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.ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-No_footnotes plainlinks metadata ambox ambox-style ambox-No_footnotes" role="presentation"><tbody><tr><td class="mbox-text"><div class="mbox-text-span">This article includes a <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">list of references</a>, <a href="/wiki/Wikipedia:Further_reading" title="Wikipedia:Further reading">related reading</a>, or <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a>, <b>but its sources remain unclear because it lacks <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">June 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>algebraic equation</b> or <b>polynomial equation</b> is an <a href="/wiki/Equation" title="Equation">equation</a> of the 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 P=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6f743f37b37ce0c2ddc1db0fdca0e577c19f51d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle P=0}"></span>, where <i>P</i> is a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> with <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> in some <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, often the field of the <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>. For example, <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^{5}-3x+1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{5}-3x+1=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/698e1afb9a1d47e492390b6a5a4612ea0dfff0cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.98ex; height:2.843ex;" alt="{\displaystyle x^{5}-3x+1=0}"></span> is an algebraic equation with <a href="/wiki/Integer" title="Integer">integer</a> coefficients and </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 y^{4}+{\frac {xy}{2}}-{\frac {x^{3}}{3}}+xy^{2}+y^{2}+{\frac {1}{7}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{4}+{\frac {xy}{2}}-{\frac {x^{3}}{3}}+xy^{2}+y^{2}+{\frac {1}{7}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cebca40bd954fec7e7a5e4bf25de98273e8451bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.977ex; height:5.843ex;" alt="{\displaystyle y^{4}+{\frac {xy}{2}}-{\frac {x^{3}}{3}}+xy^{2}+y^{2}+{\frac {1}{7}}=0}"></span></dd></dl> <p>is a <a href="/wiki/Multivariate_polynomial" class="mw-redirect" title="Multivariate polynomial">multivariate polynomial</a> equation over the rationals. For many authors, the term <i>algebraic equation</i> refers only to the <a href="/wiki/Univariate" title="Univariate">univariate</a> case, that is polynomial equations that involve only one <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a>. On the other hand, a polynomial equation may involve several variables (the <i>multivariate</i> case), in which case the term <i>polynomial equation</i> is usually preferred. </p><p>Some but not all polynomial equations with <a href="/wiki/Rational_number" title="Rational number">rational</a> coefficients have a solution that is an <a href="/wiki/Algebraic_expression" title="Algebraic expression">algebraic expression</a> that can be found using a finite number of operations that involve only those same types of coefficients (that is, can be <a href="/wiki/Algebraic_solution" class="mw-redirect" title="Algebraic solution">solved algebraically</a>). This can be done for all such equations of <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> one, two, three, or four; but for degree five or more it can only be done for some equations, <a href="/wiki/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">not all</a>. A large amount of research has been devoted to compute efficiently accurate approximations of the <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex</a> solutions of a univariate algebraic equation (see <a href="/wiki/Root-finding_algorithm" title="Root-finding algorithm">Root-finding algorithm</a>) and of the common solutions of several multivariate polynomial equations (see <a href="/wiki/System_of_polynomial_equations" title="System of polynomial equations">System of polynomial equations</a>). </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="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Terminology"><span class="tocnumber">1</span> <span class="toctext">Terminology</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#History"><span class="tocnumber">2</span> <span class="toctext">History</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Areas_of_study"><span class="tocnumber">3</span> <span class="toctext">Areas of study</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Theory"><span class="tocnumber">4</span> <span class="toctext">Theory</span></a> <ul> <li class="toclevel-2 tocsection-5"><a href="#Polynomials"><span class="tocnumber">4.1</span> <span class="toctext">Polynomials</span></a></li> <li class="toclevel-2 tocsection-6"><a href="#Existence_of_solutions_to_real_and_complex_equations"><span class="tocnumber">4.2</span> <span class="toctext">Existence of solutions to real and complex equations</span></a></li> <li class="toclevel-2 tocsection-7"><a href="#Connection_to_Galois_theory"><span class="tocnumber">4.3</span> <span class="toctext">Connection to Galois theory</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-8"><a href="#Explicit_solution_of_numerical_equations"><span class="tocnumber">5</span> <span class="toctext">Explicit solution of numerical equations</span></a> <ul> <li class="toclevel-2 tocsection-9"><a href="#Approach"><span class="tocnumber">5.1</span> <span class="toctext">Approach</span></a></li> <li class="toclevel-2 tocsection-10"><a href="#General_techniques"><span class="tocnumber">5.2</span> <span class="toctext">General techniques</span></a> <ul> <li class="toclevel-3 tocsection-11"><a href="#Factoring"><span class="tocnumber">5.2.1</span> <span class="toctext">Factoring</span></a></li> <li class="toclevel-3 tocsection-12"><a href="#Elimination_of_the_sub-dominant_term"><span class="tocnumber">5.2.2</span> <span class="toctext">Elimination of the sub-dominant term</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-13"><a href="#Quadratic_equations"><span class="tocnumber">5.3</span> <span class="toctext">Quadratic equations</span></a></li> <li class="toclevel-2 tocsection-14"><a href="#Cubic_equations"><span class="tocnumber">5.4</span> <span class="toctext">Cubic equations</span></a></li> <li class="toclevel-2 tocsection-15"><a href="#Quartic_equations"><span class="tocnumber">5.5</span> <span class="toctext">Quartic equations</span></a></li> <li class="toclevel-2 tocsection-16"><a href="#Higher-degree_equations"><span class="tocnumber">5.6</span> <span class="toctext">Higher-degree equations</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-17"><a href="#See_also"><span class="tocnumber">6</span> <span class="toctext">See also</span></a></li> <li class="toclevel-1 tocsection-18"><a href="#References"><span class="tocnumber">7</span> <span class="toctext">References</span></a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Terminology">Terminology</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=1" title="Edit section: Terminology" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>The term "algebraic equation" dates from the time when the main problem of <a href="/wiki/Algebra" title="Algebra">algebra</a> was to solve <a href="/wiki/Univariate" title="Univariate">univariate</a> polynomial equations. This problem was completely solved during the 19th century; see <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">Fundamental theorem of algebra</a>, <a href="/wiki/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a> and <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a>. </p><p>Since then, the scope of algebra has been dramatically enlarged. In particular, it includes the study of equations that involve <a href="/wiki/Nth_root" title="Nth root"><span class="texhtml mvar" style="font-style:italic;">n</span>th roots</a> and, more generally, <a href="/wiki/Algebraic_expression" title="Algebraic expression">algebraic expressions</a>. This makes the term <i>algebraic equation</i> ambiguous outside the context of the old problem. So the term <i>polynomial equation</i> is generally preferred when this ambiguity may occur, specially when considering multivariate equations. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="History">History</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=2" title="Edit section: History" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>The study of algebraic equations is probably as old as mathematics: the <a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylonian mathematicians</a>, as early as 2000 BC could solve some kinds of <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic equations</a> (displayed on <a href="/wiki/First_Babylonian_dynasty" class="mw-redirect" title="First Babylonian dynasty">Old Babylonian</a> <a href="/wiki/Clay_tablet" title="Clay tablet">clay tablets</a>). </p><p>Univariate algebraic equations over the rationals (i.e., with <a href="/wiki/Rational_number" title="Rational number">rational</a> coefficients) have a very long history. Ancient mathematicians wanted the solutions in the form of <a href="/wiki/Radical_expression" class="mw-redirect" title="Radical expression">radical expressions</a>, like <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={\frac {1+{\sqrt {5}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {1+{\sqrt {5}}}{2}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ed0a977804a1b2bc08ad4ea7db4e4f4bd2e915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.365ex; height:5.843ex;" alt="{\displaystyle x={\frac {1+{\sqrt {5}}}{2}}}"></noscript><span class="lazy-image-placeholder" style="width: 12.365ex;height: 5.843ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ed0a977804a1b2bc08ad4ea7db4e4f4bd2e915" data-alt="{\displaystyle x={\frac {1+{\sqrt {5}}}{2}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> for the positive solution of <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-1=0}"> <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> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-x-1=0}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e22ea5278827fbb7e09fb5fbeb5f50b234410f84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.818ex; height:2.843ex;" alt="{\displaystyle x^{2}-x-1=0}"></noscript><span class="lazy-image-placeholder" style="width: 14.818ex;height: 2.843ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e22ea5278827fbb7e09fb5fbeb5f50b234410f84" data-alt="{\displaystyle x^{2}-x-1=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. The ancient Egyptians knew how to solve equations of degree 2 in this manner. The Indian mathematician Brahmagupta (597–668 AD) explicitly described the quadratic formula in his treatise Brāhmasphuṭasiddhānta published in 628 AD, but written in words instead of symbols. In the 9th century <a href="/wiki/Muhammad_ibn_Musa_al-Khwarizmi" class="mw-redirect" title="Muhammad ibn Musa al-Khwarizmi">Muhammad ibn Musa al-Khwarizmi</a> and other Islamic mathematicians derived the <a href="/wiki/Quadratic_formula" title="Quadratic formula">quadratic formula</a>, the general solution of equations of degree 2, and recognized the importance of the <a href="/wiki/Discriminant" title="Discriminant">discriminant</a>. During the Renaissance in 1545, <a href="/wiki/Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a> published the solution of <a href="/wiki/Scipione_del_Ferro" title="Scipione del Ferro">Scipione del Ferro</a> and <a href="/wiki/Niccol%C3%B2_Fontana_Tartaglia" class="mw-redirect" title="Niccolò Fontana Tartaglia">Niccolò Fontana Tartaglia</a> to <a href="/wiki/Cubic_function" title="Cubic function">equations of degree 3</a> and that of <a href="/wiki/Lodovico_Ferrari" title="Lodovico Ferrari">Lodovico Ferrari</a> for <a href="/wiki/Quartic_function" title="Quartic function">equations of degree 4</a>. Finally <a href="/wiki/Niels_Henrik_Abel" title="Niels Henrik Abel">Niels Henrik Abel</a> proved, in 1824, that <a href="/wiki/Quintic_equation" class="mw-redirect" title="Quintic equation">equations of degree 5</a> and higher do not have general solutions using radicals. <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a>, named after <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a>, showed that some equations of at least degree 5 do not even have an idiosyncratic solution in radicals, and gave criteria for deciding if an equation is in fact solvable using radicals. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Areas_of_study">Areas of study</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=3" title="Edit section: Areas of study" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <p>The algebraic equations are the basis of a number of areas of modern mathematics: <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a> is the study of (univariate) algebraic equations over the rationals (that is, with <a href="/wiki/Rational_number" title="Rational number">rational</a> coefficients). <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a> was introduced by <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a> to specify criteria for deciding if an algebraic equation may be solved in terms of radicals. In <a href="/wiki/Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">field theory</a>, an <a href="/wiki/Algebraic_extension" title="Algebraic extension">algebraic extension</a> is an extension such that every element is a root of an algebraic equation over the base field. <a href="/wiki/Transcendental_number_theory" title="Transcendental number theory">Transcendental number theory</a> is the study of the real numbers which are not solutions to an algebraic equation over the rationals. A <a href="/wiki/Diophantine_equation" title="Diophantine equation">Diophantine equation</a> is a (usually multivariate) polynomial equation with integer coefficients for which one is interested in the integer solutions. <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic geometry</a> is the study of the solutions in an <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a> of multivariate polynomial equations. </p><p>Two equations are equivalent if they have the same set of <a href="/wiki/Equation" title="Equation">solutions</a>. In particular the equation <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=Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=Q}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2abc7e2c5a78e9e6cb7a2a907279953f9b4a3f52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.682ex; height:2.509ex;" alt="{\displaystyle P=Q}"></noscript><span class="lazy-image-placeholder" style="width: 6.682ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2abc7e2c5a78e9e6cb7a2a907279953f9b4a3f52" data-alt="{\displaystyle P=Q}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is equivalent to <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-Q=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>−</mo> <mi>Q</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P-Q=0}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a81eeed7e3e69684ff90fe58d5343a6f434665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.685ex; height:2.509ex;" alt="{\displaystyle P-Q=0}"></noscript><span class="lazy-image-placeholder" style="width: 10.685ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a81eeed7e3e69684ff90fe58d5343a6f434665" data-alt="{\displaystyle P-Q=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. It follows that the study of algebraic equations is equivalent to the study of polynomials. </p><p>A polynomial equation over the rationals can always be converted to an equivalent one in which the <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> are <a href="/wiki/Integer" title="Integer">integers</a>. For example, multiplying through by 42 = 2·3·7 and grouping its terms in the first member, the previously mentioned polynomial equation <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 y^{4}+{\frac {xy}{2}}={\frac {x^{3}}{3}}-xy^{2}+y^{2}-{\frac {1}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{4}+{\frac {xy}{2}}={\frac {x^{3}}{3}}-xy^{2}+y^{2}-{\frac {1}{7}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5e8a49bf6d100e7a8b65135f6faffd17e470ceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.974ex; height:5.843ex;" alt="{\displaystyle y^{4}+{\frac {xy}{2}}={\frac {x^{3}}{3}}-xy^{2}+y^{2}-{\frac {1}{7}}}"></noscript><span class="lazy-image-placeholder" style="width: 30.974ex;height: 5.843ex;vertical-align: -2.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5e8a49bf6d100e7a8b65135f6faffd17e470ceb" data-alt="{\displaystyle y^{4}+{\frac {xy}{2}}={\frac {x^{3}}{3}}-xy^{2}+y^{2}-{\frac {1}{7}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> becomes </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 42y^{4}+21xy-14x^{3}+42xy^{2}-42y^{2}+6=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>42</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>21</mn> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mn>14</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>42</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>42</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 42y^{4}+21xy-14x^{3}+42xy^{2}-42y^{2}+6=0.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45740b3d2be3fa91ee1446daa51caeca2ddd3818" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:44.74ex; height:3.009ex;" alt="{\displaystyle 42y^{4}+21xy-14x^{3}+42xy^{2}-42y^{2}+6=0.}"></noscript><span class="lazy-image-placeholder" style="width: 44.74ex;height: 3.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45740b3d2be3fa91ee1446daa51caeca2ddd3818" data-alt="{\displaystyle 42y^{4}+21xy-14x^{3}+42xy^{2}-42y^{2}+6=0.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>Because <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a>, <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a>, and 1/<i>T</i> are not polynomial functions, </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 e^{T}x^{2}+{\frac {1}{T}}xy+\sin(T)z-2=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mi>z</mi> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{T}x^{2}+{\frac {1}{T}}xy+\sin(T)z-2=0}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbc2aea972a10e959c9cff3a615277cd14c59ac8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.148ex; height:5.176ex;" alt="{\displaystyle e^{T}x^{2}+{\frac {1}{T}}xy+\sin(T)z-2=0}"></noscript><span class="lazy-image-placeholder" style="width: 31.148ex;height: 5.176ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbc2aea972a10e959c9cff3a615277cd14c59ac8" data-alt="{\displaystyle e^{T}x^{2}+{\frac {1}{T}}xy+\sin(T)z-2=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>is <i>not</i> a polynomial equation in the four variables <i>x</i>, <i>y</i>, <i>z</i>, and <i>T</i> over the rational numbers. However, it is a polynomial equation in the three variables <i>x</i>, <i>y</i>, and <i>z</i> over the field of the <a href="/wiki/Elementary_function" title="Elementary function">elementary functions</a> in the variable <i>T</i>. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Theory">Theory</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=4" title="Edit section: Theory" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <div class="mw-heading mw-heading3"><h3 id="Polynomials">Polynomials</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=5" title="Edit section: Polynomials" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Polynomial#Solving_polynomial_equations" title="Polynomial">Polynomial § Solving polynomial equations</a></div> <p>Given an equation in unknown <span class="texhtml">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 (\mathrm {E} )\qquad a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{1}x+a_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="2em"></mspace> <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> <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> <mo>⋯</mo> <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>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathrm {E} )\qquad a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{1}x+a_{0}=0}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c10f14fb76dcf89d012527aa427bf54abbd0c17c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.475ex; height:3.176ex;" alt="{\displaystyle (\mathrm {E} )\qquad a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{1}x+a_{0}=0}"></noscript><span class="lazy-image-placeholder" style="width: 46.475ex;height: 3.176ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c10f14fb76dcf89d012527aa427bf54abbd0c17c" data-alt="{\displaystyle (\mathrm {E} )\qquad a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{1}x+a_{0}=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>,</dd></dl> <p>with coefficients in a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <span class="texhtml mvar" style="font-style:italic;">K</span>, one can equivalently say that the solutions of (E) in <span class="texhtml mvar" style="font-style:italic;">K</span> are the roots in <span class="texhtml mvar" style="font-style:italic;">K</span> of the polynomial </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=a_{n}X^{n}+a_{n-1}X^{n-1}+\dots +a_{1}X+a_{0}\quad \in K[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <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> <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> <mo>⋯</mo> <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>0</mn> </mrow> </msub> <mspace width="1em"></mspace> <mo>∈</mo> <mi>K</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=a_{n}X^{n}+a_{n-1}X^{n-1}+\dots +a_{1}X+a_{0}\quad \in K[X]}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94ca967bd025982b4e757a91b0bc4e544f19e9d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.508ex; height:3.176ex;" alt="{\displaystyle P=a_{n}X^{n}+a_{n-1}X^{n-1}+\dots +a_{1}X+a_{0}\quad \in K[X]}"></noscript><span class="lazy-image-placeholder" style="width: 51.508ex;height: 3.176ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94ca967bd025982b4e757a91b0bc4e544f19e9d0" data-alt="{\displaystyle P=a_{n}X^{n}+a_{n-1}X^{n-1}+\dots +a_{1}X+a_{0}\quad \in K[X]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</dd></dl> <p>It can be shown that a polynomial of degree <span class="texhtml mvar" style="font-style:italic;">n</span> in a field has at most <span class="texhtml mvar" style="font-style:italic;">n</span> roots. The equation (E) therefore has at most <span class="texhtml mvar" style="font-style:italic;">n</span> solutions. </p><p>If <span class="texhtml mvar" style="font-style:italic;">K'</span> is a <a href="/wiki/Field_extension" title="Field extension">field extension</a> of <span class="texhtml mvar" style="font-style:italic;">K</span>, one may consider (E) to be an equation with coefficients in <span class="texhtml mvar" style="font-style:italic;">K</span> and the solutions of (E) in <span class="texhtml mvar" style="font-style:italic;">K</span> are also solutions in <span class="texhtml mvar" style="font-style:italic;">K'</span> (the converse does not hold in general). It is always possible to find a field extension of <span class="texhtml mvar" style="font-style:italic;">K</span> known as the <a href="/wiki/Rupture_field" title="Rupture field">rupture field</a> of the polynomial <span class="texhtml mvar" style="font-style:italic;">P</span>, in which (E) has at least one solution. </p> <div class="mw-heading mw-heading3"><h3 id="Existence_of_solutions_to_real_and_complex_equations">Existence of solutions to real and complex equations</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=6" title="Edit section: Existence of solutions to real and complex equations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>The <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a> states that the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> of the <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> is closed algebraically, that is, all polynomial equations with complex coefficients and degree at least one have a solution. </p><p>It follows that all polynomial equations of degree 1 or more with real coefficients have a <i>complex</i> solution. On the other hand, an equation such as <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}+1=0}"> <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> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+1=0}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e01c67127b28bb80e2102c934d0d01daa5c20a61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.648ex; height:2.843ex;" alt="{\displaystyle x^{2}+1=0}"></noscript><span class="lazy-image-placeholder" style="width: 10.648ex;height: 2.843ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e01c67127b28bb80e2102c934d0d01daa5c20a61" data-alt="{\displaystyle x^{2}+1=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> does not have a solution 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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></noscript><span class="lazy-image-placeholder" style="width: 1.678ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" data-alt="{\displaystyle \mathbb {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (the solutions are the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary units</a> <span class="texhtml">i</span> and <span class="texhtml">–i</span>). </p><p>While the real solutions of real equations are intuitive (they are the <span class="texhtml mvar" style="font-style:italic;">x</span>-coordinates of the points where the curve <span class="texhtml"><i>y</i> = <i>P</i>(<i>x</i>)</span> intersects the <span class="texhtml mvar" style="font-style:italic;">x</span>-axis), the existence of complex solutions to real equations can be surprising and less easy to visualize. </p><p>However, a <a href="/wiki/Monic_polynomial" title="Monic polynomial">monic polynomial</a> of <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">odd</a> degree must necessarily have a real root. The associated <a href="/wiki/Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial function</a> in <span class="texhtml mvar" style="font-style:italic;">x</span> is continuous, and it approaches <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><noscript><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 }"></noscript><span class="lazy-image-placeholder" style="width: 4.132ex;height: 2.176ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" data-alt="{\displaystyle -\infty }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> as <span class="texhtml mvar" style="font-style:italic;">x</span> approaches <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><noscript><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 }"></noscript><span class="lazy-image-placeholder" style="width: 4.132ex;height: 2.176ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" data-alt="{\displaystyle -\infty }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> and <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><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" 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 }"></noscript><span class="lazy-image-placeholder" style="width: 4.132ex;height: 2.176ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" data-alt="{\displaystyle +\infty }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> as <span class="texhtml mvar" style="font-style:italic;">x</span> approaches <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><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" 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 }"></noscript><span class="lazy-image-placeholder" style="width: 4.132ex;height: 2.176ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" data-alt="{\displaystyle +\infty }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. By the <a href="/wiki/Intermediate_value_theorem" title="Intermediate value theorem">intermediate value theorem</a>, it must therefore assume the value zero at some real <span class="texhtml mvar" style="font-style:italic;">x</span>, which is then a solution of the polynomial equation. </p> <div class="mw-heading mw-heading3"><h3 id="Connection_to_Galois_theory">Connection to Galois theory</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=7" title="Edit section: Connection to Galois theory" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>There exist formulas giving the solutions of real or complex polynomials of degree less than or equal to four as a function of their coefficients. <a href="/wiki/Niels_Henrik_Abel" title="Niels Henrik Abel">Abel</a> showed that it is not possible to find such a formula in general (using only the four arithmetic operations and taking roots) for equations of degree five or higher. <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a> provides a criterion which allows one to determine whether the solution to a given polynomial equation can be expressed using radicals. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Explicit_solution_of_numerical_equations">Explicit solution of numerical equations</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=8" title="Edit section: Explicit solution of numerical equations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <div class="mw-heading mw-heading3"><h3 id="Approach">Approach</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=9" title="Edit section: Approach" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>The explicit solution of a real or complex equation of degree 1 is trivial. <a href="/wiki/Equation_solving" title="Equation solving">Solving</a> an equation of higher degree <span class="texhtml mvar" style="font-style:italic;">n</span> reduces to factoring the associated polynomial, that is, rewriting (E) in the 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 a_{n}(x-z_{1})\dots (x-z_{n})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>…</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}(x-z_{1})\dots (x-z_{n})=0}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c0e26d3ab4b71fa915b3810a46c0b11796f89c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.6ex; height:2.843ex;" alt="{\displaystyle a_{n}(x-z_{1})\dots (x-z_{n})=0}"></noscript><span class="lazy-image-placeholder" style="width: 26.6ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c0e26d3ab4b71fa915b3810a46c0b11796f89c" data-alt="{\displaystyle a_{n}(x-z_{1})\dots (x-z_{n})=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>,</dd></dl> <p>where the solutions are then the <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 z_{1},\dots ,z_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{1},\dots ,z_{n}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a6f052e95c08f299bbd096baad0af0a0b29acb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.613ex; height:2.009ex;" alt="{\displaystyle z_{1},\dots ,z_{n}}"></noscript><span class="lazy-image-placeholder" style="width: 9.613ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a6f052e95c08f299bbd096baad0af0a0b29acb3" data-alt="{\displaystyle z_{1},\dots ,z_{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. The problem is then to express the <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 z_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{i}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6e920bac39ad09fff4efef16254595091a1025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.881ex; height:2.009ex;" alt="{\displaystyle z_{i}}"></noscript><span class="lazy-image-placeholder" style="width: 1.881ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6e920bac39ad09fff4efef16254595091a1025" data-alt="{\displaystyle z_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> in terms of the <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><noscript><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}}"></noscript><span class="lazy-image-placeholder" style="width: 2.029ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" data-alt="{\displaystyle a_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. </p><p>This approach applies more generally if the coefficients and solutions belong to an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a>. </p> <div class="mw-heading mw-heading3"><h3 id="General_techniques">General techniques</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=10" title="Edit section: General techniques" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <div class="mw-heading mw-heading4"><h4 id="Factoring">Factoring</h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=11" title="Edit section: Factoring" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>If an equation <span class="texhtml"><i>P</i>(<i>x</i>) = 0</span> of degree <span class="texhtml mvar" style="font-style:italic;">n</span> has a <a href="/wiki/Rational_root_theorem" title="Rational root theorem">rational root</a> <span class="texhtml">α</span>, the associated polynomial can be factored to give the form <span class="texhtml"><i>P</i>(<i>X</i>) = (<i>X</i> – α)<i>Q</i>(<i>X</i>)</span> (by <a href="/wiki/Polynomial_division" class="mw-redirect" title="Polynomial division">dividing</a> <span class="texhtml"><i>P</i>(<i>X</i>)</span> by <span class="texhtml"><i>X</i> – α</span> or by writing <span class="texhtml"><i>P</i>(<i>X</i>) – <i>P</i>(α)</span> as a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of terms of the form <span class="texhtml"><i>X<sup>k</sup></i> – α<sup><i>k</i></sup></span>, and factoring out <span class="texhtml"><i>X</i> – α</span>. Solving <span class="texhtml"><i>P</i>(<i>x</i>) = 0</span> thus reduces to solving the degree <span class="texhtml"><i>n</i> – 1</span> equation <span class="texhtml"><i>Q</i>(<i>x</i>) = 0</span>. See for example the <a href="/wiki/Cubic_function#Factorization" title="Cubic function">case <span class="texhtml"><i>n</i> = 3</span></a>. </p> <div class="mw-heading mw-heading4"><h4 id="Elimination_of_the_sub-dominant_term">Elimination of the sub-dominant term</h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=12" title="Edit section: Elimination of the sub-dominant term" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>To solve an equation of degree <span class="texhtml mvar" style="font-style:italic;">n</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 (\mathrm {E} )\qquad a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{1}x+a_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="2em"></mspace> <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> <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> <mo>⋯</mo> <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>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathrm {E} )\qquad a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{1}x+a_{0}=0}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c10f14fb76dcf89d012527aa427bf54abbd0c17c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.475ex; height:3.176ex;" alt="{\displaystyle (\mathrm {E} )\qquad a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{1}x+a_{0}=0}"></noscript><span class="lazy-image-placeholder" style="width: 46.475ex;height: 3.176ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c10f14fb76dcf89d012527aa427bf54abbd0c17c" data-alt="{\displaystyle (\mathrm {E} )\qquad a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{1}x+a_{0}=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>,</dd></dl> <p>a common preliminary step is to eliminate the degree-<span class="texhtml">n - 1</span> term: by setting <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=y-{\frac {a_{n-1}}{n\,a_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y-{\frac {a_{n-1}}{n\,a_{n}}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b25ea7fee1ee5e13de163453d69143ede6fc187" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.809ex; height:5.176ex;" alt="{\displaystyle x=y-{\frac {a_{n-1}}{n\,a_{n}}}}"></noscript><span class="lazy-image-placeholder" style="width: 13.809ex;height: 5.176ex;vertical-align: -2.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b25ea7fee1ee5e13de163453d69143ede6fc187" data-alt="{\displaystyle x=y-{\frac {a_{n-1}}{n\,a_{n}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, equation (E) becomes </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_{n}y^{n}+b_{n-2}y^{n-2}+\dots +b_{1}y+b_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}y^{n}+b_{n-2}y^{n-2}+\dots +b_{1}y+b_{0}=0}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbec294e60e73eb96630c1d83ac73df597a85d1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.228ex; height:3.009ex;" alt="{\displaystyle a_{n}y^{n}+b_{n-2}y^{n-2}+\dots +b_{1}y+b_{0}=0}"></noscript><span class="lazy-image-placeholder" style="width: 37.228ex;height: 3.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbec294e60e73eb96630c1d83ac73df597a85d1f" data-alt="{\displaystyle a_{n}y^{n}+b_{n-2}y^{n-2}+\dots +b_{1}y+b_{0}=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</dd></dl> <p><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> developed this technique for <a href="/wiki/Cubic_function#Cardano's_method" title="Cubic function">the case <span class="texhtml"><i>n</i> = 3</span></a> but it is also applicable to <a href="/wiki/Quartic_function#Euler's_solution" title="Quartic function">the case <span class="texhtml"><i>n</i> = 4</span></a>, for example. </p> <div class="mw-heading mw-heading3"><h3 id="Quadratic_equations">Quadratic equations</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=13" title="Edit section: Quadratic equations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quadratic_equation" title="Quadratic equation">Quadratic equation</a></div> <p>To solve a quadratic equation of the 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 ax^{2}+bx+c=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax^{2}+bx+c=0}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e70cfa003f402d108ec04d97983fb62f69536e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.89ex; height:2.843ex;" alt="{\displaystyle ax^{2}+bx+c=0}"></noscript><span class="lazy-image-placeholder" style="width: 16.89ex;height: 2.843ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e70cfa003f402d108ec04d97983fb62f69536e" data-alt="{\displaystyle ax^{2}+bx+c=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> one calculates the <i><a href="/wiki/Discriminant" title="Discriminant">discriminant</a></i> Δ defined by <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 \Delta =b^{2}-4ac}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =b^{2}-4ac}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8e9a3f8c6c4c62de207ffb8a59ea17c0584932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.326ex; height:2.843ex;" alt="{\displaystyle \Delta =b^{2}-4ac}"></noscript><span class="lazy-image-placeholder" style="width: 13.326ex;height: 2.843ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8e9a3f8c6c4c62de207ffb8a59ea17c0584932" data-alt="{\displaystyle \Delta =b^{2}-4ac}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. </p><p>If the polynomial has real coefficients, it has: </p> <ul><li>two distinct real roots if <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 \Delta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta >0}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5bba4bbddf69fb5d000a3d8a9daba0a36b5e720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \Delta >0}"></noscript><span class="lazy-image-placeholder" style="width: 6.197ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5bba4bbddf69fb5d000a3d8a9daba0a36b5e720" data-alt="{\displaystyle \Delta >0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> ;</li> <li>one real double root if <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 \Delta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =0}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf057da503668fa097746562ae91517330ce5b58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \Delta =0}"></noscript><span class="lazy-image-placeholder" style="width: 6.197ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf057da503668fa097746562ae91517330ce5b58" data-alt="{\displaystyle \Delta =0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> ;</li> <li>no real root if <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 \Delta <0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta <0}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc1b4ee97fd845583ecac5c0a2d151dfac284a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \Delta <0}"></noscript><span class="lazy-image-placeholder" style="width: 6.197ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc1b4ee97fd845583ecac5c0a2d151dfac284a3" data-alt="{\displaystyle \Delta <0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, but two complex conjugate roots.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Cubic_equations">Cubic equations</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=14" title="Edit section: Cubic equations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Cubic_equation" title="Cubic equation">Cubic equation</a></div> <p>The best-known method for solving cubic equations, by writing roots in terms of radicals, is <a href="/wiki/Cubic_equation#Cardano's_formula" title="Cubic equation">Cardano's formula</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Quartic_equations">Quartic equations</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=15" title="Edit section: Quartic equations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quartic_equation" title="Quartic equation">Quartic equation</a></div> <p>For detailed discussions of some solution methods see: </p> <ul><li><a href="/wiki/Tschirnhaus_transformation" title="Tschirnhaus transformation">Tschirnhaus transformation</a> (general method, not guaranteed to succeed);</li> <li><a href="/w/index.php?title=Bezout_method&action=edit&redlink=1" class="new" title="Bezout method (page does not exist)">Bezout method</a> (general method, not guaranteed to succeed);</li> <li><a href="/w/index.php?title=Ferrari_method&action=edit&redlink=1" class="new" title="Ferrari method (page does not exist)">Ferrari method</a> (solutions for degree 4);</li> <li><a href="/wiki/Euler_method" title="Euler method">Euler method</a> (solutions for degree 4);</li> <li><a href="/w/index.php?title=Lagrange_method&action=edit&redlink=1" class="new" title="Lagrange method (page does not exist)">Lagrange method</a> (solutions for degree 4);</li> <li><a href="/w/index.php?title=Descartes_method&action=edit&redlink=1" class="new" title="Descartes method (page does not exist)">Descartes method</a> (solutions for degree 2 or 4);</li></ul> <p>A quartic equation <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 ax^{4}+bx^{3}+cx^{2}+dx+e=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>e</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ccc13a4502686dfb9946be432f7ecaabb03fd90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:29.638ex; height:2.843ex;" alt="{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0}"></noscript><span class="lazy-image-placeholder" style="width: 29.638ex;height: 2.843ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ccc13a4502686dfb9946be432f7ecaabb03fd90" data-alt="{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> with <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\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f455a7f96d74aa94573d8e32da3b240ab0aa294f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.491ex; height:2.676ex;" alt="{\displaystyle a\neq 0}"></noscript><span class="lazy-image-placeholder" style="width: 5.491ex;height: 2.676ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f455a7f96d74aa94573d8e32da3b240ab0aa294f" data-alt="{\displaystyle a\neq 0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> may be reduced to a quadratic equation by a change of variable provided it is either <a href="/wiki/Quartic_function#Biquadratic_equation" title="Quartic function">biquadratic</a> (<span class="texhtml"><i>b = d</i> = 0</span>) or <a href="/wiki/Quartic_function#Quasi-palindromic_equation" title="Quartic function">quasi-palindromic</a> (<span class="texhtml"><i>e = a</i>, <i>d = b</i></span>). </p><p>Some cubic and quartic equations can be solved using <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a> or <a href="/wiki/Hyperbolic_functions" title="Hyperbolic functions">hyperbolic functions</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Higher-degree_equations">Higher-degree equations</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=16" title="Edit section: Higher-degree equations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a> and <a href="/wiki/Galois_group" title="Galois group">Galois group</a></div> <p><a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a> and <a href="/wiki/Niels_Henrik_Abel" title="Niels Henrik Abel">Niels Henrik Abel</a> showed independently that in general a polynomial of degree 5 or higher is not solvable using radicals. Some particular equations do have solutions, such as those associated with the <a href="/wiki/Cyclotomic_polynomials" class="mw-redirect" title="Cyclotomic polynomials">cyclotomic polynomials</a> of degrees 5 and 17. </p><p><a href="/wiki/Charles_Hermite" title="Charles Hermite">Charles Hermite</a>, on the other hand, showed that polynomials of degree 5 are solvable using <a href="/wiki/Elliptical_function" class="mw-redirect" title="Elliptical function">elliptical functions</a>. </p><p>Otherwise, one may find <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical approximations</a> to the roots using <a href="/wiki/Root-finding_algorithms" class="mw-redirect" title="Root-finding algorithms">root-finding algorithms</a>, such as <a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a>. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="See_also">See also</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=17" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <ul><li><a href="/wiki/Algebraic_function" title="Algebraic function">Algebraic function</a></li> <li><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic number</a></li> <li><a href="/wiki/Root_finding" class="mw-redirect" title="Root finding">Root finding</a></li> <li><a href="/wiki/Linear_equation" title="Linear equation">Linear equation</a> (degree = 1)</li> <li><a href="/wiki/Quadratic_equation" title="Quadratic equation">Quadratic equation</a> (degree = 2)</li> <li><a href="/wiki/Cubic_equation" title="Cubic equation">Cubic equation</a> (degree = 3)</li> <li><a href="/wiki/Quartic_equation" title="Quartic equation">Quartic equation</a> (degree = 4)</li> <li><a href="/wiki/Quintic_equation" class="mw-redirect" title="Quintic equation">Quintic equation</a> (degree = 5)</li> <li><a href="/wiki/Sextic_equation" title="Sextic equation">Sextic equation</a> (degree = 6)</li> <li><a href="/wiki/Septic_equation" title="Septic equation">Septic equation</a> (degree = 7)</li> <li><a href="/wiki/System_of_linear_equations" title="System of linear equations">System of linear equations</a></li> <li><a href="/wiki/System_of_polynomial_equations" title="System of polynomial equations">System of polynomial equations</a></li> <li><a href="/wiki/Linear_Diophantine_equation" class="mw-redirect" title="Linear Diophantine equation">Linear Diophantine equation</a></li> <li><a href="/wiki/Linear_equation_over_a_ring" title="Linear equation over a ring">Linear equation over a ring</a></li> <li><a href="/wiki/Cramer%27s_theorem_(algebraic_curves)" title="Cramer's theorem (algebraic curves)">Cramer's theorem (algebraic curves)</a>, on the number of points usually sufficient to determine a bivariate <i>n</i>-th degree curve</li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="References">References</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Algebraic_equation&action=edit&section=18" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-7 collapsible-block" id="mf-section-7"> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Algebraic_equation">"Algebraic equation"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Algebraic+equation&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DAlgebraic_equation&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+equation" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Algebraic_Equation"><link 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ekvacio – Esperanto" lang="eo" hreflang="eo" data-title="Algebra ekvacio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Ekuazio_aljebraiko" title="Ekuazio aljebraiko – Basque" lang="eu" hreflang="eu" data-title="Ekuazio aljebraiko" data-language-autonym="Euskara" data-language-local-name="Basque" 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/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D9%87_%D8%AC%D8%A8%D8%B1%DB%8C" title="معادله جبری – Persian" lang="fa" hreflang="fa" data-title="معادله جبری" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89quation_polynomiale" title="Équation polynomiale – French" lang="fr" hreflang="fr" data-title="Équation polynomiale" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Cothrom%C3%B3id_ailg%C3%A9abrach" title="Cothromóid ailgéabrach – Irish" lang="ga" hreflang="ga" data-title="Cothromóid ailgéabrach" data-language-autonym="Gaeilge" data-language-local-name="Irish" 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/Ecuaci%C3%B3n_alx%C3%A9brica" title="Ecuación alxébrica – Galician" lang="gl" hreflang="gl" data-title="Ecuación alxébrica" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8C%80%EC%88%98_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="대수 방정식 – Korean" lang="ko" hreflang="ko" data-title="대수 방정식" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%B6%D6%80%D5%A1%D5%B0%D5%A1%D5%B7%D5%BE%D5%A1%D5%AF%D5%A1%D5%B6_%D5%B0%D5%A1%D5%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%B4" title="Հանրահաշվական հավասարում – Armenian" lang="hy" hreflang="hy" data-title="Հանրահաշվական հավասարում" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A5%80%E0%A4%9C%E0%A5%80%E0%A4%AF_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3" title="बीजीय समीकरण – Hindi" lang="hi" hreflang="hi" data-title="बीजीय समीकरण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Algebral_equaciono" title="Algebral equaciono – Ido" lang="io" hreflang="io" data-title="Algebral equaciono" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Persamaan_aljabar" title="Persamaan aljabar – Indonesian" lang="id" hreflang="id" data-title="Persamaan aljabar" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Equazione_algebrica" title="Equazione algebrica – Italian" lang="it" hreflang="it" data-title="Equazione algebrica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AC%E0%B3%88%E0%B2%9C%E0%B2%BF%E0%B2%95_%E0%B2%B8%E0%B2%AE%E0%B3%80%E0%B2%95%E0%B2%B0%E0%B2%A3" title="ಬೈಜಿಕ ಸಮೀಕರಣ – Kannada" lang="kn" hreflang="kn" data-title="ಬೈಜಿಕ ಸಮೀಕರಣ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%BB%D1%8B%D0%BA_%D1%82%D0%B5%D2%A3%D0%B4%D0%B5%D0%BC%D0%B5" title="Алгебралык теңдеме – Kyrgyz" lang="ky" hreflang="ky" data-title="Алгебралык теңдеме" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Algebrin%C4%97_lygtis" title="Algebrinė lygtis – Lithuanian" lang="lt" hreflang="lt" data-title="Algebrinė lygtis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BB%A3%E6%95%B0%E6%96%B9%E7%A8%8B%E5%BC%8F" title="代数方程式 – Japanese" lang="ja" hreflang="ja" data-title="代数方程式" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Algebraisk_likning" title="Algebraisk likning – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Algebraisk likning" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Algebraik_tenglama" title="Algebraik tenglama – Uzbek" lang="uz" hreflang="uz" data-title="Algebraik tenglama" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Equassion_alg%C3%A9brica" title="Equassion algébrica – Piedmontese" lang="pms" hreflang="pms" data-title="Equassion algébrica" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/R%C3%B3wnanie_algebraiczne" title="Równanie algebraiczne – Polish" lang="pl" hreflang="pl" data-title="Równanie algebraiczne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Equa%C3%A7%C3%A3o_alg%C3%A9brica" title="Equação algébrica – Portuguese" lang="pt" hreflang="pt" data-title="Equação algébrica" data-language-autonym="Português" data-language-local-name="Portuguese" 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/Ecua%C8%9Bie_polinomial%C4%83" title="Ecuație polinomială – Romanian" lang="ro" hreflang="ro" data-title="Ecuație polinomială" data-language-autonym="Română" data-language-local-name="Romanian" 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%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5" title="Алгебраическое уравнение – Russian" lang="ru" hreflang="ru" data-title="Алгебраическое уравнение" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Ekuacionet_e_shkall%C3%ABs_s%C3%AB_p%C3%ABrgjithshme" title="Ekuacionet e shkallës së përgjithshme – Albanian" lang="sq" hreflang="sq" data-title="Ekuacionet e shkallës së përgjithshme" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Algebraic_equation" title="Algebraic equation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Algebraic equation" data-language-autonym="Simple English" data-language-local-name="Simple English" 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/Algebrick%C3%A1_rovnica" title="Algebrická rovnica – Slovak" lang="sk" hreflang="sk" data-title="Algebrická rovnica" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Algebraisk_ekvation" title="Algebraisk ekvation – Swedish" lang="sv" hreflang="sv" data-title="Algebraisk ekvation" data-language-autonym="Svenska" data-language-local-name="Swedish" 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%87%E0%AE%AF%E0%AE%B1%E0%AF%8D%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%9A%E0%AF%8D_%E0%AE%9A%E0%AE%AE%E0%AE%A9%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81" title="இயற்கணிதச் சமன்பாடு – Tamil" lang="ta" hreflang="ta" data-title="இயற்கணிதச் சமன்பாடு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B8%D1%87%D0%BD%D0%B5_%D1%80%D1%96%D0%B2%D0%BD%D1%8F%D0%BD%D0%BD%D1%8F" title="Алгебричне рівняння – Ukrainian" lang="uk" hreflang="uk" data-title="Алгебричне рівняння" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C6%B0%C6%A1ng_tr%C3%ACnh_%C4%91%E1%BA%A1i_s%E1%BB%91" title="Phương trình đại số – Vietnamese" lang="vi" hreflang="vi" data-title="Phương trình đại số" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BB%A3%E6%95%B8%E6%96%B9%E7%A8%8B" title="代數方程 – Cantonese" lang="yue" hreflang="yue" data-title="代數方程" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BB%A3%E6%95%B0%E6%96%B9%E7%A8%8B" title="代数方程 – Chinese" lang="zh" hreflang="zh" data-title="代数方程" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li></ul> </section> </div> <div class="minerva-footer-logo"><img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"/> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod"> This page was last edited on 22 February 2025, at 23:25<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Content is available under <a class="external" rel="nofollow" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">CC BY-SA 4.0</a> unless otherwise noted.</li> </ul> <ul id="footer-places" class="footer-places hlist hlist-separated"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li 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