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Polynom – Wikipedia
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(de)"> <link rel="EditURI" type="application/rsd+xml" href="//de.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://de.wikipedia.org/wiki/Polynom"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de"> <link rel="alternate" type="application/atom+xml" title="Atom-Feed für „Wikipedia“" href="/w/index.php?title=Spezial:Letzte_%C3%84nderungen&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin-vector-legacy mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Polynom rootpage-Polynom skin-vector action-view"><div id="mw-page-base" class="noprint"></div> <div id="mw-head-base" class="noprint"></div> <div id="content" class="mw-body" role="main"> <a id="top"></a> <div id="siteNotice"><!-- CentralNotice --></div> <div class="mw-indicators"> </div> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Polynom</span></h1> <div id="bodyContent" class="vector-body"> <div id="siteSub" class="noprint">aus Wikipedia, der freien Enzyklopädie</div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Zur Navigation springen</a> <a class="mw-jump-link" href="#searchInput">Zur Suche springen</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Ein <b>Polynom</b> ist ein <a href="/wiki/Algebra" title="Algebra">algebraischer</a> <a href="/wiki/Term" title="Term">Term</a>, der sich als Summe von Vielfachen von <a href="/wiki/Potenz_(Mathematik)" title="Potenz (Mathematik)">Potenzen</a> einer <a href="/wiki/Variable_(Mathematik)" title="Variable (Mathematik)">Variablen</a> bzw. <a href="/wiki/Unbestimmte" title="Unbestimmte">Unbestimmten</a> darstellen lässt: </p> <dl><dd><span class="mwe-math-element"><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)=a_{0}+a_{1}x+a_{2}x^{2}+\dotsb +a_{n}x^{n},\quad n\in \mathbb {N} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=a_{0}+a_{1}x+a_{2}x^{2}+\dotsb +a_{n}x^{n},\quad n\in \mathbb {N} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2ee95dce0cb9a7d11841622b47faaf6260d82b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.954ex; height:3.176ex;" alt="{\displaystyle P(x)=a_{0}+a_{1}x+a_{2}x^{2}+\dotsb +a_{n}x^{n},\quad n\in \mathbb {N} _{0}}"></span></dd></dl> <p>oder kurz mit dem <a href="/wiki/Summenzeichen" class="mw-redirect" title="Summenzeichen">Summenzeichen</a>: </p> <dl><dd><span class="mwe-math-element"><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},\quad n\in \mathbb {N} _{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=\sum _{i=0}^{n}a_{i}x^{i},\quad n\in \mathbb {N} _{0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f8463b70c6967413ee4f08bb1843ecf6bc32588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.855ex; height:6.843ex;" alt="{\displaystyle P(x)=\sum _{i=0}^{n}a_{i}x^{i},\quad n\in \mathbb {N} _{0}.}"></span></dd></dl> <p>Dabei ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑<!-- ∑ --></mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92795a77657ae4e5746d1e5d8aa40151e176e723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.454ex; height:2.843ex;" alt="{\displaystyle \textstyle \sum }"></span> das Summenzeichen, die Zahlen <span class="mwe-math-element"><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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"></span> sind die Koeffizienten (das können beispielsweise reelle Zahlen oder allgemeiner Elemente aus einem beliebigen <a href="/wiki/Ring_(Algebra)" title="Ring (Algebra)">Ring</a> sein) und <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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}"></span> ist die Unbestimmte. </p><p>Exponenten der Potenzen sind <a href="/wiki/Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürliche Zahlen</a>. Die Summe ist außerdem stets endlich. Unendliche Summen von Vielfachen von Potenzen mit natürlichzahligen Exponenten einer Unbestimmten heißen <a href="/wiki/Formale_Potenzreihe" title="Formale Potenzreihe">formale Potenzreihen</a>. </p><p>Für Mathematik und Physik gibt es einige wichtige <a href="/wiki/Liste_spezieller_Polynome" title="Liste spezieller Polynome">spezielle Polynome</a>. </p><p>In der <a href="/wiki/Elementare_Algebra" title="Elementare Algebra">elementaren Algebra</a> identifiziert man diesen Ausdruck mit einer Funktion in <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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}"></span> (einer <a href="/wiki/Polynomfunktion" class="mw-redirect" title="Polynomfunktion">Polynomfunktion</a>). In der abstrakten Algebra unterscheidet man streng zwischen einer Polynomfunktion und einem Polynom als Element eines <a href="/wiki/Polynomring" title="Polynomring">Polynomrings</a>. In der Schulmathematik wird eine Polynomfunktion oft auch als <a href="/wiki/Ganzrationale_Funktion" title="Ganzrationale Funktion">ganzrationale Funktion</a> bezeichnet. </p><p>Dieser Artikel erklärt außerdem die mathematischen Begriffe: <i>Leitkoeffizient,</i> <i>Normieren</i> eines Polynoms und <i>Absolutglied</i>. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Etymologie"><span class="tocnumber">1</span> <span class="toctext">Etymologie</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Polynome_in_der_elementaren_Algebra"><span class="tocnumber">2</span> <span class="toctext">Polynome in der elementaren Algebra</span></a> <ul> <li class="toclevel-2 tocsection-3"><a href="#Definition"><span class="tocnumber">2.1</span> <span class="toctext">Definition</span></a></li> <li class="toclevel-2 tocsection-4"><a href="#Einfaches_Beispiel"><span class="tocnumber">2.2</span> <span class="toctext">Einfaches Beispiel</span></a></li> <li class="toclevel-2 tocsection-5"><a href="#Bezeichnung_spezieller_Polynomfunktionen"><span class="tocnumber">2.3</span> <span class="toctext">Bezeichnung spezieller Polynomfunktionen</span></a></li> <li class="toclevel-2 tocsection-6"><a href="#Nullstellen"><span class="tocnumber">2.4</span> <span class="toctext">Nullstellen</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-7"><a href="#Polynome_in_der_abstrakten_Algebra"><span class="tocnumber">3</span> <span class="toctext">Polynome in der abstrakten Algebra</span></a> <ul> <li class="toclevel-2 tocsection-8"><a href="#Definition_2"><span class="tocnumber">3.1</span> <span class="toctext">Definition</span></a></li> <li class="toclevel-2 tocsection-9"><a href="#Konstruktion"><span class="tocnumber">3.2</span> <span class="toctext">Konstruktion</span></a></li> <li class="toclevel-2 tocsection-10"><a href="#Zusammenhang_mit_der_analytischen_Definition"><span class="tocnumber">3.3</span> <span class="toctext">Zusammenhang mit der analytischen Definition</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-11"><a href="#Verallgemeinerungen"><span class="tocnumber">4</span> <span class="toctext">Verallgemeinerungen</span></a> <ul> <li class="toclevel-2 tocsection-12"><a href="#Polynome_in_mehreren_Unbestimmten"><span class="tocnumber">4.1</span> <span class="toctext">Polynome in mehreren Unbestimmten</span></a></li> <li class="toclevel-2 tocsection-13"><a href="#Formale_Potenzreihen"><span class="tocnumber">4.2</span> <span class="toctext">Formale Potenzreihen</span></a></li> <li class="toclevel-2 tocsection-14"><a href="#Laurent-Polynome_und_Laurent-Reihen"><span class="tocnumber">4.3</span> <span class="toctext">Laurent-Polynome und Laurent-Reihen</span></a></li> <li class="toclevel-2 tocsection-15"><a href="#Posynomialfunktionen"><span class="tocnumber">4.4</span> <span class="toctext">Posynomialfunktionen</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-16"><a href="#Literatur"><span class="tocnumber">5</span> <span class="toctext">Literatur</span></a></li> <li class="toclevel-1 tocsection-17"><a href="#Weblinks"><span class="tocnumber">6</span> <span class="toctext">Weblinks</span></a></li> <li class="toclevel-1 tocsection-18"><a href="#Einzelnachweise"><span class="tocnumber">7</span> <span class="toctext">Einzelnachweise</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Etymologie">Etymologie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=1" title="Abschnitt bearbeiten: Etymologie" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Etymologie"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Das Wort <i>Polynom</i> bedeutet so viel wie „mehrnamig“. Es entstammt dem griech. πολύ <i>polý</i> „viel“ und όνομα <i>onoma</i> „Name“. Diese Bezeichnung geht zurück bis auf <a href="/wiki/Elemente_(Euklid)" title="Elemente (Euklid)">Euklids Elemente</a>. In Buch X nennt er eine zweigliedrige Summe <span class="mwe-math-element"><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+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2391acf09244b9dba74eb940e871a6be7e7973a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}"></span> ἐκ δύο ὀνομάτων (ek dýo onomátōn): „aus zwei Namen (bestehend)“. Die Bezeichnung <i>Polynom</i> geht auf <a href="/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">Viëta</a> zurück: In seiner <i>Isagoge</i> (1591) verwendet er den Ausdruck <i>polynomia magnitudo</i> für eine mehrgliedrige Größe.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Polynome_in_der_elementaren_Algebra">Polynome in der elementaren Algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=2" title="Abschnitt bearbeiten: Polynome in der elementaren Algebra" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Polynome in der elementaren Algebra"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Polynomialdeg5.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Polynomialdeg5.svg/220px-Polynomialdeg5.svg.png" decoding="async" width="220" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Polynomialdeg5.svg/330px-Polynomialdeg5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Polynomialdeg5.svg/440px-Polynomialdeg5.svg.png 2x" data-file-width="233" data-file-height="179" /></a><figcaption>Graph einer Polynomfunktion 5. Grades</figcaption></figure> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Polynomfunktion" class="mw-redirect" title="Polynomfunktion">Polynomfunktion</a></i></div> <p>Im Gegensatz zur abstrakten Algebra werden Polynome in der elementaren Algebra als Funktionen aufgefasst. Daher wird in diesem Abschnitt der Begriff Polynomfunktion anstatt Polynom verwendet. </p> <div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=3" title="Abschnitt bearbeiten: Definition" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Definition"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In der <a href="/wiki/Elementare_Algebra" title="Elementare Algebra">elementaren Algebra</a> ist eine <i>Polynomfunktion</i> eine <a href="/wiki/Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</a> <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" 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}"></span>, die durch einen Ausdruck der Form </p> <dl><dd><span class="mwe-math-element"><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}=a_{0}+a_{1}x+a_{2}x^{2}+\dotsb +a_{n-1}x^{n-1}+a_{n}x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=\sum _{i=0}^{n}a_{i}x^{i}=a_{0}+a_{1}x+a_{2}x^{2}+\dotsb +a_{n-1}x^{n-1}+a_{n}x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0e1771f452452ce868b5dc868edce7195f2c00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:60.667ex; height:6.843ex;" alt="{\displaystyle P(x)=\sum _{i=0}^{n}a_{i}x^{i}=a_{0}+a_{1}x+a_{2}x^{2}+\dotsb +a_{n-1}x^{n-1}+a_{n}x^{n}}"></span></dd></dl> <p>mit <span class="mwe-math-element"><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\in \mathbb {N} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf5c8ad993619a325ca57a25c22cdc75a460f88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.968ex; height:2.509ex;" alt="{\displaystyle n\in \mathbb {N} _{0}}"></span> gegeben ist, wobei als <a href="/wiki/Definitionsbereich" class="mw-redirect" title="Definitionsbereich">Definitionsbereich</a> jede beliebige <span style="white-space:nowrap"><span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>-<a href="/wiki/Algebra_(Struktur)" class="mw-redirect" title="Algebra (Struktur)">Algebra</a></span> in Frage kommt, wenn <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> der Wertebereich der Koeffizienten ist (siehe unten). Häufig ist dieser jedoch die Menge der <a href="/wiki/Ganze_Zahlen" class="mw-redirect" title="Ganze Zahlen">ganzen</a>, der <a href="/wiki/Reelle_Zahlen" class="mw-redirect" title="Reelle Zahlen">reellen</a> oder der <a href="/wiki/Komplexe_Zahlen" class="mw-redirect" title="Komplexe Zahlen">komplexen Zahlen</a>. Die <span class="mwe-math-element"><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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"></span> stammen aus einem <a href="/wiki/Ring_(Algebra)" title="Ring (Algebra)">Ring</a> <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>, zum Beispiel einem <a href="/wiki/K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> oder einem <a href="/wiki/Restklassenring" title="Restklassenring">Restklassenring</a>, und werden <i><a href="/wiki/Koeffizient" title="Koeffizient">Koeffizienten</a></i> genannt. </p> <ul><li>Alle <a href="/wiki/Exponent_(Mathematik)" class="mw-redirect" title="Exponent (Mathematik)">Exponenten</a> sind <a href="/wiki/Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürliche Zahlen</a>.</li></ul> <p><span id="Leitkoeffizient"></span> </p> <ul><li>Als <a href="/wiki/Grad_(Polynom)" title="Grad (Polynom)">Grad</a> des Polynoms wird der höchste <a href="/wiki/Exponent_(Mathematik)" class="mw-redirect" title="Exponent (Mathematik)">Exponent</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \{0,\ldots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈<!-- ∈ --></mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \{0,\ldots ,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8f9040497045263b5ed05c4dfe76fb80f2a946e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.703ex; height:2.843ex;" alt="{\displaystyle i\in \{0,\ldots ,n\}}"></span> bezeichnet, für den der Koeffizient <span class="mwe-math-element"><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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"></span> des <a href="/wiki/Monom" title="Monom">Monoms</a> <span class="mwe-math-element"><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_{i}x^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}x^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdfa736714ce14762a9ecb42c79763e41a303acf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.159ex; height:3.009ex;" alt="{\displaystyle a_{i}x^{i}}"></span> nicht null ist. Dieser Koeffizient heißt <b>Leitkoeffizient</b> (auch: <i>führender Koeffizient</i>). (Die Schreibweise <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>⁡<!-- --></mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \deg f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f194fcb353ccf18cf61c0b9533e2016ac974501b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.153ex; height:2.509ex;" alt="{\displaystyle \deg f}"></span> für den Grad des Polynoms <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> ist vom englischen Begriff <i><span lang="en">degree</span></i> abgeleitet. In der deutschsprachigen Literatur findet sich häufig auch die aus dem Deutschen kommende Schreibweise <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {grad} \,f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> <mspace width="thinmathspace" /> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {grad} \,f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/884fc1acbf8aababf53d74d4d0d7dff18346e6cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.195ex; height:2.509ex;" alt="{\displaystyle \mathrm {grad} \,f}"></span> oder <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Grad} \,f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> <mspace width="thinmathspace" /> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Grad} \,f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/378a7886971f45214577ff4bedcf46971e2d13c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.857ex; height:2.509ex;" alt="{\displaystyle \mathrm {Grad} \,f}"></span>.)</li> <li>Das <b>reverse Polynom</b> zu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle P(x)=\sum _{i=0}^{n}a_{i}x^{i};\;a_{0},a_{n}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>;</mo> <mspace width="thickmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠<!-- ≠ --></mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle P(x)=\sum _{i=0}^{n}a_{i}x^{i};\;a_{0},a_{n}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0f3fcfd4b8a0a499e7d41972ca98cb678c5f61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.589ex; height:3.176ex;" alt="{\displaystyle \textstyle P(x)=\sum _{i=0}^{n}a_{i}x^{i};\;a_{0},a_{n}\neq 0}"></span> ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle P_{\text{rev}}(x)=\sum _{i=0}^{n}a_{n-i}x^{i}=a_{n}+a_{n-1}x+a_{n-2}x^{2}+\dotsb +a_{1}x^{n-1}+a_{0}x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rev</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle P_{\text{rev}}(x)=\sum _{i=0}^{n}a_{n-i}x^{i}=a_{n}+a_{n-1}x+a_{n-2}x^{2}+\dotsb +a_{1}x^{n-1}+a_{0}x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2c3ee10dc98f4725436ce94b1c1f029f4d7d589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:69.417ex; height:3.176ex;" alt="{\displaystyle \textstyle P_{\text{rev}}(x)=\sum _{i=0}^{n}a_{n-i}x^{i}=a_{n}+a_{n-1}x+a_{n-2}x^{2}+\dotsb +a_{1}x^{n-1}+a_{0}x^{n}}"></span>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li>Die Menge aller reellen Polynomfunktionen beliebigen (aber endlichen) Grades ist ein <a href="/wiki/Vektorraum" title="Vektorraum">Vektorraum</a>, der sich nicht offensichtlich mittels <a href="/wiki/Vektor#Geometrie" title="Vektor">geometrischer Vorstellungen</a> veranschaulichen lässt.</li> <li>Für das <b>Nullpolynom</b>, bei dem alle <span class="mwe-math-element"><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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"></span> Null sind, wird der Grad als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−<!-- − --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }"></span> definiert.<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></li> <li>Ist der Leitkoeffizient 1, dann heißt das Polynom <i>normiert</i> oder auch <i>monisch</i>.</li> <li>Sind die Koeffizienten <a href="/wiki/Teilerfremdheit" title="Teilerfremdheit">teilerfremd</a>, bzw. ist der <a href="/wiki/Inhalt_(Polynom)" title="Inhalt (Polynom)">Inhalt</a> 1, dann heißt das Polynom <i>primitiv.</i></li></ul> <p>Der Koeffizient <span class="mwe-math-element"><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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"></span> heißt <i>Absolutglied.</i> <span class="mwe-math-element"><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_{1}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47bed450fae78bb6ec7eae7ce31a5961bff5ebdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.614ex; height:2.009ex;" alt="{\displaystyle a_{1}x}"></span> wird als <i>lineares Glied</i> bezeichnet, <span class="mwe-math-element"><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}x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{2}x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d64ee465c5def316b07678c3dc48587adde46bca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.668ex; height:3.009ex;" alt="{\displaystyle a_{2}x^{2}}"></span> als <i>quadratisches Glied</i> und <span class="mwe-math-element"><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}x^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{3}x^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b353b9cb35c8036fb3193d339a99a399b175934c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.668ex; height:3.009ex;" alt="{\displaystyle a_{3}x^{3}}"></span> als <i>kubisches.</i> </p> <div class="mw-heading mw-heading3"><h3 id="Einfaches_Beispiel">Einfaches Beispiel</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=4" title="Abschnitt bearbeiten: Einfaches Beispiel" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Einfaches Beispiel"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Durch </p> <dl><dd><span class="mwe-math-element"><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):=9x^{3}+x^{2}+7x-3{,}8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mn>9</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>7</mn> <mi>x</mi> <mo>−<!-- − --></mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x):=9x^{3}+x^{2}+7x-3{,}8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec6b29d12aca3b5a3b0b6d88c2772d386f766c0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.545ex; height:3.176ex;" alt="{\displaystyle P(x):=9x^{3}+x^{2}+7x-3{,}8}"></span></dd></dl> <p>ist ein Polynom dritten Grades gegeben (der höchste vorkommende Exponent ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}"></span>). In diesem Beispiel ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d3d1e1f9dfe0254c628379e69a69711fe4eabd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 9}"></span> der <i>Leitkoeffizient</i> (als Faktor vor der höchsten Potenz von <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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}"></span>), die weiteren Koeffizienten lauten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,7}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ee442587258001ae95c7968a9f8e670d4905ca7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.359ex; height:2.509ex;" alt="{\displaystyle 1,7}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -3{,}8.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−<!-- − --></mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>8.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -3{,}8.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73ac974a8bfd3f74e095383de7fc29a88be09e5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.427ex; height:2.509ex;" alt="{\displaystyle -3{,}8.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Bezeichnung_spezieller_Polynomfunktionen">Bezeichnung spezieller Polynomfunktionen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=5" title="Abschnitt bearbeiten: Bezeichnung spezieller Polynomfunktionen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Bezeichnung spezieller Polynomfunktionen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Polynome des Grades </p> <ul><li>0 werden <i><a href="/wiki/Konstante_Funktion" title="Konstante Funktion">konstante Funktionen</a></i> genannt (z. B. <span class="mwe-math-element"><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)=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−<!-- − --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/640ba9d2aaba9d13f3b52b6ea0e0e44fb9795cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.953ex; height:2.843ex;" alt="{\displaystyle P(x)=-1}"></span>).</li> <li>1 werden <i><a href="/wiki/Lineare_Funktion" title="Lineare Funktion">lineare Funktionen</a></i> oder genauer <i>affin lineare Funktionen</i> genannt (z. B. <span class="mwe-math-element"><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)=3x+5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=3x+5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a1deefd7c56c566ab4fc2c9fe2a9d0287e29da0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.478ex; height:2.843ex;" alt="{\displaystyle P(x)=3x+5}"></span>).</li> <li>2 werden <i><a href="/wiki/Quadratische_Funktion" title="Quadratische Funktion">quadratische Funktionen</a></i> genannt (z. B. <span class="mwe-math-element"><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)=-3x^{2}-4x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−<!-- − --></mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=-3x^{2}-4x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f3e1e0d9d55ddc508b107e68ced429b88896f1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.673ex; height:3.176ex;" alt="{\displaystyle P(x)=-3x^{2}-4x+1}"></span>).</li> <li>3 werden <i><a href="/wiki/Kubische_Funktion" title="Kubische Funktion">kubische Funktionen</a></i> genannt (z. B. <span class="mwe-math-element"><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)=4x^{3}-2x^{2}+7x+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>7</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=4x^{3}-2x^{2}+7x+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa3a4e8162b9cd1fead658778692e3038c6e892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.251ex; height:3.176ex;" alt="{\displaystyle P(x)=4x^{3}-2x^{2}+7x+2}"></span>).</li> <li>4 werden <i><a href="/wiki/Polynom_4._Grades" class="mw-redirect" title="Polynom 4. Grades">quartische Funktionen</a></i> genannt (z. B. <span class="mwe-math-element"><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)=6x^{4}-x^{3}+4x^{2}+2x+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−<!-- − --></mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=6x^{4}-x^{3}+4x^{2}+2x+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c939f1209f6b7ef1e02282f13bee5cd42cdc3fc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.476ex; height:3.176ex;" alt="{\displaystyle P(x)=6x^{4}-x^{3}+4x^{2}+2x+2}"></span>).</li></ul> <p><span id="Nullstelle"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Nullstellen">Nullstellen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=6" title="Abschnitt bearbeiten: Nullstellen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Nullstellen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Als <i><a href="/wiki/Nullstelle" title="Nullstelle">Nullstellen</a></i> einer Polynomfunktion oder <i>Wurzeln</i> bzw. <i><a href="/wiki/L%C3%B6sung_(Mathematik)" title="Lösung (Mathematik)">Lösungen</a></i> einer Polynomgleichung werden jene Werte von <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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}"></span> bezeichnet, für die der Funktionswert <span class="mwe-math-element"><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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89833156eff2c51bfb8750db3306a0544ce34e14" 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)}"></span> null ist, das heißt, die die Gleichung <span class="mwe-math-element"><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)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81b206fb7efdcf09cecb110d09e4543295673ef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.145ex; height:2.843ex;" alt="{\displaystyle P(x)=0}"></span> erfüllen. Eine Polynomfunktion über einem Körper (oder allgemeiner einem <a href="/wiki/Integrit%C3%A4tsring" title="Integritätsring">Integritätsring</a>) hat stets höchstens so viele Nullstellen, wie sein Grad angibt. </p><p>Weiterhin besagt der <a href="/wiki/Fundamentalsatz_der_Algebra" title="Fundamentalsatz der Algebra">Fundamentalsatz der Algebra</a>, dass eine komplexe Polynomfunktion (das heißt eine Polynomfunktion mit <a href="/wiki/Komplexe_Zahl" title="Komplexe Zahl">komplexen</a> Koeffizienten) vom Grad <span class="mwe-math-element"><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\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥<!-- ≥ --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"></span> mindestens eine <a href="/wiki/Komplexe_Zahlen" class="mw-redirect" title="Komplexe Zahlen">komplexe</a> Nullstelle hat (reiner Existenzsatz). Dann gibt es genau <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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}"></span> Nullstellen (<a href="/wiki/Polynomdivision" title="Polynomdivision">Polynomdivision</a>), wenn die Nullstellen entsprechend ihrer <i>Vielfachheit</i> gezählt werden. So ist beispielsweise die Nullstelle <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f39b6e42e5ffb81ac7b051b9e48b9a91d0713c7" 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}"></span> der Polynomfunktion <span class="mwe-math-element"><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)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−<!-- − --></mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-2)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80ee44ffa5764a8e1f25ee0dc8adf6e66a22ac29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.196ex; height:3.176ex;" alt="{\displaystyle (x-2)^{2}}"></span> eine <i>doppelte</i>. Im Ergebnis lässt sich jede komplexe Polynomfunktion positiven Grades in ein <a href="/wiki/Produkt_(Mathematik)" title="Produkt (Mathematik)">Produkt</a> von <a href="/wiki/Linearfaktor" class="mw-redirect" title="Linearfaktor">Linearfaktoren</a> zerlegen. Allgemein kann man zu jedem Körper <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> eine <a href="/wiki/Algebraische_Erweiterung" title="Algebraische Erweiterung">algebraische Körpererweiterung</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> finden, in der alle Polynome positiven Grades mit Koeffizienten in <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> als Polynome über <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> in Linearfaktoren zerfallen. In diesem Fall nennt man <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> den <a href="/wiki/Algebraischer_Abschluss" title="Algebraischer Abschluss">algebraischen Abschluss</a> von <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>. </p><p>Die Nullstellen von Polynomen ersten, zweiten, dritten und vierten Grades lassen sich mit Formeln exakt berechnen (zum Beispiel durch die <a href="/wiki/Pq-Formel" class="mw-redirect" title="Pq-Formel">pq-Formel</a> für quadratische Gleichungen), dagegen lassen sich Polynomfunktionen höheren Grades nur in Spezialfällen mit Hilfe von <a href="/wiki/Wurzelzeichen" title="Wurzelzeichen">Wurzelzeichen</a> exakt faktorisieren. Dies ist die Aussage des <a href="/wiki/Satz_von_Abel-Ruffini" title="Satz von Abel-Ruffini">Satzes von Abel-Ruffini</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Polynome_in_der_abstrakten_Algebra">Polynome in der abstrakten Algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=7" title="Abschnitt bearbeiten: Polynome in der abstrakten Algebra" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Polynome in der abstrakten Algebra"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Polynomring" title="Polynomring">Polynomring</a></i></div> <div class="mw-heading mw-heading3"><h3 id="Definition_2">Definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=8" title="Abschnitt bearbeiten: Definition" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Definition"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In der <a href="/wiki/Abstrakte_Algebra" title="Abstrakte Algebra">abstrakten Algebra</a> definiert man ein Polynom als ein Element eines Polynomringes <span class="mwe-math-element"><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[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afeccd52deabb878398a8485755c3ceea80caf9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.038ex; height:2.843ex;" alt="{\displaystyle R[X]}"></span>. Dieser wiederum ist die <a href="/wiki/Adjunktion_(Algebra)" title="Adjunktion (Algebra)">Erweiterung</a> des Koeffizientenringes <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> durch ein unbestimmtes, (algebraisch) freies Element <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>. Damit enthält <span class="mwe-math-element"><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[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afeccd52deabb878398a8485755c3ceea80caf9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.038ex; height:2.843ex;" alt="{\displaystyle R[X]}"></span> die Potenzen <span class="mwe-math-element"><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^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/268db8293666fefd75cfb00513706171948edf09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.215ex; height:2.343ex;" alt="{\displaystyle X^{n}}"></span> (<span class="mwe-math-element"><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\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></span>) und deren Linearkombinationen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle a_{0}+\sum _{k=1}^{n}a_{k}X^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle a_{0}+\sum _{k=1}^{n}a_{k}X^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dfb20d0a39fdd5279b622bc4550a88473ddbd64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.559ex; height:3.176ex;" alt="{\displaystyle \textstyle a_{0}+\sum _{k=1}^{n}a_{k}X^{k}}"></span> mit <span class="mwe-math-element"><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_{k}\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈<!-- ∈ --></mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{k}\in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47318ec7ec2bf99a4d3c49e3838f66b26477162c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.923ex; height:2.509ex;" alt="{\displaystyle a_{k}\in R}"></span>. Dies sind auch schon alle Elemente, d. h., jedes Polynom ist eindeutig durch die <a href="/wiki/Folge_(Mathematik)" title="Folge (Mathematik)">Folge</a> </p> <dl><dd><span class="mwe-math-element"><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_{0},a_{1},\dots ,a_{n},0,0,\dots )\in R\times R\times R\times \dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…<!-- … --></mo> <mo stretchy="false">)</mo> <mo>∈<!-- ∈ --></mo> <mi>R</mi> <mo>×<!-- × --></mo> <mi>R</mi> <mo>×<!-- × --></mo> <mi>R</mi> <mo>×<!-- × --></mo> <mo>…<!-- … --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{0},a_{1},\dots ,a_{n},0,0,\dots )\in R\times R\times R\times \dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1389d56fe1849d8080bfc6c802b66937ea2ec737" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.565ex; height:2.843ex;" alt="{\displaystyle (a_{0},a_{1},\dots ,a_{n},0,0,\dots )\in R\times R\times R\times \dots }"></span></dd></dl> <p>seiner Koeffizienten charakterisiert. </p> <div class="mw-heading mw-heading3"><h3 id="Konstruktion">Konstruktion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=9" title="Abschnitt bearbeiten: Konstruktion" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Konstruktion"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Umgekehrt kann ein Modell des Polynomrings <span class="mwe-math-element"><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[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afeccd52deabb878398a8485755c3ceea80caf9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.038ex; height:2.843ex;" alt="{\displaystyle R[X]}"></span> durch die Menge der endlichen Folgen in <span class="mwe-math-element"><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\times R\times R\times \dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>×<!-- × --></mo> <mi>R</mi> <mo>×<!-- × --></mo> <mi>R</mi> <mo>×<!-- × --></mo> <mo>…<!-- … --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\times R\times R\times \dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85406946223a87f66b5966178dd5a5b10c0fca04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.536ex; height:2.176ex;" alt="{\displaystyle R\times R\times R\times \dots }"></span> konstruiert werden. Dazu wird auf <span class="mwe-math-element"><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[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afeccd52deabb878398a8485755c3ceea80caf9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.038ex; height:2.843ex;" alt="{\displaystyle R[X]}"></span> eine Addition „<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" 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 +}"></span>“ als gliedweise Summe der Folgen und eine Multiplikation „<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅<!-- ⋅ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }"></span>“ durch Faltung der Folgen definiert. Ist also <span class="mwe-math-element"><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=(a_{n})_{n\in \mathbb {N} _{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=(a_{n})_{n\in \mathbb {N} _{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/742852b52b364a8dca5de536b3aaff065eca3010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.919ex; height:3.009ex;" alt="{\displaystyle a=(a_{n})_{n\in \mathbb {N} _{0}}}"></span> und <span class="mwe-math-element"><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=(b_{n})_{n\in \mathbb {N} _{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=(b_{n})_{n\in \mathbb {N} _{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61b006a8619d87493be3bd41aa9a89cca95e778a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.454ex; height:3.009ex;" alt="{\displaystyle b=(b_{n})_{n\in \mathbb {N} _{0}}}"></span>, so ist </p> <dl><dd><span class="mwe-math-element"><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+b:=(a_{n}+b_{n})_{n\in \mathbb {N} _{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>:=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b:=(a_{n}+b_{n})_{n\in \mathbb {N} _{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6cc0dc90644aea49d99ce0aef52ef55a5d2ca16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.46ex; height:3.009ex;" alt="{\displaystyle a+b:=(a_{n}+b_{n})_{n\in \mathbb {N} _{0}}}"></span></dd></dl> <p>und </p> <dl><dd><span class="mwe-math-element"><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\cdot b:=\left(\sum _{i=0}^{n}a_{i}b_{n-i}\right)_{n\in \mathbb {N} _{0}}=\left(\sum _{i+j=n}a_{i}b_{j}\right)_{n\in \mathbb {N} _{0}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅<!-- ⋅ --></mo> <mi>b</mi> <mo>:=</mo> <msub> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> <mo>=</mo> <mi>n</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b:=\left(\sum _{i=0}^{n}a_{i}b_{n-i}\right)_{n\in \mathbb {N} _{0}}=\left(\sum _{i+j=n}a_{i}b_{j}\right)_{n\in \mathbb {N} _{0}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2c8f27ec6612c25c262a698f1435f4b10dd0b0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:46.371ex; height:8.176ex;" alt="{\displaystyle a\cdot b:=\left(\sum _{i=0}^{n}a_{i}b_{n-i}\right)_{n\in \mathbb {N} _{0}}=\left(\sum _{i+j=n}a_{i}b_{j}\right)_{n\in \mathbb {N} _{0}},}"></span></dd></dl> <p><span class="mwe-math-element"><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[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afeccd52deabb878398a8485755c3ceea80caf9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.038ex; height:2.843ex;" alt="{\displaystyle R[X]}"></span> mit diesen Verknüpfungen ist nun selbst ein kommutativer Ring, der <a href="/wiki/Polynomring" title="Polynomring">Polynomring</a> (in einer Unbestimmten) über <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>. </p><p>Identifiziert man die Unbestimmte als Folge <span class="mwe-math-element"><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:=(0,1,0,0,\dotsc )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>:=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…<!-- … --></mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X:=(0,1,0,0,\dotsc )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7824c6c993923200ed8a4f48dcce07be7778efa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.043ex; height:2.843ex;" alt="{\displaystyle X:=(0,1,0,0,\dotsc )}"></span>, so dass <span class="mwe-math-element"><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\cdot X=(0,0,1,0,0,\dotsc )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>X</mi> <mo>⋅<!-- ⋅ --></mo> <mi>X</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…<!-- … --></mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{2}=X\cdot X=(0,0,1,0,0,\dotsc )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5194951e2d7719d5bd3adc85390f9cb6c526d78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.402ex; height:3.176ex;" alt="{\displaystyle X^{2}=X\cdot X=(0,0,1,0,0,\dotsc )}"></span>, <span class="mwe-math-element"><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}=X^{2}\cdot X=(0,0,0,1,0,0,\dotsc )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅<!-- ⋅ --></mo> <mi>X</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…<!-- … --></mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{3}=X^{2}\cdot X=(0,0,0,1,0,0,\dotsc )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339485da5406c1f0f15712bba3c6dfe19b1ec4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.669ex; height:3.176ex;" alt="{\displaystyle X^{3}=X^{2}\cdot X=(0,0,0,1,0,0,\dotsc )}"></span> etc., so kann jede Folge <span class="mwe-math-element"><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_{0},a_{1},a_{2},\dotsc )\in R[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…<!-- … --></mo> <mo stretchy="false">)</mo> <mo>∈<!-- ∈ --></mo> <mi>R</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{0},a_{1},a_{2},\dotsc )\in R[X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e53ddb980d40bc739046a179cfcb53c05e8bd2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.365ex; height:2.843ex;" alt="{\displaystyle (a_{0},a_{1},a_{2},\dotsc )\in R[X]}"></span> wieder im intuitiven Sinne als Polynom dargestellt werden als </p> <dl><dd><span class="mwe-math-element"><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_{0},a_{1},a_{2},\dotsc )=a_{0}+a_{1}\cdot X+a_{2}\cdot X^{2}+\dotsb =a_{0}+\sum _{n\in \mathbb {N} _{>0}}a_{n}\cdot X^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…<!-- … --></mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅<!-- ⋅ --></mo> <mi>X</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅<!-- ⋅ --></mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>></mo> <mn>0</mn> </mrow> </msub> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅<!-- ⋅ --></mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{0},a_{1},a_{2},\dotsc )=a_{0}+a_{1}\cdot X+a_{2}\cdot X^{2}+\dotsb =a_{0}+\sum _{n\in \mathbb {N} _{>0}}a_{n}\cdot X^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18b4691ed9214afe3bb8758aa21ef1cb722a1cf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:65.809ex; height:6.009ex;" alt="{\displaystyle (a_{0},a_{1},a_{2},\dotsc )=a_{0}+a_{1}\cdot X+a_{2}\cdot X^{2}+\dotsb =a_{0}+\sum _{n\in \mathbb {N} _{>0}}a_{n}\cdot X^{n}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Zusammenhang_mit_der_analytischen_Definition">Zusammenhang mit der analytischen Definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=10" title="Abschnitt bearbeiten: Zusammenhang mit der analytischen Definition" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Zusammenhang mit der analytischen Definition"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bedenkt man nun, dass nach der Voraussetzung eine natürliche Zahl <span class="mwe-math-element"><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\in \mathbb {N} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf5c8ad993619a325ca57a25c22cdc75a460f88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.968ex; height:2.509ex;" alt="{\displaystyle n\in \mathbb {N} _{0}}"></span> existiert, so dass <span class="mwe-math-element"><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_{i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d2e283d81c74e9c6742dcbbcad8c622ef0c3c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.29ex; height:2.509ex;" alt="{\displaystyle a_{i}=0}"></span> für alle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i>n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>></mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i>n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88568023bcb42b163446ab1cc4b343f4ba079e95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.296ex; height:2.176ex;" alt="{\displaystyle i>n}"></span> gilt, so lässt sich nach den obigen Überlegungen jedes Polynom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in R[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈<!-- ∈ --></mo> <mi>R</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in R[X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e815fc021f3aa97f1d3eeba7806422803ccd9b93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.157ex; height:2.843ex;" alt="{\displaystyle f\in R[X]}"></span> über einem kommutativen unitären Ring eindeutig schreiben als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=a_{0}+a_{1}\cdot X+\dotsb +a_{n}\cdot X^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅<!-- ⋅ --></mo> <mi>X</mi> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅<!-- ⋅ --></mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=a_{0}+a_{1}\cdot X+\dotsb +a_{n}\cdot X^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e46b4dc154c9554fa321979a5173c11cbf5e3411" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.191ex; height:2.676ex;" alt="{\displaystyle f=a_{0}+a_{1}\cdot X+\dotsb +a_{n}\cdot X^{n}}"></span>. Dabei ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> jedoch keine Funktion wie in der Analysis oder elementaren Algebra, sondern eine unendliche Folge (ein Element des Ringes <span class="mwe-math-element"><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[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afeccd52deabb878398a8485755c3ceea80caf9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.038ex; height:2.843ex;" alt="{\displaystyle R[X]}"></span>) und <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> ist keine „Unbekannte“, sondern die Folge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1,0,0,\dotsc )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…<!-- … --></mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1,0,0,\dotsc )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17d5d1f320e25fc0d8deb555b3f11e1a6653e919" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.318ex; height:2.843ex;" alt="{\displaystyle (0,1,0,0,\dotsc )}"></span>. Man kann jedoch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> als „Muster“ benutzen, um danach eine Polynomfunktion (d. h. ein Polynom im gewöhnlichen analytischen Sinne) zu bilden. Dazu benutzt man den sogenannten <a href="/wiki/Satz_%C3%BCber_den_Einsetzungshomomorphismus" title="Satz über den Einsetzungshomomorphismus">Einsetzungshomomorphismus</a>. </p><p>Man sollte allerdings beachten, dass verschiedene Polynome dieselbe Polynomfunktion induzieren können. Ist beispielsweise <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> der <a href="/wiki/Restklassenring" title="Restklassenring">Restklassenring</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /3\mathbb {Z} =\{{\bar {0}},{\bar {1}},{\bar {2}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">¯<!-- ¯ --></mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>1</mn> <mo stretchy="false">¯<!-- ¯ --></mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>2</mn> <mo stretchy="false">¯<!-- ¯ --></mo> </mover> </mrow> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /3\mathbb {Z} =\{{\bar {0}},{\bar {1}},{\bar {2}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a21d321d0fad4050e237508368da1d90ceab3e11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.404ex; height:3.009ex;" alt="{\displaystyle \mathbb {Z} /3\mathbb {Z} =\{{\bar {0}},{\bar {1}},{\bar {2}}\}}"></span>, so induzieren die Polynome <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\in (\mathbb {Z} /3\mathbb {Z} )[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈<!-- ∈ --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\in (\mathbb {Z} /3\mathbb {Z} )[X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c84ae324d088f587e988aae44e2b4e6913595c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.778ex; height:2.843ex;" alt="{\displaystyle f,g\in (\mathbb {Z} /3\mathbb {Z} )[X]}"></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=X(X-{\bar {1}})(X-{\bar {2}})=X^{3}-{\bar {3}}X^{2}+{\bar {2}}X=X^{3}-X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>1</mn> <mo stretchy="false">¯<!-- ¯ --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>2</mn> <mo stretchy="false">¯<!-- ¯ --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo stretchy="false">¯<!-- ¯ --></mo> </mover> </mrow> </mrow> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>2</mn> <mo stretchy="false">¯<!-- ¯ --></mo> </mover> </mrow> </mrow> <mi>X</mi> <mo>=</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=X(X-{\bar {1}})(X-{\bar {2}})=X^{3}-{\bar {3}}X^{2}+{\bar {2}}X=X^{3}-X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f91e2b3886e9e3aaf2803d077ef7d78dbd01134" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.098ex; height:3.176ex;" alt="{\displaystyle f=X(X-{\bar {1}})(X-{\bar {2}})=X^{3}-{\bar {3}}X^{2}+{\bar {2}}X=X^{3}-X}"></span></dd></dl> <p>und </p> <dl><dd>das Nullpolynom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0607b4f563220901d455691e4a1705d598fdbaeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.377ex; height:2.509ex;" alt="{\displaystyle g=0}"></span></dd></dl> <p>beide die <a href="/wiki/Nullabbildung" class="mw-redirect" title="Nullabbildung">Nullabbildung</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in \operatorname {Abb} \left(\mathbb {Z} /3\mathbb {Z} ,\mathbb {Z} /3\mathbb {Z} \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>∈<!-- ∈ --></mo> <mi>Abb</mi> <mo>⁡<!-- --></mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\in \operatorname {Abb} \left(\mathbb {Z} /3\mathbb {Z} ,\mathbb {Z} /3\mathbb {Z} \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c0406a60a132801302ad46eb4b8a232c32759c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.026ex; height:2.843ex;" alt="{\displaystyle 0\in \operatorname {Abb} \left(\mathbb {Z} /3\mathbb {Z} ,\mathbb {Z} /3\mathbb {Z} \right)}"></span>, das heißt: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=g(x)={\bar {0}}=0(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">¯<!-- ¯ --></mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=g(x)={\bar {0}}=0(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1feea28c510cdccdfbc7fcbcc9df9aec6ca23678" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.432ex; height:3.009ex;" alt="{\displaystyle f(x)=g(x)={\bar {0}}=0(x)}"></span> für alle <span class="mwe-math-element"><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\in \mathbb {Z} /3\mathbb {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {Z} /3\mathbb {Z} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55465f5ef3ae737a1dd42a6d52a314a1854a7974" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.243ex; height:2.843ex;" alt="{\displaystyle x\in \mathbb {Z} /3\mathbb {Z} .}"></span> </p><p>Für Polynome über den reellen oder ganzen Zahlen oder allgemein jedem unendlichen <a href="/wiki/Integrit%C3%A4tsring" title="Integritätsring">Integritätsring</a> ist ein Polynom jedoch durch die induzierte Polynomfunktion bestimmt. </p><p>Auch die Menge der Polynomfunktionen mit Werten in <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> bildet einen Ring (<a href="/wiki/Unterring" class="mw-redirect" title="Unterring">Unterring</a> des <a href="/wiki/Funktionenring" title="Funktionenring">Funktionenrings</a>), der jedoch nur selten betrachtet wird. Es gibt einen natürlichen Ring-<a href="/wiki/Homomorphismus" title="Homomorphismus">Homomorphismus</a> von <span class="mwe-math-element"><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[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afeccd52deabb878398a8485755c3ceea80caf9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.038ex; height:2.843ex;" alt="{\displaystyle R[X]}"></span> in den Ring der Polynomfunktionen, dessen <a href="/wiki/Kern_(Algebra)" title="Kern (Algebra)">Kern</a> die Menge der Polynome ist, die die Nullfunktion induzieren. </p> <div class="mw-heading mw-heading2"><h2 id="Verallgemeinerungen">Verallgemeinerungen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=11" title="Abschnitt bearbeiten: Verallgemeinerungen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Verallgemeinerungen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Polynome_in_mehreren_Unbestimmten"><span id="multivariates_Polynom"></span> Polynome in mehreren Unbestimmten</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=12" title="Abschnitt bearbeiten: Polynome in mehreren Unbestimmten" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Polynome in mehreren Unbestimmten"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Allgemein versteht man jede Summe von <a href="/wiki/Monom" title="Monom">Monomen</a> der Form <span class="mwe-math-element"><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_{i_{1},\dotsc ,i_{n}}X_{1}^{i_{1}}\dotsm X_{n}^{i_{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯<!-- ⋯ --></mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i_{1},\dotsc ,i_{n}}X_{1}^{i_{1}}\dotsm X_{n}^{i_{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8662be124e4af5170c9163a98fe1c931d8b56c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.121ex; height:3.509ex;" alt="{\displaystyle a_{i_{1},\dotsc ,i_{n}}X_{1}^{i_{1}}\dotsm X_{n}^{i_{n}}}"></span> als <a href="/wiki/Multivariat" class="mw-redirect" title="Multivariat">multivariates</a> Polynom (in mehreren Unbestimmten): </p> <dl><dd><span class="mwe-math-element"><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_{1},\dotsc ,X_{n})=\sum _{i_{1},\dotsc ,i_{n}}a_{i_{1},\dotsc ,i_{n}}X_{1}^{i_{1}}\dotsm X_{n}^{i_{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯<!-- ⋯ --></mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X_{1},\dotsc ,X_{n})=\sum _{i_{1},\dotsc ,i_{n}}a_{i_{1},\dotsc ,i_{n}}X_{1}^{i_{1}}\dotsm X_{n}^{i_{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bd87d973f8e234f7634cf78d615c5a06e4077c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:42.232ex; height:5.843ex;" alt="{\displaystyle P(X_{1},\dotsc ,X_{n})=\sum _{i_{1},\dotsc ,i_{n}}a_{i_{1},\dotsc ,i_{n}}X_{1}^{i_{1}}\dotsm X_{n}^{i_{n}}}"></span></dd> <dd><small>Lies: <i>„Groß-p von Groß-x-1 bis Groß-x-n (ist) gleich die Summe über alle i-1 bis i-n von a-i-1-bis-i-n mal Groß-x-1 hoch i-1 bis Groß-x-n hoch i-n“</i></small></dd></dl> <p>Durch eine <a href="/wiki/Monomordnung" title="Monomordnung">Monomordnung</a> ist es möglich, die Monome in einem solchen Polynom anzuordnen und dadurch Begriffe wie <i>Leitkoeffizient</i> zu verallgemeinern. </p><p>Die Größe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{1}+\dotsb +i_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>+</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{1}+\dotsb +i_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b83c86bd5dbce5aae5917bacd402499ccb79d072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.282ex; height:2.509ex;" alt="{\displaystyle i_{1}+\dotsb +i_{n}}"></span> heißt der <i>Totalgrad</i> eines Monoms <span class="mwe-math-element"><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_{1}^{i_{1}}\dotsm X_{n}^{i_{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯<!-- ⋯ --></mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}^{i_{1}}\dotsm X_{n}^{i_{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8030e4ed3e0afcf3aff91e53799eab50c8483057" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.887ex; height:3.509ex;" alt="{\displaystyle X_{1}^{i_{1}}\dotsm X_{n}^{i_{n}}}"></span>. Haben alle (nichtverschwindenden) Monome in einem Polynom denselben Totalgrad, so heißt es <a href="/wiki/Homogenes_Polynom" title="Homogenes Polynom">homogen</a>. Der maximale Totalgrad aller nichtverschwindenden Monome ist der <i>Grad des Polynoms.</i> </p><p>Die maximale Anzahl der möglichen Monome eines bestimmten Grades ist<sup id="cite_ref-Kunz_4-0" class="reference"><a href="#cite_note-Kunz-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><a href="/wiki/Binomialkoeffizient" title="Binomialkoeffizient"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {n+k-1}{k}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n+k-1}{k}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d9ba2e754ad7746682ad34371874c9fdfca521a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.517ex; height:6.176ex;" alt="{\displaystyle {\binom {n+k-1}{k}},}"></span></a></dd> <dd><small>Lies: <i><a href="/wiki/Binomialkoeffizient" title="Binomialkoeffizient">„n + k − 1 über k“</a></i> oder <i><a href="/wiki/Binomialkoeffizient" title="Binomialkoeffizient">„k aus n + k − 1“</a></i></small></dd></dl> <p>wobei <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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}"></span> die Anzahl der vorkommenden Unbestimmten und <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> der Grad ist. Anschaulich wird hier ein Problem von Kombinationen mit Wiederholung (Zurücklegen) betrachtet. </p><p>Summiert man die Anzahl der möglichen Monome des Grades <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> bis <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>, erhält man für die Anzahl der <i>möglichen Monome in einem Polynom bestimmten Grades</i>: </p> <dl><dd><a href="/wiki/Binomialkoeffizient" title="Binomialkoeffizient"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {n+k}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {n+k}{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc8b2f75d8724f64416133b84c44cb820b79cfbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.868ex; height:6.176ex;" alt="{\displaystyle {\binom {n+k}{k}}}"></span></a></dd> <dd><small>Lies: <i><a href="/wiki/Binomialkoeffizient" title="Binomialkoeffizient">„n + k über k“</a></i> oder <i><a href="/wiki/Binomialkoeffizient" title="Binomialkoeffizient">„k aus n + k“</a></i></small></dd></dl> <p>Sind alle Unbestimmten in gewisser Weise „gleichberechtigt“, so heißt das Polynom <a href="/wiki/Symmetrisches_Polynom" title="Symmetrisches Polynom">symmetrisch</a>. Gemeint ist, dass das Polynom sich bei Vertauschungen der Unbestimmten nicht ändert. </p><p>Auch die Polynome in den <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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}"></span> Unbestimmten <span class="mwe-math-element"><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_{1},\dotsc ,X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\dotsc ,X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae25602d4b1df2584ef4dd94474b2db32c4e970e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.299ex; height:2.509ex;" alt="{\displaystyle X_{1},\dotsc ,X_{n}}"></span>über dem Ring <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> bilden einen Polynomring, geschrieben als <span class="mwe-math-element"><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[X_{1},\dotsc ,X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[X_{1},\dotsc ,X_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a163f23c4ea9575b3da5875cedfae1da278689b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.357ex; height:2.843ex;" alt="{\displaystyle R[X_{1},\dotsc ,X_{n}]}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Formale_Potenzreihen">Formale Potenzreihen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=13" title="Abschnitt bearbeiten: Formale Potenzreihen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Formale Potenzreihen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Geht man zu <a href="/wiki/Unendliche_Reihe" class="mw-redirect" title="Unendliche Reihe">unendlichen Reihen</a> der Form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=\sum _{i=0}^{\infty }a_{i}X^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\sum _{i=0}^{\infty }a_{i}X^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4f7fe4484ddeec31cfb1af01c81c5fab39653d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.945ex; height:6.843ex;" alt="{\displaystyle f=\sum _{i=0}^{\infty }a_{i}X^{i}}"></span></dd> <dd><small>Lies: <i>„f (ist) gleich die Summe von i gleich Null bis Unendlich von a-i (mal) (Groß-) x hoch i“</i></small></dd></dl> <p>über, erhält man <a href="/wiki/Formale_Potenzreihe" title="Formale Potenzreihe">formale Potenzreihen</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Laurent-Polynome_und_Laurent-Reihen">Laurent-Polynome und Laurent-Reihen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=14" title="Abschnitt bearbeiten: Laurent-Polynome und Laurent-Reihen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Laurent-Polynome und Laurent-Reihen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lässt man auch in einem Polynom auch negative Exponenten zu, so erhält man ein <a href="/wiki/Laurent-Polynom" title="Laurent-Polynom">Laurent-Polynom</a>. Entsprechend zu den formalen Potenzreihen können auch formale <a href="/wiki/Laurent-Reihe" title="Laurent-Reihe">Laurent-Reihen</a> betrachtet werden. Es handelt sich dabei um Objekte der Form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=\sum _{i=-N}^{\infty }a_{i}X^{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mo>−<!-- − --></mo> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\sum _{i=-N}^{\infty }a_{i}X^{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24cdc6b794cdc1829113b767dce81834625ed4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.821ex; height:7.009ex;" alt="{\displaystyle f=\sum _{i=-N}^{\infty }a_{i}X^{i}.}"></span><br /><small>Lies: <i>„f (ist) gleich die Summe von i gleich minus (Groß-) n bis Unendlich von a-i (mal) (Groß-) x hoch i“</i></small></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Posynomialfunktionen">Posynomialfunktionen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=15" title="Abschnitt bearbeiten: Posynomialfunktionen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Posynomialfunktionen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lässt man mehrere Variablen und beliebige reelle Potenzen zu, so erhält man den Begriff der <a href="/wiki/Posynomialfunktion" title="Posynomialfunktion">Posynomialfunktion</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=16" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=16" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Albrecht Beutelspacher: <i>Lineare Algebra.</i> 8. Auflage, <a href="/wiki/Spezial:ISBN-Suche/9783658024130" class="internal mw-magiclink-isbn">ISBN 978-3-658-02413-0</a>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-658-02413-0">10.1007/978-3-658-02413-0</a></span></li> <li>Michael Holz & Detlef Wille: <i>Repetitorium der Linearen Algebra, Teil 2</i>, <a href="/wiki/Spezial:ISBN-Suche/9783923923427" class="internal mw-magiclink-isbn">ISBN 978-3-923923-42-7</a></li> <li>Gerd Fischer: <i>Lehrbuch der Algebra</i>, <a href="/wiki/Spezial:ISBN-Suche/9783658022211" class="internal mw-magiclink-isbn">ISBN 978-3-658-02221-1</a>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-658-02221-1">10.1007/978-3-658-02221-1</a></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=17" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=17" title="Quellcode des Abschnitts bearbeiten: Weblinks"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><span 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="Wiktionary"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/16px-Wiktfavicon_en.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/24px-Wiktfavicon_en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/32px-Wiktfavicon_en.svg.png 2x" data-file-width="16" data-file-height="16" /></span></span></span><b><a href="https://de.wiktionary.org/wiki/Polynom" class="extiw" title="wikt:Polynom">Wiktionary: Polynom</a></b> – Bedeutungserklärungen, Wortherkunft, Synonyme, Übersetzungen</div> <ul><li><a rel="nofollow" class="external text" href="http://www.arndt-bruenner.de/mathe/java/nullstellen.htm">Java-Applet</a> zur Berechnung der (auch komplexen) Nullstellen von Polynomen maximal 24. Grades (nach dem <a href="/wiki/Newton-Verfahren" class="mw-redirect" title="Newton-Verfahren">Newton-Verfahren</a>)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynom&veaction=edit&section=18" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynom&action=edit&section=18" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">cf. Barth, Federle, Haller: <i>Algebra 1.</i> Ehrenwirth-Verlag, München 1980, S. 187, Fußnote **, dort Erklärung zur Bezeichnung „Binomische Formel“</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www3.math.tu-berlin.de/algebra/teaching/15-ws/algebra1/exercise-11.pdf">math.tu-berlin.de</a> Prof. Bürgisser / Dr. Lairez / P. Breiding, TU Berlin, Institut für Mathematik, Fakultät II, Algebra I, WS 2015/2016, Blatt 11, Aufgabe 2, Bemerkung, abgerufen am 8. April 2023</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Für die Zweckmäßigkeit dieser Setzung siehe <a href="/wiki/Division_mit_Rest#Polynome" title="Division mit Rest">Division mit Rest</a>.</span> </li> <li id="cite_note-Kunz-4"><span class="mw-cite-backlink"><a href="#cite_ref-Kunz_4-0">↑</a></span> <span class="reference-text">Ernst Kunz: <i>Einführung in die algebraische Geometrie.</i> S. 213, Vieweg+Teubner, Wiesbaden 1997, <a href="/wiki/Spezial:ISBN-Suche/3528072873" class="internal mw-magiclink-isbn">ISBN 3-528-07287-3</a>.</span> </li> </ol></div><!--esi <esi:include src="/esitest-fa8a495983347898/content" /> --><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Polynom&oldid=246806436">https://de.wikipedia.org/w/index.php?title=Polynom&oldid=246806436</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:Polynom" title="Kategorie:Polynom">Polynom</a></li><li><a href="/wiki/Kategorie:Theorie_der_Polynome" title="Kategorie:Theorie der Polynome">Theorie der Polynome</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Meine Werkzeuge</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Benutzerseite der IP-Adresse, von der aus du Änderungen durchführst">Nicht angemeldet</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n"><span>Diskussionsseite</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y"><span>Beiträge</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Polynom" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q43260" title="Link zum verbundenen Objekt im Datenrepositorium [g]" accesskey="g"><span>Wikidata-Datenobjekt</span></a></li> </ul> </div> </nav> <nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label" > <h3 id="p-lang-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">In anderen Sprachen</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Polinoom" title="Polinoom – Afrikaans" lang="af" hreflang="af" data-title="Polinoom" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AA%D8%B9%D8%AF%D8%AF%D8%A9_%D8%A7%D9%84%D8%AD%D8%AF%D9%88%D8%AF" title="متعددة الحدود – Arabisch" lang="ar" hreflang="ar" data-title="متعددة الحدود" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Polinomiu" title="Polinomiu – Asturisch" lang="ast" hreflang="ast" data-title="Polinomiu" data-language-autonym="Asturianu" data-language-local-name="Asturisch" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C3%87oxh%C9%99dli" title="Çoxhədli – Aserbaidschanisch" lang="az" hreflang="az" data-title="Çoxhədli" data-language-autonym="Azərbaycanca" data-language-local-name="Aserbaidschanisch" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D2%AF%D0%BF%D0%B1%D1%8B%D1%83%D1%8B%D0%BD" title="Күпбыуын – Baschkirisch" lang="ba" hreflang="ba" data-title="Күпбыуын" data-language-autonym="Башҡортса" data-language-local-name="Baschkirisch" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%BD%D0%B0%D0%B3%D0%B0%D1%87%D0%BB%D0%B5%D0%BD" title="Мнагачлен – Belarussisch" lang="be" hreflang="be" data-title="Мнагачлен" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9C%D0%BD%D0%B0%D0%B3%D0%B0%D1%81%D0%BA%D0%BB%D0%B0%D0%B4" title="Мнагасклад – Weißrussisch (Taraschkewiza)" lang="be-tarask" hreflang="be-tarask" data-title="Мнагасклад" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Weißrussisch (Taraschkewiza)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D1%87%D0%BB%D0%B5%D0%BD" title="Многочлен – Bulgarisch" lang="bg" hreflang="bg" data-title="Многочлен" data-language-autonym="Български" data-language-local-name="Bulgarisch" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%B9%E0%A7%81%E0%A6%AA%E0%A6%A6%E0%A7%80" title="বহুপদী – Bengalisch" lang="bn" hreflang="bn" data-title="বহুপদী" data-language-autonym="বাংলা" data-language-local-name="Bengalisch" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Polinom" title="Polinom – Bosnisch" lang="bs" hreflang="bs" data-title="Polinom" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Polinomi" title="Polinomi – Katalanisch" lang="ca" hreflang="ca" data-title="Polinomi" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%D8%A7%D8%AF%DB%95%D8%AF%D8%A7%D8%B1" title="ڕادەدار – Zentralkurdisch" lang="ckb" hreflang="ckb" data-title="ڕادەدار" data-language-autonym="کوردی" data-language-local-name="Zentralkurdisch" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Polynom" title="Polynom – Tschechisch" lang="cs" hreflang="cs" data-title="Polynom" data-language-autonym="Čeština" data-language-local-name="Tschechisch" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B8%D0%BD%D0%BE%D0%BC" title="Полином – Tschuwaschisch" lang="cv" hreflang="cv" data-title="Полином" data-language-autonym="Чӑвашла" data-language-local-name="Tschuwaschisch" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Polynomial" title="Polynomial – Walisisch" lang="cy" hreflang="cy" data-title="Polynomial" data-language-autonym="Cymraeg" data-language-local-name="Walisisch" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Polynomium" title="Polynomium – Dänisch" lang="da" hreflang="da" data-title="Polynomium" data-language-autonym="Dansk" data-language-local-name="Dänisch" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Polinom" title="Polinom – Zazaki" lang="diq" hreflang="diq" data-title="Polinom" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%BF%CE%BB%CF%85%CF%8E%CE%BD%CF%85%CE%BC%CE%BF" title="Πολυώνυμο – Griechisch" lang="el" hreflang="el" data-title="Πολυώνυμο" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Polynomial" title="Polynomial – Englisch" lang="en" hreflang="en" data-title="Polynomial" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Polinomo" title="Polinomo – Esperanto" lang="eo" hreflang="eo" data-title="Polinomo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Polinomio" title="Polinomio – Spanisch" lang="es" hreflang="es" data-title="Polinomio" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Pol%C3%BCnoom" title="Polünoom – Estnisch" lang="et" hreflang="et" data-title="Polünoom" data-language-autonym="Eesti" data-language-local-name="Estnisch" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Polinomio" title="Polinomio – Baskisch" lang="eu" hreflang="eu" data-title="Polinomio" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%86%D9%86%D8%AF%D8%AC%D9%85%D9%84%D9%87%E2%80%8C%D8%A7%DB%8C" title="چندجملهای – Persisch" lang="fa" hreflang="fa" data-title="چندجملهای" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Polynomi" title="Polynomi – Finnisch" lang="fi" hreflang="fi" data-title="Polynomi" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Polyn%C3%B4me" title="Polynôme – Französisch" lang="fr" hreflang="fr" data-title="Polynôme" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Mearterm" title="Mearterm – Westfriesisch" lang="fy" hreflang="fy" data-title="Mearterm" data-language-autonym="Frysk" data-language-local-name="Westfriesisch" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Ilt%C3%A9armach" title="Iltéarmach – Irisch" lang="ga" hreflang="ga" data-title="Iltéarmach" data-language-autonym="Gaeilge" data-language-local-name="Irisch" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Polinomio" title="Polinomio – Galicisch" lang="gl" hreflang="gl" data-title="Polinomio" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%9C%D7%99%D7%A0%D7%95%D7%9D" title="פולינום – Hebräisch" lang="he" hreflang="he" data-title="פולינום" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A4%B9%E0%A5%81%E0%A4%AA%E0%A4%A6" title="बहुपद – Hindi" lang="hi" hreflang="hi" data-title="बहुपद" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Polinom" title="Polinom – Kroatisch" lang="hr" hreflang="hr" data-title="Polinom" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Polinom" title="Polinom – Ungarisch" lang="hu" hreflang="hu" data-title="Polinom" data-language-autonym="Magyar" data-language-local-name="Ungarisch" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%A1%D5%A6%D5%B4%D5%A1%D5%B6%D5%A4%D5%A1%D5%B4" title="Բազմանդամ – Armenisch" lang="hy" hreflang="hy" data-title="Բազմանդամ" data-language-autonym="Հայերեն" data-language-local-name="Armenisch" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Polinomial" title="Polinomial – Iban" lang="iba" hreflang="iba" data-title="Polinomial" data-language-autonym="Jaku Iban" data-language-local-name="Iban" class="interlanguage-link-target"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Polinomial" title="Polinomial – Indonesisch" lang="id" hreflang="id" data-title="Polinomial" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Polinomio" title="Polinomio – Ido" lang="io" hreflang="io" data-title="Polinomio" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Margli%C3%B0a" title="Margliða – Isländisch" lang="is" hreflang="is" data-title="Margliða" data-language-autonym="Íslenska" data-language-local-name="Isländisch" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Polinomio" title="Polinomio – Italienisch" lang="it" hreflang="it" data-title="Polinomio" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%A4%9A%E9%A0%85%E5%BC%8F" title="多項式 – Japanisch" lang="ja" hreflang="ja" data-title="多項式" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%A0%E1%83%90%E1%83%95%E1%83%90%E1%83%9A%E1%83%AC%E1%83%94%E1%83%95%E1%83%A0%E1%83%98" title="მრავალწევრი – Georgisch" lang="ka" hreflang="ka" data-title="მრავალწევრი" data-language-autonym="ქართული" data-language-local-name="Georgisch" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D3%A9%D0%BF%D0%BC%D2%AF%D1%88%D0%B5%D0%BB%D1%96%D0%BA" title="Көпмүшелік – Kasachisch" lang="kk" hreflang="kk" data-title="Көпмүшелік" data-language-autonym="Қазақша" data-language-local-name="Kasachisch" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%96%E1%9E%A0%E1%9E%BB%E1%9E%92%E1%9E%B6" title="ពហុធា – Khmer" lang="km" hreflang="km" data-title="ពហុធា" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A4%ED%95%AD%EC%8B%9D" title="다항식 – Koreanisch" lang="ko" hreflang="ko" data-title="다항식" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Polynomium" title="Polynomium – Latein" lang="la" hreflang="la" data-title="Polynomium" data-language-autonym="Latina" data-language-local-name="Latein" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Polinomas" title="Polinomas – Litauisch" lang="lt" hreflang="lt" data-title="Polinomas" data-language-autonym="Lietuvių" data-language-local-name="Litauisch" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Polinoms" title="Polinoms – Lettisch" lang="lv" hreflang="lv" data-title="Polinoms" data-language-autonym="Latviešu" data-language-local-name="Lettisch" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Maromiantoana" title="Maromiantoana – Malagasy" lang="mg" hreflang="mg" data-title="Maromiantoana" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B8%D0%BD%D0%BE%D0%BC" title="Полином – Mazedonisch" lang="mk" hreflang="mk" data-title="Полином" data-language-autonym="Македонски" data-language-local-name="Mazedonisch" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AC%E0%B4%B9%E0%B5%81%E0%B4%AA%E0%B4%A6%E0%B4%82" title="ബഹുപദം – Malayalam" lang="ml" hreflang="ml" data-title="ബഹുപദം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AC%E0%A4%B9%E0%A5%81%E0%A4%AA%E0%A4%A6%E0%A5%80" title="बहुपदी – Marathi" lang="mr" hreflang="mr" data-title="बहुपदी" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Polinomial" title="Polinomial – Malaiisch" lang="ms" hreflang="ms" data-title="Polinomial" data-language-autonym="Bahasa Melayu" data-language-local-name="Malaiisch" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Polynoom" title="Polynoom – Niederländisch" lang="nl" hreflang="nl" data-title="Polynoom" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Polynom" title="Polynom – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Polynom" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegisch (Nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Polynom" title="Polynom – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Polynom" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wielomian" title="Wielomian – Polnisch" lang="pl" hreflang="pl" data-title="Wielomian" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt badge-Q70893996 mw-list-item" title=""><a href="https://pt.wikipedia.org/wiki/Polin%C3%B3mio" title="Polinómio – Portugiesisch" lang="pt" hreflang="pt" data-title="Polinómio" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Polinom" title="Polinom – Rumänisch" lang="ro" hreflang="ro" data-title="Polinom" data-language-autonym="Română" data-language-local-name="Rumänisch" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D1%87%D0%BB%D0%B5%D0%BD" title="Многочлен – Russisch" lang="ru" hreflang="ru" data-title="Многочлен" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Polinom" title="Polinom – Serbokroatisch" lang="sh" hreflang="sh" data-title="Polinom" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbokroatisch" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B6%E0%B7%84%E0%B7%94_%E0%B6%B4%E0%B6%AF%E0%B6%BA" title="බහු පදය – Singhalesisch" lang="si" hreflang="si" data-title="බහු පදය" data-language-autonym="සිංහල" data-language-local-name="Singhalesisch" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Polynomial" title="Polynomial – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Polynomial" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Mnoho%C4%8Dlen" title="Mnohočlen – Slowakisch" lang="sk" hreflang="sk" data-title="Mnohočlen" data-language-autonym="Slovenčina" data-language-local-name="Slowakisch" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Polinom" title="Polinom – Slowenisch" lang="sl" hreflang="sl" data-title="Polinom" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Polinomet" title="Polinomet – Albanisch" lang="sq" hreflang="sq" data-title="Polinomet" data-language-autonym="Shqip" data-language-local-name="Albanisch" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B8%D0%BD%D0%BE%D0%BC" title="Полином – Serbisch" lang="sr" hreflang="sr" data-title="Полином" data-language-autonym="Српски / srpski" data-language-local-name="Serbisch" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Polynom" title="Polynom – Schwedisch" lang="sv" hreflang="sv" data-title="Polynom" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%B2%E0%AF%8D%E0%AE%B2%E0%AF%81%E0%AE%B1%E0%AF%81%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%B5%E0%AF%88" title="பல்லுறுப்புக்கோவை – Tamil" lang="ta" hreflang="ta" data-title="பல்லுறுப்புக்கோவை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%91%D0%B8%D1%81%D1%91%D1%80%D1%8A%D1%83%D0%B7%D0%B2%D0%B0" title="Бисёръузва – Tadschikisch" lang="tg" hreflang="tg" data-title="Бисёръузва" data-language-autonym="Тоҷикӣ" data-language-local-name="Tadschikisch" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9E%E0%B8%AB%E0%B8%B8%E0%B8%99%E0%B8%B2%E0%B8%A1" title="พหุนาม – Thailändisch" lang="th" hreflang="th" data-title="พหุนาม" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Polynomial" title="Polynomial – Tagalog" lang="tl" hreflang="tl" data-title="Polynomial" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Polinom" title="Polinom – Türkisch" lang="tr" hreflang="tr" data-title="Polinom" data-language-autonym="Türkçe" data-language-local-name="Türkisch" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D1%87%D0%BB%D0%B5%D0%BD" title="Многочлен – Ukrainisch" lang="uk" hreflang="uk" data-title="Многочлен" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DA%A9%D8%AB%DB%8C%D8%B1_%D8%A7%D9%84%D8%A7%D8%B3%D9%85%DB%8C" title="کثیر الاسمی – Urdu" lang="ur" hreflang="ur" data-title="کثیر الاسمی" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Ko%CA%BBphad" title="Koʻphad – Usbekisch" lang="uz" hreflang="uz" data-title="Koʻphad" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Usbekisch" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90a_th%E1%BB%A9c" title="Đa thức – Vietnamesisch" lang="vi" hreflang="vi" data-title="Đa thức" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%A4%9A%E9%A1%B9%E5%BC%8F" title="多项式 – Wu" lang="wuu" hreflang="wuu" data-title="多项式" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A4%D7%90%D7%9C%D7%99%D7%A0%D7%90%D7%9D" title="פאלינאם – Jiddisch" lang="yi" hreflang="yi" data-title="פאלינאם" data-language-autonym="ייִדיש" data-language-local-name="Jiddisch" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo badge-Q17437798 badge-goodarticle mw-list-item" title="lesenswerter Artikel"><a href="https://yo.wikipedia.org/wiki/On%C3%ADr%C3%BAiyep%C3%BAp%E1%BB%8D%CC%80" title="Onírúiyepúpọ̀ – Yoruba" lang="yo" hreflang="yo" data-title="Onírúiyepúpọ̀" data-language-autonym="Yorùbá" data-language-local-name="Yoruba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A4%9A%E9%A0%85%E5%BC%8F" title="多項式 – Chinesisch" lang="zh" hreflang="zh" data-title="多項式" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%A4%9A%E9%A0%85%E5%BC%8F" title="多項式 – Klassisches Chinesisch" lang="lzh" hreflang="lzh" data-title="多項式" data-language-autonym="文言" data-language-local-name="Klassisches Chinesisch" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%A4%9A%E9%A0%85%E5%BC%8F" title="多項式 – Kantonesisch" lang="yue" hreflang="yue" data-title="多項式" data-language-autonym="粵語" data-language-local-name="Kantonesisch" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span 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