b53b752e39419df88d9c65341.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.596ex; height:2.509ex;" alt="{\displaystyle -d_{0}\,e_{1}}" loading="lazy"></span></td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -d_{0}\,e_{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0179df7b209d52bbf3c32fa738cc1c99110f329.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.596ex; height:2.509ex;" alt="{\displaystyle -d_{0}\,e_{0}}" loading="lazy"></span>

</td></tr>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {-d_{1}\,})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e654880dc324d99b1b7ea2d382d036986518726.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.363ex; height:2.843ex;" alt="{\displaystyle {-d_{1}\,})}" loading="lazy"></span>

</td>

</td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -d_{1}\,e_{n-2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72d955c7575544dae87dc8331542af1c6107fe8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.861ex; height:2.509ex;" alt="{\displaystyle -d_{1}\,e_{n-2}}" loading="lazy"></span>

</td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -d_{1}\,e_{n-3}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b13e641127b6a80dbc25398f4c1b24f637020ab2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.861ex; height:2.509ex;" alt="{\displaystyle -d_{1}\,e_{n-3}}" loading="lazy"></span>

</td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -d_{1}\,e_{n-4}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69d3aef604e2a01107150ea29c503aed8e17d1b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.861ex; height:2.509ex;" alt="{\displaystyle -d_{1}\,e_{n-4}}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dotso }</annotation>

</semantics>

</td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -d_{1}\,e_{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/644ac28fbb981566ebfaa711c53f9f49be11f532.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.596ex; height:2.509ex;" alt="{\displaystyle -d_{1}\,e_{2}}" loading="lazy"></span>

</td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -d_{1}\,e_{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/004149e1f6f94be8a9bc7303c6c8f91b7f4baab2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.596ex; height:2.509ex;" alt="{\displaystyle -d_{1}\,e_{1}}" loading="lazy"></span>

</td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -d_{1}\,e_{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3fa370534f7c4439564f94ebc46fbfe5c93aa081.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.596ex; height:2.509ex;" alt="{\displaystyle -d_{1}\,e_{0}}" loading="lazy"></span>

</td>

</td></tr>

<td>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,e_{n-2}}</annotation>

</semantics>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,e_{n-3}}</annotation>

</semantics>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,e_{n-4}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/192d315c1afc87aec5e2a4393abf51b86abfee09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.79ex; height:2.009ex;" alt="{\displaystyle \,e_{n-4}}" loading="lazy"></span>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,e_{n-5}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8200f963a0881592cf703f61c800a622458ed39a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.79ex; height:2.009ex;" alt="{\displaystyle \,e_{n-5}}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dotso }</annotation>

</semantics>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,e_{1}}</annotation>

</semantics>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,e_{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d6e5b56bd57650ae6a2701558e08e6ef581f6d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.525ex; height:2.009ex;" alt="{\displaystyle \,e_{0}}" loading="lazy"></span>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,r_{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6f49c17719ffff4375990f7bdc5b7279113b6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.49ex; height:2.009ex;" alt="{\displaystyle \,r_{1}}" loading="lazy"></span>

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,r_{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0b4943b1b4b540120e51189a50b8ab5bc4e49cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.49ex; height:2.009ex;" alt="{\displaystyle \,r_{0}}" loading="lazy"></span>

</td></tr></tbody></table>

<div class="mw-heading mw-heading5"><h5 id="Ein_Beispiel">Ein Beispiel</h5></div>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-6x^{6}+14x^{5}-8x^{4}-2x^{3}+8x-6)\,:\,(x^{2}-2x+1)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1a3ae39f5908f86df7d4730ee3bf5ef7b1679bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.062ex; height:3.176ex;" alt="{\displaystyle (-6x^{6}+14x^{5}-8x^{4}-2x^{3}+8x-6)\,:\,(x^{2}-2x+1)}" loading="lazy"></span></dd></dl>

<p>Im Horner-Schema:

</p>

<td>

</td>

<td style="width:16px;">−6

</td>

<td style="width:25px;">14

</td>

<td style="width:25px;">−8

</td>

<td style="width:25px;">−2

</td>

<td style="width:25px;">0

</td>

<td style="width:25px;">8

</td>

<td style="width:25px;">−6

</td></tr>

<tr>

<td>−1)

</td>

<td>

</td>

<td>

</td>

<td>6

</td>

<td>−2

</td>

<td>−2

</td>

<td>0

</td>

<td>2

</td></tr>

<tr>

<td>2)

</td>

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">−12

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">4

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">4

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">0

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">−4

</td>

</td></tr>

<tr>

<td>

</td>

<td style="border-bottom-width:medium;border-bottom-style:double">−6

</td>

<td style="border-bottom-width:medium;border-bottom-style:double">2

</td>

<td style="border-bottom-width:medium;border-bottom-style:double">2

</td>

<td style="border-bottom-width:medium;border-bottom-style:double">0

</td>

<td style="border-bottom-width:medium;border-bottom-style:double">−2

</td>

<td>4

</td>

<td>−4

</td></tr></tbody></table>

<p>Daraus ergibt sich:

</p>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {(-6x^{6}+14x^{5}-8x^{4}-2x^{3}+8x-6)}{(x^{2}-2x+1)}}\,=\,-6x^{4}+2x^{3}+2x^{2}-2{\mbox{ Rest }}4x-4}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/014facfffae91885eca5f5b45572c344944e7b5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:75.882ex; height:6.676ex;" alt="{\displaystyle {\frac {(-6x^{6}+14x^{5}-8x^{4}-2x^{3}+8x-6)}{(x^{2}-2x+1)}}\,=\,-6x^{4}+2x^{3}+2x^{2}-2{\mbox{ Rest }}4x-4}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Lineartransformation">Lineartransformation</h3></div>

<p>In einigen Fällen, beispielsweise zur Verbesserung der Konvergenz beim <a href="Newton-Verfahren" class="mw-redirect" title="Newton-Verfahren">Newton-Verfahren</a>, kann es sehr hilfreich sein, ein Polynom <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> in ein Polynom <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{a}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5830e67f78d703f1bc6ff6d691691cba661ef48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.594ex; height:2.509ex;" alt="{\displaystyle P_{a}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> konstant, zu transformieren, so dass mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a+y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=a+y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f977d52fd866eeee2811ae0551461c02607a3633.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.654ex; height:2.343ex;" alt="{\displaystyle x=a+y}" loading="lazy"></span> gilt:

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(x)=P(a+y)\;=\;P_{a}(y)=P_{a}(x-a)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c56162444113cb1e4583698d3a6d90f6f21986db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.613ex; height:2.843ex;" alt="{\displaystyle P(x)=P(a+y)\;=\;P_{a}(y)=P_{a}(x-a)}" loading="lazy"></span></dd></dl>

<p>Eine solche Lineartransformation kann man durch Einsetzen von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a+y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de9de6f92501bfce64d6eacb79ef52a77177aacc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.226ex; height:2.343ex;" alt="{\displaystyle a+y}" loading="lazy"></span> anstelle von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> und anschließendes Ausmultiplizieren erhalten. Wesentlich effizienter lässt sich diese Rechnung mit dem <i>vollständigen</i> Horner-Schema durchführen.

</p><p>Betrachten wir das Polynom <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=\sum _{i=0}^{n}a_{i}x^{i}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(x)=\sum _{i=0}^{n}a_{i}x^{i}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f695568eae8df4a96e805d6032b59758030c61d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.884ex; height:6.843ex;" alt="{\displaystyle P(x)=\sum _{i=0}^{n}a_{i}x^{i}}" loading="lazy"></span> vom Grad <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>, welches wir nach Potenzen von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x-a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=x-a}</annotation>

</semantics>

Hierzu dividieren wir das Polynom <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89833156eff2c51bfb8750db3306a0544ce34e14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}" loading="lazy"></span> mittels des Horner-Schemas durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-a)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x-a)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d517d2bab79d156c742a2ca3652fff7d1dfd4d9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.209ex; height:2.843ex;" alt="{\displaystyle (x-a)}" loading="lazy"></span>. Wie oben gezeigt, können wir aus dem Schema das Polynom <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{1}(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94eab84009032485394374ca4c468caef8a390f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.908ex; height:2.843ex;" alt="{\displaystyle E_{1}(x)}" loading="lazy"></span> und den Rest <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb12fcfddb65e3d1e6a044215f6e833f0cd4337b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{0}}" loading="lazy"></span> ablesen, so dass gilt:

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(x)\;=E_{1}(x)(x-a)+r_{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7a449ace9a246791fbca234acddcb20954ca565.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.689ex; height:2.843ex;" alt="{\displaystyle P(x)\;=E_{1}(x)(x-a)+r_{0}}" loading="lazy"></span></dd></dl>

<p>Nun wird die Division auf dem Ergebnis-Polynom <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{1}(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94eab84009032485394374ca4c468caef8a390f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.908ex; height:2.843ex;" alt="{\displaystyle E_{1}(x)}" loading="lazy"></span> durchgeführt, und wir erhalten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e6ee346e54f38302f47b5cf3016d8718f2040c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{2}}" loading="lazy"></span> bzw. den Rest <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea214f2b31fb3869344bb9311da41c5cc38a99e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" loading="lazy"></span>:

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{1}(x)\;=E_{2}(x)(x-a)+r_{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a7186f65b07441a012299cc136a059680b4c335.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.713ex; height:2.843ex;" alt="{\displaystyle E_{1}(x)\;=E_{2}(x)(x-a)+r_{1}}" loading="lazy"></span></dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{n-1}(x)\;=E_{n}(x)(x-a)+r_{n-1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7af18c70273c46c4a91d453a6657ab343e86ad72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.407ex; height:2.843ex;" alt="{\displaystyle E_{n-1}(x)\;=E_{n}(x)(x-a)+r_{n-1}}" loading="lazy"></span> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}(x)\;=a_{n}=:r_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{n}(x)\;=a_{n}=:r_{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4ce293cd9c6f3478e3869e06387aa5d11830fb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.277ex; height:2.843ex;" alt="{\displaystyle E_{n}(x)\;=a_{n}=:r_{n}}" loading="lazy"></span></dd></dl>

<p>Es folgt:

</p>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mrow>

<mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}P(x)&=&E_{1}(x)(x-a)+r_{0}\\&=&\left(E_{2}(x)(x-a)+r_{1}\right)(x-a)+r_{0}\\&=&\left(\dotso \left(r_{n}(x-a)+r_{n-1}\right)(x-a)+\dotsb +r_{1}\right)(x-a)+r_{0}\\\end{matrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbeea0cdb4ba9455127bfa1eb7225d47cf27996d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:66.145ex; height:9.843ex;" alt="{\displaystyle {\begin{matrix}P(x)&=&E_{1}(x)(x-a)+r_{0}\\&=&\left(E_{2}(x)(x-a)+r_{1}\right)(x-a)+r_{0}\\&=&\left(\dotso \left(r_{n}(x-a)+r_{n-1}\right)(x-a)+\dotsb +r_{1}\right)(x-a)+r_{0}\\\end{matrix}}}" loading="lazy"></span></dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=x-a}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06912767259e3cc11901eaa95e92458f6c3e9e3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.654ex; height:2.343ex;" alt="{\displaystyle y=x-a}" loading="lazy"></span> ist dann die Lineartransformation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{a}}</annotation>

</semantics>

</p>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<mrow>

<mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}P_{a}(y)&=&\left(\dotso \left(r_{n}y+r_{n-1}\right)y+\dotsb +r_{1}\right)y+r_{0}\\&=&\sum _{i=0}^{n}r_{i}y^{i}\end{matrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f9ac318919f154e82c1701bae5ce8e1ce7a8c34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.659ex; height:6.509ex;" alt="{\displaystyle {\begin{matrix}P_{a}(y)&=&\left(\dotso \left(r_{n}y+r_{n-1}\right)y+\dotsb +r_{1}\right)y+r_{0}\\&=&\sum _{i=0}^{n}r_{i}y^{i}\end{matrix}}}" loading="lazy"></span></dd></dl>

<p>D. h. die Reste bei der fortgesetzten Division mit dem Linearfaktor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-a)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x-a)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d517d2bab79d156c742a2ca3652fff7d1dfd4d9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.209ex; height:2.843ex;" alt="{\displaystyle (x-a)}" loading="lazy"></span> bilden die Koeffizienten des transformierten Polynoms <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{a}}</annotation>

</semantics>

</p>

<div class="mw-heading mw-heading4"><h4 id="Beispiel_3">Beispiel</h4></div>

<p>Möchte man z. B. die Nullstelle des Polynoms <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)\,=\,x^{3}-2x-5}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(x)\,=\,x^{3}-2x-5}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0dff7eb9d6a48a01f1dcba25713566235a33fd37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.476ex; height:3.176ex;" alt="{\displaystyle P(x)\,=\,x^{3}-2x-5}" loading="lazy"></span> berechnen, so kann man leicht den Punkt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=2}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f39b6e42e5ffb81ac7b051b9e48b9a91d0713c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=2}" loading="lazy"></span> als erste Näherung raten. Für die weitere Berechnung ist es nun hilfreich, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89833156eff2c51bfb8750db3306a0544ce34e14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}" loading="lazy"></span> nach <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x-2}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbad8f67d6fc0c608bd6f91db5db4c65bbe3f60b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.333ex; height:2.343ex;" alt="{\displaystyle x-2}" loading="lazy"></span> zu entwickeln (siehe <a href="Newton-Verfahren#Historisches_über_das_Newtonverfahren" class="mw-redirect" title="Newton-Verfahren">Newtonverfahren/„Methodus fluxionum et serierum infinitarum“</a>). Gesucht ist also das Polynom <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}(x)=P(2+x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{2}(x)=P(2+x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ad357ec9ecb53ef34ed0875c4bef6cc28fae250.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.671ex; height:2.843ex;" alt="{\displaystyle P_{2}(x)=P(2+x)}" loading="lazy"></span>.

</p>

<td>

</td>

<td style="width:16px;">1

</td>

<td style="width:25px;">0

</td>

<td style="width:25px;">−2

</td>

<td style="width:25px;">−5

</td></tr>

<tr>

<td>2)

</td>

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">2

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">4

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">4

</td></tr>

<tr>

<td>

</td>

<td>1

</td>

<td>2

</td>

<td>2

</td>

<td style="border-bottom-width:medium;border-bottom-style:double">−1

</td></tr>

<tr>

<td>2)

</td>

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">2

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">8

</td>

<td>

</td></tr>

<tr>

<td>

</td>

<td>1

</td>

<td>4

</td>

<td style="border-bottom-width:medium;border-bottom-style:double">10

</td></tr>

<tr>

<td>2)

</td>

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">2

</td>

<td>

</td></tr>

<tr>

<td>

</td>

<td>1

</td>

<td style="border-bottom-width:medium;border-bottom-style:double">6

</td></tr></tbody></table>

<p>Das gesuchte Polynom ist also <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}(x)\,=\,x^{3}+6x^{2}+10x-1}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{2}(x)\,=\,x^{3}+6x^{2}+10x-1}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0a355176da790f2cbc5db438ac902defed582d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.827ex; height:3.176ex;" alt="{\displaystyle P_{2}(x)\,=\,x^{3}+6x^{2}+10x-1}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Berechnung_der_Ableitung">Berechnung der Ableitung</h3></div>

<p>Eine weitere Eigenschaft des Horner-Schemas ist, dass man recht schnell die erste Ableitung an der Stelle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span> berechnen kann.

</p><p>Betrachten wir die Division

</p>

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {P(x)}{(x-x_{0})}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad87721940615cc0c118aa6620b96761cd1aa892.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.199ex; height:6.509ex;" alt="{\displaystyle {\frac {P(x)}{(x-x_{0})}}}" loading="lazy"></span></dd></dl>

<p>mit dem Ergebnis

</p>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{e}(x)+{\frac {r}{(x-x_{0})}},}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5204069d3ccd4e28c945644da8f9f4d8207d0cff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.316ex; height:5.509ex;" alt="{\displaystyle P_{e}(x)+{\frac {r}{(x-x_{0})}},}" loading="lazy"></span></dd></dl>

<p>welches wir aus dem Horner-Schema ablesen können. Weiter oben konnte man auch sehen, dass

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r=P(x_{0})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1153ed1439880a704430d072d8c2146910a36a9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.086ex; height:2.843ex;" alt="{\displaystyle r=P(x_{0})}" loading="lazy"></span>

ist.

Es gilt also:

</p>

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {P(x)}{(x-x_{0})}}=P_{e}(x)+{\frac {P(x_{0})}{(x-x_{0})}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c7e31183e5dde95f95b8bc07043fb9f61d719a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.967ex; height:6.509ex;" alt="{\displaystyle {\frac {P(x)}{(x-x_{0})}}=P_{e}(x)+{\frac {P(x_{0})}{(x-x_{0})}}}" loading="lazy"></span></dd></dl>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Rightarrow \;{\frac {P(x)-P(x_{0})}{(x-x_{0})}}=P_{e}(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06b2ee77c0f38dd8bb01aa20c221d5f154df66ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.842ex; height:6.509ex;" alt="{\displaystyle \Rightarrow \;{\frac {P(x)-P(x_{0})}{(x-x_{0})}}=P_{e}(x)}" loading="lazy"></span>

</p><p>Die Ableitung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{\prime }(x)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P^{\prime }(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ba3c0fabac19defe279eef8d4599a8b4b47e3da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.645ex; height:3.009ex;" alt="{\displaystyle P^{\prime }(x)}" loading="lazy"></span> lässt sich mit dem <a href="Differenzenquotient" title="Differenzenquotient">Differenzenquotienten</a> berechnen. Es gilt:

</p>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P^{\prime }(x_{0})={\frac {\mathrm {d} P(x_{0})}{\mathrm {d} x}}=\lim _{x\rightarrow x_{0}}{\frac {P(x)-P(x_{0})}{(x-x_{0})}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be023d8a97cc0675f676ba93823118dd994246b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.205ex; height:6.509ex;" alt="{\displaystyle P^{\prime }(x_{0})={\frac {\mathrm {d} P(x_{0})}{\mathrm {d} x}}=\lim _{x\rightarrow x_{0}}{\frac {P(x)-P(x_{0})}{(x-x_{0})}}}" loading="lazy"></span></dd></dl>

<p>Daraus folgt:

</p>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P^{\prime }(x_{0})=P_{e}(x_{0})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8764d9d72ffd2426534722e9e70ed94a8de3126.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.482ex; height:3.009ex;" alt="{\displaystyle P^{\prime }(x_{0})=P_{e}(x_{0})}" loading="lazy"></span></dd></dl>

<p>D. h. die Zahlen in der dritten Zeile des Horner-Schemas bilden die Koeffizienten für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{e}(x_{0})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{e}(x_{0})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b4202120041d7d19e5b7e2c0fa2033f1c86e58e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.684ex; height:2.843ex;" alt="{\displaystyle P_{e}(x_{0})}" loading="lazy"></span>. Durch nochmalige Anwendung des Horner-Schemas kann dann schließlich der Wert der Ableitung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{\prime }(x_{0})}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P^{\prime }(x_{0})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/870d5a3f3605f9cd9b414d3ce95e76f2aaf51721.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.699ex; height:3.009ex;" alt="{\displaystyle P^{\prime }(x_{0})}" loading="lazy"></span> berechnet werden.

</p>

<div class="mw-heading mw-heading4"><h4 id="Beispiel_4">Beispiel</h4></div>

<p>Betrachten wir das Polynom <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=x^{5}-4x^{4}+4x^{3}+3x^{2}-8x+4}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(x)=x^{5}-4x^{4}+4x^{3}+3x^{2}-8x+4}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f807616b509cf3baa6128de3144edf159c59878.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.863ex; height:3.176ex;" alt="{\displaystyle P(x)=x^{5}-4x^{4}+4x^{3}+3x^{2}-8x+4}" loading="lazy"></span> an der Stelle x=2

</p>

<td>

</td>

<td style="width:16px;">1

</td>

<td style="width:25px;">−4

</td>

<td style="width:25px;">4

</td>

<td style="width:25px;">3

</td>

<td style="width:25px;">−8

</td>

<td style="width:25px;">4

</td></tr>

<tr>

<td>2)

</td>

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">2

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">−4

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">0

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">6

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">−4

</td></tr>

<tr>

<td>

</td>

<td>1

</td>

<td>−2

</td>

<td>0

</td>

<td>3

</td>

<td>−2

</td>

<td style="border-bottom-width:medium;border-bottom-style:double">0

</td></tr>

<tr>

<td>2)

</td>

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">2

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">0

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">0

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">6

</td>

<td>

</td></tr>

<tr>

<td>

</td>

<td>1

</td>

<td>0

</td>

<td>0

</td>

<td>3

</td>

<td style="border-bottom-width:medium;border-bottom-style:double">4

</td>

<td>

</td></tr></tbody></table>

<p>Aus dem Schema kann man nun ablesen: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(2)\,=\,0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(2)\,=\,0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ecd682b4feaed232a9e4e681221d8fa2f455638.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.752ex; height:2.843ex;" alt="{\displaystyle P(2)\,=\,0}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{\prime }(2)\,=4}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P^{\prime }(2)\,=4}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d23c8287a06dd09dfd081e37a74f443bc46780.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.126ex; height:3.009ex;" alt="{\displaystyle P^{\prime }(2)\,=4}" loading="lazy"></span>

</p>

<div class="mw-heading mw-heading5"><h5 id="Probe">Probe</h5></div>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P^{\prime }(x)=5x^{4}-16x^{3}+12x^{2}+6x-8}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71f9cfd76515382d24c4ad2315649f664e0c7b97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.724ex; height:3.176ex;" alt="{\displaystyle P^{\prime }(x)=5x^{4}-16x^{3}+12x^{2}+6x-8}" loading="lazy"></span>

</p><p>Aus dem Horner-Schema

</p>

<td>

</td>

<td width="30">5

</td>

<td width="30">−16

</td>

<td width="30">12

</td>

<td width="30">6

</td>

<td width="30">−8

</td></tr>

<tr>

<td>2)

</td>

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">10

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">−12

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">0

</td>

<td style="border-bottom-width:thin;border-bottom-style:solid">12

</td></tr>

<tr>

<td>

</td>

<td>5

</td>

<td>−6

</td>

<td>0

</td>

<td>6

</td>

<td style="border-bottom-width:medium;border-bottom-style:double">4

</td></tr></tbody></table>

<p>folgt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{\prime }(2)\,=4}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P^{\prime }(2)\,=4}</annotation>

</semantics>

</p>

<div class="mw-heading mw-heading4"><h4 id="Mehrfache_Ableitungen">Mehrfache Ableitungen</h4></div>

<p>Auch die Werte der weiteren Ableitungen lassen sich aus dem Horner-Schema ablesen. Sei

</p>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(x)=\sum _{i=0}^{n}a_{i}x^{i}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f695568eae8df4a96e805d6032b59758030c61d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.884ex; height:6.843ex;" alt="{\displaystyle P(x)=\sum _{i=0}^{n}a_{i}x^{i}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{a}(y)=\sum _{i=0}^{n}r_{i}y^{i}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{a}(y)=\sum _{i=0}^{n}r_{i}y^{i}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b887aad1fbed598c8c8f1ec47acbf6dfa8e94c86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.208ex; height:6.843ex;" alt="{\displaystyle P_{a}(y)=\sum _{i=0}^{n}r_{i}y^{i}}" loading="lazy"></span>, mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a+y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=a+y}</annotation>

</semantics>

<p>das Polynom, welches wir aus dem vollständigen Horner-Schema ablesen können (<a href="#Lineartransformation">siehe oben</a>), so ist

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P^{(k)}(a)=P^{(k)}(a+0)=P_{a}^{(k)}(0)=k!\,r_{k}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd910485e7c6d81bfc09e84512b2f2cf6e475447.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.3ex; height:3.509ex;" alt="{\displaystyle P^{(k)}(a)=P^{(k)}(a+0)=P_{a}^{(k)}(0)=k!\,r_{k}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Nullstellenbestimmung">Nullstellenbestimmung</h3></div>

<p>Das Horner-Schema lässt sich in verschiedenen <a href="Numerische_Mathematik" title="Numerische Mathematik">numerischen</a> Verfahren zur Nullstellenbestimmung von Polynomen einsetzen.

</p><p>Hat man z. B. eine <a href="Nullstelle" title="Nullstelle">Nullstelle</a> <i>„erraten“</i>, so kann man, wie oben gezeigt wurde, schnell überprüfen, ob die Vermutung stimmt.

</p><p>Um das „Erraten“ der Nullstelle in manchen einfachen Aufgaben zu verkürzen, kann man den <a href="Satz_%C3%BCber_rationale_Nullstellen" title="Satz über rationale Nullstellen">Satz über rationale Nullstellen</a> verwenden. Aus diesem folgt, dass eine ganzzahlige Nullstelle ein Teiler von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e425056f502ca07b103ffbf6ac4720e0f8a01f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{0}}" loading="lazy"></span> ist. Sollte ein Faktor vor der höchsten Potenz von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> stehen (z. B. 3 bei: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x^{3}-18x^{2}-3x+18=0}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3x^{3}-18x^{2}-3x+18=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/950eed78a9326d0781b1b928bc1040ddd8ff7eb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:25.854ex; height:2.843ex;" alt="{\displaystyle 3x^{3}-18x^{2}-3x+18=0}" loading="lazy"></span>), so sind auch die Teiler von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e425056f502ca07b103ffbf6ac4720e0f8a01f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{0}}" loading="lazy"></span> und am besten die gesamte Funktion durch diesen zu teilen (→ <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}-6x^{2}-x+6=0}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{3}-6x^{2}-x+6=0}</annotation>

</semantics>

</p><p>Beispiel: Man betrachte <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}-6x^{2}-x+6=0}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{3}-6x^{2}-x+6=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7095b096d0ec120bb054a800eee448a3d42af01f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.205ex; height:2.843ex;" alt="{\displaystyle x^{3}-6x^{2}-x+6=0}" loading="lazy"></span> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}=6}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0}=6}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff51b78d6e4a7819e7e55700ae16948de03fde19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.313ex; height:2.509ex;" alt="{\displaystyle b_{0}=6}" loading="lazy"></span>. Die möglichen Teiler von 6 und damit Kandidaten für Nullstellen sind 1, 2, 3, 6, und auch −1, −2, −3, −6. Mit dem Hornerschema kann man nun die Funktionswerte an diesen Stellen berechnen und so die tatsächlichen Nullstellen bestimmen. Als Nullstellen erhält man dann −1, +1, +6. Hat man eine Nullstelle bestimmt, kann man mit dem Hornerschema zudem, wie weiter oben erläutert, auch einen Linearfaktor abspalten.

</p><p>Ein weiteres Einsatzgebiet ist das <a href="Newtonverfahren" title="Newtonverfahren">newtonsche Näherungsverfahren</a>. Für das Newton-Verfahren benötigt man in jedem <a href="Iteration" title="Iteration">Iterations</a>-Schritt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x_{n})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(x_{n})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbea07a3b032a1a82d6051e876aca7d6457a3443.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.103ex; height:2.843ex;" alt="{\displaystyle P(x_{n})}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P'(x_{n})}">

<msup>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P'(x_{n})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8d36833712bd6b23d2137748c1b1d6aa5b53e15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.864ex; height:3.009ex;" alt="{\displaystyle P'(x_{n})}" loading="lazy"></span>. Diese Werte lassen sich, wie oben beschrieben, recht schnell mit dem Horner-Schema berechnen.

</p>

<div class="mw-heading mw-heading2"><h2 id="Geschichte">Geschichte</h2></div>

<p><a href="William_George_Horner" title="William George Horner">William George Horner</a> war nicht der erste, der dieses Verfahren entdeckte. Er hatte es vor allem <a href="Augustus_De_Morgan" title="Augustus De Morgan">De Morgan</a> zu verdanken, dass das Verfahren unter seinem Namen bekannt wurde. <a href="Paolo_Ruffini_(Mathematiker)" title="Paolo Ruffini (Mathematiker)">Paolo Ruffini</a> veröffentlichte 15 Jahre vor Horner bereits ein entsprechendes Verfahren; es wird in Spanien daher auch als <i>regla de Ruffini</i> bezeichnet. Erste bekannte Beschreibungen des Verfahrens reichen bis ins 11. Jahrhundert zurück (<a href="Jia_Xian" title="Jia Xian">Jia Xian</a> in China und <a href="As-Samaw'al" title="As-Samaw'al">as-Samaw'al</a> im Nahen Osten).<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Der Chinese <a href="Zhu_Shijie" title="Zhu Shijie">Zhu Shijie</a> beschrieb 1303 in seinem Buch <i>Siyuan yujian</i> eine Umwandlungsmethode zur Lösung von Gleichungen, die er <i>fan fa</i> nannte. Auch die Araber verwendeten die Methode (as-Samawal, <a href="Sharaf_al-Din_al-Tusi" class="mw-redirect" title="Sharaf al-Din al-Tusi">Sharaf al-Din al-Tusi</a>).

</p>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>William George Horner: <a rel="nofollow" class="external text" href="http://www.google.de/books?id=VOU_AAAAYAAJ&pg=RA1-PA308&source=gbs_toc_r&cad=4#v=onepage&q=&f=false"><i>A new method of solving numerical equations of all orders, by continuous approximation.</i></a> In: <i>Philosophical Transactions of the Royal Society of London.</i> 1819, S. 308–335.</li>

<li>Charles D. Miller, Margaret L. Lial, David I. Schneider: <i>Fundamentals of College Algebra.</i> 3., überarbeitete Auflage. Scott & Foresman/Little & Brown Higher Education, 1990, ISBN 0-673-38638-4, S. 204–209.</li>

<li><a href="Gisela_Engeln-M%C3%BCllges" title="Gisela Engeln-Müllges">Gisela Engeln-Müllges</a>, Klaus Niederdrenk, Reinhard Wodicka: <i>Numerik-Algorithmen: Verfahren, Beispiele, Anwendungen.</i> Springer 2005, ISBN 3-540-62669-7, S. 92–100 (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=cHMzw7t7D84C&pg=PA92#v=onepage">Auszug</a> in der Google-Buchsuche)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.matheplanet.com/default3.html?call=article.php?sid=477"><i>Das Hornerschema und andere Tricks</i></a> auf <a href="Matroids_Matheplanet" title="Matroids Matheplanet">Matroids Matheplanet</a> am 27. Juli 2003</li></ul>

<div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Wikibooks"></span></span></div><b><a href="https://de.wikibooks.org/wiki/Algorithmensammlung:_Numerik:_Horner-Schema" class="extiw external" title="b:Algorithmensammlung: Numerik: Horner-Schema">Wikibooks: Horner-Schema</a></b> – Implementierungen in der Algorithmensammlung</div>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise_und_Anmerkungen">Einzelnachweise und Anmerkungen</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Josef Stoer: <cite style="font-style:italic">Numerische Mathematik 1</cite>. 9. Auflage. Springer, 2004.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Horner-Schema&rft.au=Josef+Stoer&rft.btitle=Numerische+Mathematik+1&rft.date=2004&rft.edition=9.&rft.genre=book&rft.pub=Springer" style="display:none"> </span></span>

</li>

<li id="cite_note-AnzMult-2"><span class="mw-cite-backlink"><a href="#cite_ref-AnzMult_2-0">↑</a></span> <span class="reference-text">Bei der Berechnung der Potenzen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{k}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/525e1133440e1565055dec6243aaf0f27d4d4e9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.418ex; height:2.676ex;" alt="{\displaystyle x^{k}}" loading="lazy"></span>, k≥2, kann man zuerst die niedrigen und dann die höheren Potenzen berechnen. Damit macht man sich jeweils zunutze, dass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{k-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{k-1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71a40cf24bf2496bae8605b470ab5c46f0973f3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.519ex; height:2.676ex;" alt="{\displaystyle x^{k-1}}" loading="lazy"></span> schon berechnet ist, wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{k}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/525e1133440e1565055dec6243aaf0f27d4d4e9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.418ex; height:2.676ex;" alt="{\displaystyle x^{k}}" loading="lazy"></span> gebraucht wird. Für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{k}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/525e1133440e1565055dec6243aaf0f27d4d4e9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.418ex; height:2.676ex;" alt="{\displaystyle x^{k}}" loading="lazy"></span> braucht man daher nur eine weitere Multiplikation und nicht deren <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k-1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (k-1)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e69f74fa2adbbab50f6969acb2af719045435461.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.023ex; height:2.843ex;" alt="{\displaystyle (k-1)}" loading="lazy"></span>.</span>

</li>

/* start https://de.wikipedia.org/ */

.mw-parser-output .webarchiv-memento a{color:inherit}

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</style><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140521181854/https://www.bmf-steuerrechner.de/web/infotexte/08_01_hilfe.jsp">Interaktiver Lohn- und Einkommensteuerrechner: Rundungsvorschrift</a> (<span class="webarchiv-memento"><a href="Webarchivierung#Begrifflichkeiten" title="Webarchivierung">Memento</a></span> vom 21. Mai 2014 im <i><a href="Internet_Archive" title="Internet Archive">Internet Archive</a></i>) Bundesministerium der Finanzen: Interaktiver Lohn- und Einkommensteuerrechner: Rundungsvorschrift</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.parmentier.de/steuer/steuer.htm?steuersatz.htm">Die Einkommensteuertarif-Formeln seit 1958</a> Wolfgang & Johannes Parmentier: Die Einkommensteuertarif-Formeln seit 1958.</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">John J. O’Connor, <a href="Edmund_Robertson" title="Edmund Robertson">Edmund F. Robertson</a>: <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Jia_Xian/"><i>Horner-Schema.</i></a> In: <span lang="en"><i><a href="MacTutor_History_of_Mathematics_archive" title="MacTutor History of Mathematics archive">MacTutor History of Mathematics archive</a></i></span> (englisch).</span>

</li>

</ol></div><div><div class="zim-footer">

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