@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-V91WMW66-Q
  skos:prefLabel "équation polynomiale"@fr, "polynomial equation"@en ;
  a skos:Concept ;
  skos:narrower psr:-LPBF743P-0 .

psr:-W7M3M400-M
  skos:prefLabel "cubic function"@en, "fonction cubique"@fr ;
  a skos:Concept ;
  skos:related psr:-LPBF743P-0 .

psr:-BDTFRKZ3-5
  skos:prefLabel "nombre plastique"@fr, "plastic number"@en ;
  a skos:Concept ;
  skos:related psr:-LPBF743P-0 .

psr: a skos:ConceptScheme .
psr:-W96QGKZX-0
  skos:prefLabel "elementary algebra"@en, "algèbre élémentaire"@fr ;
  a skos:Concept ;
  skos:narrower psr:-LPBF743P-0 .

psr:-LPBF743P-0
  skos:broader psr:-V91WMW66-Q, psr:-W96QGKZX-0 ;
  dc:created "2023-08-01"^^xsd:date ;
  a skos:Concept ;
  skos:prefLabel "équation cubique"@fr, "cubic equation"@en ;
  skos:related psr:-BDTFRKZ3-5, psr:-W7M3M400-M ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Cubic_equation>, <https://fr.wikipedia.org/wiki/%C3%89quation_cubique> ;
  skos:inScheme psr: ;
  skos:definition """En mathématiques, une <b>équation cubique</b> est une équation polynomiale de degré 3, de la forme <span class="texhtml"><i>ax</i><sup>3</sup> + <i>bx</i><sup>2</sup> + <i>cx </i>+ <i>d </i>= 0</span> avec <span class="texhtml mvar" style="font-style:italic;">a</span> non nul, où les coefficients <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, <span class="texhtml mvar" style="font-style:italic;">c</span> et <span class="texhtml mvar" style="font-style:italic;">d</span> sont en général supposés réels ou complexes.  
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/%C3%89quation_cubique">https://fr.wikipedia.org/wiki/%C3%89quation_cubique</a>)"""@fr, """In algebra, a <b>cubic equation</b> in one variable is an equation of the form  <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 ax^{3}+bx^{2}+cx+d=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>3</mn>           </mrow>         </msup>         <mo>+</mo>         <mi>b</mi>         <msup>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>+</mo>         <mi>c</mi>         <mi>x</mi>         <mo>+</mo>         <mi>d</mi>         <mo>=</mo>         <mn>0</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle ax^{3}+bx^{2}+cx+d=0}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c6a382654ad8c94dc3bfea84bf4a869ab5c68cb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.33ex; height:2.843ex;" alt="{\\\\displaystyle ax^{3}+bx^{2}+cx+d=0}"></span></dd></dl> in which <span class="texhtml mvar" style="font-style:italic;">a</span> is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the coefficients <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, <span class="texhtml mvar" style="font-style:italic;">c</span>, and <span class="texhtml mvar" style="font-style:italic;">d</span> of the cubic equation are real numbers, then it has at least one real root (this is true for all odd-degree polynomial functions). All of the roots of the cubic equation can be found by the following means:  
<br/>- algebraically: more precisely, they can be expressed by a <i>cubic formula</i> involving the four coefficients, the four basic arithmetic operations, square roots and cube roots. (This is also true of quadratic (second-degree) and quartic (fourth-degree) equations, but not for higher-degree equations, by the Abel–Ruffini theorem.)
<br/>- trigonometrically 
<br/>- numerical approximations of the roots can be found using root-finding algorithms such as Newton's method.
<br/> The coefficients do not need to be real numbers. Much of what is covered below is valid for coefficients in any field with characteristic other than 2 and 3. The solutions of the cubic equation do not necessarily belong to the same field as the coefficients. For example, some cubic equations with rational coefficients have roots that are irrational (and even non-real) complex numbers. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Cubic_equation">https://en.wikipedia.org/wiki/Cubic_equation</a>)"""@en .