{"id":6123,"date":"2021-05-18T15:50:16","date_gmt":"2021-05-18T13:50:16","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=6123"},"modified":"2025-05-30T17:25:47","modified_gmt":"2025-05-30T15:25:47","slug":"trinome-du-second-degre-et-python","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2021\/05\/18\/trinome-du-second-degre-et-python\/","title":{"rendered":"Trin\u00f4me du second degr\u00e9 et Python"},"content":{"rendered":"\n<p>Nous allons aborder ici la notion de trin\u00f4me du second degr\u00e9 en Python sous un angle POO (<a href=\"https:\/\/www.mathweb.fr\/euclide\/2019\/09\/06\/les-classes-en-python\/\">Programmation Orient\u00e9e Objet<\/a>). J&#8217;avais en effet envie de construire une classe Python repr\u00e9sentant un tel trin\u00f4me et pouvant effectuer toutes les op\u00e9rations classiques du lyc\u00e9e.<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 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\/2021\/05\/18\/trinome-du-second-degre-et-python\/#Trinome_du_second_degre_et_Python_preliminaires_mathematiques\" >Trin\u00f4me du second degr\u00e9 et Python: pr\u00e9liminaires math\u00e9matiques<\/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\/2021\/05\/18\/trinome-du-second-degre-et-python\/#Trinome_du_second_degre_et_Python_preliminaires_informatiques\" >Trin\u00f4me du second degr\u00e9 et Python: pr\u00e9liminaires informatiques<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2021\/05\/18\/trinome-du-second-degre-et-python\/#Python_et_giac_Xcas\" >Python et giac Xcas<\/a><\/li><\/ul><\/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\/2021\/05\/18\/trinome-du-second-degre-et-python\/#Trinome_du_second_degre_et_Python_la_classe\" >Trin\u00f4me du second degr\u00e9 et Python: la classe<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2021\/05\/18\/trinome-du-second-degre-et-python\/#Le_constructeur\" >Le constructeur<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2021\/05\/18\/trinome-du-second-degre-et-python\/#Les_methodes\" >Les m\u00e9thodes<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.mathweb.fr\/euclide\/2021\/05\/18\/trinome-du-second-degre-et-python\/#Exemples\" >Exemples<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.mathweb.fr\/euclide\/2021\/05\/18\/trinome-du-second-degre-et-python\/#Telechargement_de_la_classe\" >T\u00e9l\u00e9chargement de la classe<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Trinome_du_second_degre_et_Python_preliminaires_mathematiques\"><\/span>Trin\u00f4me du second degr\u00e9 et Python: pr\u00e9liminaires math\u00e9matiques<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Un trin\u00f4me du second degr\u00e9 est un polyn\u00f4me de la forme:$$P(x)=ax^2+bx+c,\\quad a\\neq0.$$<\/p>\n\n\n\n<p>\u00c0 ce polyn\u00f4me, on peut lui attribuer le nombre:$$\\Delta=b^2-4ac$$qui est son <em>discriminant<\/em>.<\/p>\n\n\n\n<p>Si \\(\\Delta &gt; 0\\) alors il existe deux valeurs distinctes \\(x_1\\) et \\(x_2\\) de <em>x<\/em> telles que P(<em>x<\/em>) = 0: ce sont les <em>racines<\/em> de P. Dans ce sas, P(<em>x<\/em>) peut se factoriser sous la forme \\( P(x) = a(x-x_1)(x-x_2) \\).<\/p>\n\n\n\n<p>Si \\(\\Delta = 0\\) alors il existe une unique valeur (x_0\\) pour laquelle \\(P(x_0) = 0\\): c&#8217;est sa <em>racine double<\/em>. Sans ce cas, \\( P(x)=a(x-x_0)^2\\).<\/p>\n\n\n\n<p>Si \\(\\Delta &lt; 0\\) alors l&#8217;\u00e9quation P(<em>x<\/em>) = 0 n&#8217;admet aucune racine r\u00e9elle, et P(<em>x<\/em>) ne se factorise pas dans l&#8217;ensemble des nombres r\u00e9els.<\/p>\n\n\n\n<p>\u00c0 l&#8217;aide de la forme factoris\u00e9e de P(<em>x<\/em>), si elle existe, on peut d\u00e9duire le signe de P(<em>x<\/em>).<\/p>\n\n\n\n<p>Je vous invite \u00e0 regarder la fiche de cours sur le second degr\u00e9 <a href=\"https:\/\/www.mathweb.fr\/euclide\/fiches-de-cours-de-mathematiques-classe-de-premiere-enseignement-de-specialite\/\" target=\"_blank\" rel=\"noreferrer noopener\">sur cette page<\/a>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Trinome_du_second_degre_et_Python_preliminaires_informatiques\"><\/span>Trin\u00f4me du second degr\u00e9 et Python: pr\u00e9liminaires informatiques<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>J&#8217;avais d&#8217;abord publi\u00e9 un article dans lequel j&#8217;exposais des m\u00e9thodes plut\u00f4t longues&#8230; Et puis, apr\u00e8s quelques recherches, j&#8217;ai d\u00e9couvert que je pouvais utiliser <a href=\"https:\/\/www-fourier.ujf-grenoble.fr\/~parisse\/giac_fr.html\" target=\"_blank\" rel=\"noreferrer noopener\">Xcas<\/a> avec Python ! Cela tombe bien car cela facilite beaucoup la construction des m\u00e9thodes.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Python_et_giac_Xcas\"><\/span>Python et giac Xcas<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>En ligne de commande, il faut avant tout installer les modules Python suivants. Sous windows:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>python.exe -m pip install wheel\npython.exe -m pip install giacpy<\/code><\/pre>\n\n\n\n<p>Pour les autres plateformes (MacOS et Linux), allez sur la page suivante pour plus d&#8217;informations:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><a href=\"https:\/\/www-fourier.ujf-grenoble.fr\/~parisse\/giac_fr.html\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/www-fourier.ujf-grenoble.fr\/~parisse\/giac_fr.html<\/a><\/p>\n\n\n\n<p>Maintenant, on peut importer les fonctions n\u00e9cessaires du module <em>giacpy<\/em>:<\/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=\"\">from giacpy import latex, solve, simplifier, evalf, factoriser<\/pre>\n\n\n\n<p>Ce qu&#8217;il y a d&#8217;int\u00e9ressant avec ce module, c&#8217;est que l&#8217;on va pouvoir effectuer des calculs alg\u00e9briques. Par exemple:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>&gt;&gt;&gt; solve('x*x-3*x+2=0')<\/code><\/pre>\n\n\n\n<p>est une liste contenant toutes les solution de l&#8217;\u00e9quation \\(x^2-3x+2=0\\).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Trinome_du_second_degre_et_Python_la_classe\"><\/span>Trin\u00f4me du second degr\u00e9 et Python: la classe<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_constructeur\"><\/span>Le constructeur<span class=\"ez-toc-section-end\"><\/span><\/h3>\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=\"\">class Trinome:\n    def __init__(self,a,b,c):\n        self.a = a\n        self.b = b\n        self.c = c\n        self.delta = self.b**2 - 4*self.a*self.c\n        self.giac_equation = '{}*x*x+{}*x+{}=0'.format(a,b,c)\n        self.giac_trinome = '{}*x*x+{}*x+{}'.format(a,b,c)<\/pre>\n\n\n\n<p>J&#8217;ai opt\u00e9 pour d\u00e9finir le discriminant d\u00e8s le constructeur car il ne m&#8217;a pas sembl\u00e9 utile d&#8217;\u00e9crire une m\u00e9thode rien que pour cela.<\/p>\n\n\n\n<p>Les variables <em>self.giac_&#8230;<\/em> me serviront pour trouver les racines et la factorisation.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Les_methodes\"><\/span>Les m\u00e9thodes<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>La plupart des m\u00e9thodes comporte un argument bool\u00e9en optionnel <em>tex<\/em> qui permet un retour au format \\(\\LaTeX\\).<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>__str__ : pour afficher le polyn\u00f4me<\/li>\n\n\n\n<li>racines(tex,approx) : pour calculer les racines (en valeur exacte ou approch\u00e9e)<\/li>\n\n\n\n<li>factorise(tex) : pour factoriser le trin\u00f4me<\/li>\n\n\n\n<li>canonique(tex) : pour avoir la forme canonique du trinome<\/li>\n\n\n\n<li>signe(tex) pour avoir un tableau de signes<\/li>\n\n\n\n<li>draw(tex,e,namepic) pour dessiner la parabole sur l&#8217;intervalle \\(\\left[-\\frac{b}{2a}-e~;~-\\frac{b}{2a}+e\\right]\\) (si tex=True, cela enregistre l&#8217;image avec le nom <em>namepic<\/em> qui, par d\u00e9faut, est &#8220;parabole.png&#8221;)<\/li>\n\n\n\n<li>export_latex(filename) : pour cr\u00e9er un fichier \\(\\LaTeX\\) avec toutes les informations, et qui sera compil\u00e9 via pdflatex.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Exemples\"><\/span>Exemples<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<pre class=\"wp-block-code\"><code>&gt;&gt;&gt; P = Trinome(-2,3,4)\n&gt;&gt;&gt; P.racines()\n('(-sqrt(41)+3)\/4', '(sqrt(41)+3)\/4')\n&gt;&gt;&gt; print(P)\n-2x\u00b2+3x+4\n&gt;&gt;&gt; P.canonique()\n'-2(x-3\/4)\u00b2 + 41\/8'\n&gt;&gt;&gt; print(P.signe())\n--------------------------------\n|  x   |-\u221e     x\u2081     x\u2082     +\u221e|\n--------------------------------\n| P(x) |    -   0  +   0  -    |\n--------------------------------\n&gt;&gt;&gt; P.draw()<\/code><\/pre>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"640\" height=\"480\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/05\/trinome_python.jpg\" alt=\"trinome second degr\u00e9 python\" class=\"wp-image-6137\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/05\/trinome_python.jpg 640w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/05\/trinome_python-300x225.jpg 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/05\/trinome_python-600x450.jpg 600w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/figure>\n<\/div>\n\n\n<pre class=\"wp-block-code\"><code>&gt;&gt;&gt; P.export_latex()<\/code><\/pre>\n\n\n\n<p>qui cr\u00e9e un document .tex et qui g\u00e9n\u00e8re le PDF suivant:<\/p>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/05\/trinome.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Contenu embarqu\u00e9 trinome.\"><\/object><a id=\"wp-block-file--media-96ad274c-b274-4f31-8ce3-3a057238f845\"  href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/05\/trinome.pdf\" data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">trinome<\/a><a  href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/05\/trinome.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-96ad274c-b274-4f31-8ce3-3a057238f845\" data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">T\u00e9l\u00e9charger<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Telechargement_de_la_classe\"><\/span>T\u00e9l\u00e9chargement de la classe<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<div class=\"wp-block-file aligncenter um_article\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/05\/Trinome.py\">Classe trinome<\/a><a  href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/05\/trinome.pdf\" class=\"wp-block-file__button wp-element-button\" download data-fancybox data-type=\"iframe\" data-width=\"90%\" data-height=\"100%\" data-preload=\"false\">T\u00e9l\u00e9charger<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Nous allons aborder ici la notion de trin\u00f4me du second degr\u00e9 en Python sous un angle POO (Programmation Orient\u00e9e Objet). J&#8217;avais en effet envie de construire une classe Python repr\u00e9sentant un tel trin\u00f4me et pouvant effectuer toutes les op\u00e9rations classiques du lyc\u00e9e.<\/p>\n","protected":false},"author":1,"featured_media":6125,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,6,5],"tags":[140,287],"class_list":["post-6123","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-enseignement","category-mathematiques","category-python","tag-poo","tag-trinome"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Trin\u00f4me du second degr\u00e9 et Python - Mathweb.fr - POO<\/title>\n<meta name=\"description\" content=\"Trin\u00f4me du second degr\u00e9 en Python vu comme un objet. \u00c0 l&#039;aide de cela, on peut impl\u00e9menter des m\u00e9thodes pour effectuer tout type de calculs.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.mathweb.fr\/euclide\/2021\/05\/18\/trinome-du-second-degre-et-python\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Trin\u00f4me du second degr\u00e9 et Python - 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