{"id":443,"date":"2018-08-27T12:00:20","date_gmt":"2018-08-27T10:00:20","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=443"},"modified":"2024-06-09T17:01:04","modified_gmt":"2024-06-09T15:01:04","slug":"equations-polynomiales-de-degre-3","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/27\/equations-polynomiales-de-degre-3\/","title":{"rendered":"\u00c9quations polynomiales de degr\u00e9 3"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Les \u00e9quations polynomiales de degr\u00e9 3 sont de la forme : \\[&nbsp; ax^3+bx^2+cx+d=0.\\qquad(1) \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ce dont nous pouvons \u00eatre assur\u00e9.e.s, c&rsquo;est qu&rsquo;elle admet au moins une solution r\u00e9elle. En effet, la fonction :&nbsp;\\[ f(x)=ax^3+bx^2+cx+d\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">est continue sur \\(\\mathbb{R}\\) et, de plus,&nbsp;\\[&nbsp;\\left\\{&nbsp;\\begin{array}{l}&nbsp;\\lim\\limits_{x\\to-\\infty} f(x)=\\lim\\limits_{x\\to-\\infty} (ax^3)=\\text{sgn}(-a)\\infty\\\\\\lim\\limits_{x\\to+\\infty} f(x)=\\lim\\limits_{x\\to+\\infty} (ax^3)=\\text{sgn}(a)\\infty\\end{array}\\right.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">o\u00f9 \\(\\text{sgn}(a)\\) d\u00e9signe le signe de <em>a<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi, d&rsquo;apr\u00e8s le th\u00e9or\u00e8me des valeurs interm\u00e9diaires, l&rsquo;\u00e9quation \\(f(x)=0\\) admet au moins une solution sur \\(\\mathbb{R}\\).<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/27\/equations-polynomiales-de-degre-3\/#Equations_polynomiales_de_degre_3_transformation_de_Tschirnhaus\" >\u00c9quations polynomiales de degr\u00e9 3: transformation de Tschirnhaus<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/27\/equations-polynomiales-de-degre-3\/#Equations_polynomiales_de_degre_3_methode_de_Hudde\" >\u00c9quations polynomiales de degr\u00e9 3: m\u00e9thode de Hudde<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/27\/equations-polynomiales-de-degre-3\/#Equations_polynomiales_de_degre_3_formule_de_Cardan-Tartaglia\" >\u00c9quations polynomiales de degr\u00e9 3: formule de Cardan-Tartaglia<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/27\/equations-polynomiales-de-degre-3\/#Equations_polynomiales_de_degre_3_un_exemple_au_hasard_ou_presque_%E2%80%A6\" >\u00c9quations polynomiales de degr\u00e9 3: un exemple au hasard (ou presque &#8230;)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/27\/equations-polynomiales-de-degre-3\/#Une_etude_de_Bombelli\" >Une \u00e9tude de Bombelli<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/08\/27\/equations-polynomiales-de-degre-3\/#Algorithmique\" >Algorithmique<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Equations_polynomiales_de_degre_3_transformation_de_Tschirnhaus\"><\/span>\u00c9quations polynomiales de degr\u00e9 3: transformation de Tschirnhaus<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Dans un premier temps, nous allons transformer l&rsquo;\u00e9quation (1) : \\[&nbsp;(1)&nbsp; \\iff x^3+\\frac{b}{a}x^2+\\frac{c}{a}x+\\frac{d}{a}=0. \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On pose alors \\(x=X-\\displaystyle\\frac{b}{3a}\\) : \\[ \\begin{align*} &amp; (1)\\\\ \\iff &amp; \\left(X-\\frac{b}{3a}\\right)^3+\\frac{b}{a}\\left(X-\\frac{b}{3a}\\right)^2+\\frac{c}{a}\\left(X-\\frac{b}{3a}\\right)+\\frac{d}{a}=0\\\\ \\iff &amp; X^3-\\frac{b}{a}X^2+\\frac{b^2}{3a^2}X-\\frac{b^3}{27a^3}+\\frac{b}{a}\\left(X^2-\\frac{2b}{3a}X+\\frac{b^2}{9a^2}\\right)+\\frac{c}{a}X\\\\ &amp; -\\frac{bc}{3a^2}+\\frac{d}{a}=0\\\\ \\iff &amp;&nbsp; X^3+\\left(\\frac{b^2}{3a^2}-\\frac{2b^2}{3a^2}+\\frac{c}{a}\\right)X-\\frac{b^3}{27a^3}+\\frac{b^3}{9a^3}-\\frac{bc}{3a^2}+\\frac{d}{a}=0\\\\ \\iff &amp; X^3+\\left(\\frac{3ac-b^2}{3a^2}\\right)X+\\frac{2b^3-9abc+27a^2d}{27a^3}=0\\end{align*} \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On peut donc simplifier l&rsquo;\u00e9quation (1) en : \\[&nbsp;X^3+pX+q=0\\qquad (2) \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">avec :&nbsp;\\[&nbsp;\\left\\{\\begin{array}{l}p=\\frac{3ac-b^2}{3a^2}\\\\[10pt]q=\\frac{2b^3-9abc+27a^2d}{27a^3}&nbsp;\\end{array}\\right.\\]<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Equations_polynomiales_de_degre_3_methode_de_Hudde\"><\/span>\u00c9quations polynomiales de degr\u00e9 3: m\u00e9thode de Hudde<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n<div class=\"wp-block-image is-style-default\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/7\/7b\/Hudde.jpg\" alt=\"Hudde \u00e9quations polynomiales de degr\u00e9 3\" style=\"width:313px;height:364px\"\/><figcaption class=\"wp-element-caption\">Johan Hudde, math\u00e9maticien n\u00e9erlandais<\/figcaption><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">La m\u00e9thode de Hudde consiste \u00e0 poser dans l&rsquo;\u00e9quation (2) :&nbsp;\\[ X = u+v \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">en remarquant l&rsquo;\u00e9galit\u00e9 suivante :&nbsp;\\[ (u+v)^3=u^3+3u^2v+3uv^2+v^3.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On a alors :&nbsp;\\[ (u+v)^3-u^3-3u^2v-3uv^2-v^3=0.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On en d\u00e9duit alors :&nbsp;\\[ (u+v)^3-3uv(u+v)-(u^3+v^3)=0,\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">soit : \\[&nbsp;X^3-3uvX-(u^3+v^3)=0.\\qquad (3) \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pour que les \u00e9quations (2) et (3) soient semblables, on pose alors :&nbsp;\\[\\left\\{\\begin{array}{l}p=-3uv\\\\q=-(u^3+v^3)\\end{array}\\right.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">soit :&nbsp;\\[\\left\\{\\begin{array}{l}uv=-\\frac{p}{3}\\\\u^3+v^3=-q\\end{array}\\right.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">ou encore :&nbsp;\\[\\left\\{\\begin{array}{l}u^3v^3=-\\frac{p^3}{27}\\\\u^3+v^3=-q\\end{array}\\right.\\]<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Equations_polynomiales_de_degre_3_formule_de_Cardan-Tartaglia\"><\/span>\u00c9quations polynomiales de degr\u00e9 3: formule de Cardan-Tartaglia<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-7387b849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img decoding=\"async\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/0\/03\/Cardano.jpg\" alt=\"jerome cardan \u00e9quations polynomiales de degr\u00e9 3\"\/><figcaption class=\"wp-element-caption\">J\u00e9r\u00f4me Cardan<\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/0\/0b\/Niccol%C3%B2_Tartaglia.jpg\" alt=\"tartaglia \u00e9quations polynomiales de degr\u00e9 3\" style=\"width:283px;height:330px\"\/><figcaption class=\"wp-element-caption\">Nicolas Tartaglia<\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">En posant \\(U=u^3\\) et \\(V=v^3\\), on s&rsquo;aper\u00e7oit que l&rsquo;on doit chercher deux nombres <em>U<\/em>&nbsp;et <em>V<\/em>&nbsp;connaissant leur somme <em>S<\/em>&nbsp;et leur produit <em>P<\/em>. Ainsi, <em>U<\/em>&nbsp;et <em>V<\/em>&nbsp;sont solutions de l&rsquo;\u00e9quation :&nbsp;\\[ Y^2-SY+P=0,\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">soit : \\[&nbsp;Y^2+qY-\\frac{p^3}{27}=0.\\qquad (4) \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Le discriminant du polyn\u00f4me \\(Y^2+qY-\\frac{p^3}{27}\\) est :&nbsp;\\[ \\Delta = q^2+\\frac{4p^3}{27}.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pour que l&rsquo;\u00e9quation (4) ait au moins une solution, il faut que \\(\\Delta\\geq 0\\), soit :&nbsp;\\[ 27q^2+4p^3 \\geq 0.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sous cette derni\u00e8re condition, on a :&nbsp;\\[ U=\\frac{-q-\\sqrt{\\frac{27q^2+4p^3}{27}}}{2}\\qquad\\text{et}\\qquad V=\\frac{-q+\\sqrt{\\frac{27q^2+4p^3}{27}}}{2}.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Et donc :&nbsp;\\[&nbsp;u=\\sqrt[3]{\\frac{-q-\\sqrt{\\frac{27q^2+4p^3}{27}}}{2}}\\qquad\\text{et}\\qquad v=\\sqrt[3]{\\frac{-q+\\sqrt{\\frac{27q^2+4p^3}{27}}}{2}}.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi, une solution \u00e0 l&rsquo;\u00e9quation (2) est :&nbsp;\\[ X=u+v=\\sqrt[3]{\\frac{-q-\\sqrt{\\frac{27q^2+4p^3}{27}}}{2}}+\\sqrt[3]{\\frac{-q+\\sqrt{\\frac{27q^2+4p^3}{27}}}{2}}.\\]<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Equations_polynomiales_de_degre_3_un_exemple_au_hasard_ou_presque_%E2%80%A6\"><\/span>\u00c9quations polynomiales de degr\u00e9 3: un exemple au hasard (ou presque &#8230;)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Consid\u00e9rons l&rsquo;\u00e9quation :&nbsp;\\[ (E)\\ :\\ 2x^3-5x^2+4x-21=0.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On pose alors :&nbsp;\\[ \\left\\{\\begin{array}{l}p=\\frac{3\\times2\\times4-(-5)^2}{3\\times2^2}=\\frac{-1}{12}\\\\q=\\frac{2\\times(-5)^3-9\\times2\\times(-5)\\times4+27\\times2^2\\times(-21)}{27\\times2^3}=-\\frac{2158}{216}\\end{array}\\right.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi :&nbsp;\\[ (E)\\Leftrightarrow X^3-\\frac{1}{12}X-\\frac{1079}{108}=0.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On a alors :&nbsp;\\[\\Delta=\\sqrt{\\frac{2695}{27}}\\approx 9,99073645.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Et donc : \\[\\begin{align*}u \\approx 0,012897285\\\\v \\approx 2,153769382\\end{align*}\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">soit :&nbsp;\\[ X=u=v\\approx 2,166666667\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">et donc, finalement :&nbsp;\\[ x=X+\\frac{5}{6}=3.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Si on remplace&nbsp;<em>x<\/em> par 3 dans (E), on v\u00e9rifie bien que <em>x&nbsp;<\/em>= 3 est une de ses solutions.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_etude_de_Bombelli\"><\/span>Une \u00e9tude de Bombelli<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img decoding=\"async\" src=\"https:\/\/3.bp.blogspot.com\/-2HzZjvNGuzE\/XERDvhl8ivI\/AAAAAAAAMZk\/yWTC5ThNFtUfrofxprJWJEN3zV2pSl6fgCLcBGAs\/s1600\/GltQizxi_400x400.jpeg\" alt=\"bombelli \u00e9quations polynomiales de degr\u00e9 3\"\/><figcaption class=\"wp-element-caption\">Rapha\u00ebl Bombelli<\/figcaption><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Bombelli a \u00e9tudi\u00e9 l&rsquo;\u00e9quation :&nbsp;\\[ X^3-15X-4=0.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Il a obtenu :&nbsp;\\[ \\Delta = -13608\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">tt donc :&nbsp;\\[ X=\\sqrt[3]{2-\\sqrt{-121}}+\\sqrt[3]{2+\\sqrt{-121}}\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">que l&rsquo;on peut aussi \u00e9crire, en faisant fi du fait que l&rsquo;on ait un radicant n\u00e9gatif pour la racine carr\u00e9e :&nbsp;\\[ X=\\sqrt[3]{2-11\\sqrt{-1}}+\\sqrt[3]{2+11\\sqrt{-1}}.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Bombelli s&rsquo;est alors aper\u00e7u que : \\[&nbsp;\\sqrt[3]{2-11\\sqrt{-1}}=2-\\sqrt{-1}\\qquad\\text{et}\\qquad \\sqrt[3]{2+11\\sqrt{-1}}=2+\\sqrt{-1}.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi :&nbsp;\\[ X=2-\\sqrt{-1}+2+\\sqrt{-1}=4.\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi, en concevant que \\(\\sqrt{-1}\\) existe, il s&rsquo;aper\u00e7ut que cela ne g\u00eanait pas les calculs, qui menaient tout de m\u00eame \u00e0 une solution.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">C&rsquo;est ainsi que les <em>nombres complexes<\/em>&nbsp;firent leur apparition, nombres s&rsquo;\u00e9crivant sous la forme \\(a+b\\sqrt{-1}\\)&nbsp;ou, en posant \\(\\text{i}=\\sqrt{-1}\\), \\(a+\\text{i}b\\).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Algorithmique\"><\/span>Algorithmique<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Comme vous l&rsquo;avez constat\u00e9, les calculs peuvent \u00eatre assez fastidieux. Nous pouvons donc avoir recours \u00e0 un algorithme pour d\u00e9terminer une solution. Je vous en propose un dans le fichier PDF. Cet algorithme (sous Algobox) dans le fichier zipp\u00e9.<\/p>\n\n\n\n<div class=\"wp-block-file aligncenter um_article\"><a id=\"wp-block-file--media-c970ce98-54a6-45b3-be37-5274eeae4fb8\"  href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/08\/Equations-polynomiales-de-degre-3.pdf\" data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">Document PDF<\/a><a  href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/08\/Equations-polynomiales-de-degre-3.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-c970ce98-54a6-45b3-be37-5274eeae4fb8\" data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">T\u00e9l\u00e9charger<\/a><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Les sources \\(\\LaTeX\\) du document PDF :<\/p>\n\n\n\n<div class=\"wp-block-file aligncenter um_article\"><a id=\"wp-block-file--media-429bbcc4-7e8f-41fc-861c-17d3a9814b0e\" href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/08\/Equations-polynomiales-de-degr\u00e9-3.zip\">Source LaTeX<\/a><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/08\/Equations-polynomiales-de-degr\u00e9-3.zip\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-429bbcc4-7e8f-41fc-861c-17d3a9814b0e\">T\u00e9l\u00e9charger<\/a><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Une fonction Python pour trouver une solution (dans le cas o\u00f9 le discriminant est positif ou nul) est:<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"false\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">def resol(a,b,c,d):\n    p = (3*a*c - b*b) \/ (3*a*a)\n    q = (2*(b**3)-9*a*b*c + 27*a*a*d)\/(27*(a**3))\n    delta = (27*q*q + 4*(p**3))\/27\n    if delta >= 0:\n        u = (-q - delta**0.5)\/2\n        v = (-q + delta**0.5)\/2\n        s = u\/2\n        for _ in range(50):\n            s = ( 2*s + u\/(s*s) )\/3\n        u = s\n        s = v\/2\n        for _ in range(50):\n            s = ( 2*s + v\/(s*s) )\/3\n        v = s\n        s = u + v - b\/(3*a)\n        \n        return s\n    else:\n        return \"Le recours aux nombres complexes s'impose.\"<\/pre>\n\n\n\n<pre class=\"wp-block-code\"><code>&gt;&gt;&gt; resol(2,-5,4,-21)\n2.9999999999992792<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">On retrouve (presque) la solution pr\u00e9c\u00e9demment trouv\u00e9e, qui \u00e9tait \u00ab\u00a03\u00a0\u00bb.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Les \u00e9quations polynomiales de degr\u00e9 3 sont de la forme : \\[&nbsp; ax^3+bx^2+cx+d=0.\\qquad(1) \\] Ce dont nous pouvons \u00eatre assur\u00e9.e.s, c&rsquo;est qu&rsquo;elle admet au moins une solution r\u00e9elle. En effet, la fonction :&nbsp;\\[ f(x)=ax^3+bx^2+cx+d\\] est continue sur \\(\\mathbb{R}\\) et, de plus,&nbsp;\\[&nbsp;\\left\\{&nbsp;\\begin{array}{l}&nbsp;\\lim\\limits_{x\\to-\\infty} f(x)=\\lim\\limits_{x\\to-\\infty} (ax^3)=\\text{sgn}(-a)\\infty\\\\\\lim\\limits_{x\\to+\\infty} f(x)=\\lim\\limits_{x\\to+\\infty} (ax^3)=\\text{sgn}(a)\\infty\\end{array}\\right.\\] o\u00f9 \\(\\text{sgn}(a)\\) d\u00e9signe le signe [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":7677,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[31,32,34,33],"class_list":["post-443","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","tag-bombelli","tag-cardan","tag-equations","tag-tartaglia"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>\u00c9quations polynomiales de degr\u00e9 3 - 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