{"id":1084,"date":"2019-03-18T11:12:27","date_gmt":"2019-03-18T10:12:27","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=1084"},"modified":"2022-02-16T10:59:42","modified_gmt":"2022-02-16T09:59:42","slug":"la-formule-de-viete-sur-les-polynomes","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2019\/03\/18\/la-formule-de-viete-sur-les-polynomes\/","title":{"rendered":"La formule de Vi\u00e8te sur les polyn\u00f4mes"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">La formule de Vi\u00e8te sur les polyn\u00f4mes nous donne la valeur de la somme des racines complexes d&rsquo;un polyn\u00f4me. Cette somme est l&rsquo;oppos\u00e9e du quotient de deux coefficients cons\u00e9cutifs du polyn\u00f4me; elle est donc r\u00e9elle. Regardons cela de plus pr\u00e8s&#8230;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Cet article est accessible aux \u00e9l\u00e8ves de lyc\u00e9e d\u00e8s la classe de Terminale.<\/em><\/p>\n\n\n\n<!--more-->\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"360\" height=\"540\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/formule-viete.jpg\" alt=\"formule vi\u00e8te\" class=\"wp-image-3667\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/formule-viete.jpg 360w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/formule-viete-300x450.jpg 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/formule-viete-200x300.jpg 200w\" sizes=\"auto, (max-width: 360px) 100vw, 360px\" \/><figcaption>Selfie de Fran\u00e7ois Vi\u00e8te<\/figcaption><\/figure><\/div>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_84 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\/2019\/03\/18\/la-formule-de-viete-sur-les-polynomes\/#Formule_de_Viete_histoire_du_bonhomme\" >Formule de Vi\u00e8te: histoire du bonhomme<\/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\/2019\/03\/18\/la-formule-de-viete-sur-les-polynomes\/#En_route_vers_la_formule_de_Viete_sur_les_polynomes\" >En route vers la formule de Vi\u00e8te sur les polyn\u00f4mes<\/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\/2019\/03\/18\/la-formule-de-viete-sur-les-polynomes\/#Demonstration\" >D\u00e9monstration<\/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\/2019\/03\/18\/la-formule-de-viete-sur-les-polynomes\/#Un_exemple_pour_la_route%E2%80%A6\" >Un exemple pour la route&#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\/2019\/03\/18\/la-formule-de-viete-sur-les-polynomes\/#A_quoi_sert_la_formule_de_Viete\" >\u00c0 quoi sert la formule de Vi\u00e8te ?<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2019\/03\/18\/la-formule-de-viete-sur-les-polynomes\/#Polynome_de_degre_2\" >Polyn\u00f4me de degr\u00e9 2<\/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\/2019\/03\/18\/la-formule-de-viete-sur-les-polynomes\/#Polynome_de_degre_3\" >Polyn\u00f4me de degr\u00e9 3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.mathweb.fr\/euclide\/2019\/03\/18\/la-formule-de-viete-sur-les-polynomes\/#Vers_la_theorie_de_Galois\" >Vers la th\u00e9orie de Galois<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"formule-de-viete-histoire-du-bonhomme\"><span class=\"ez-toc-section\" id=\"Formule_de_Viete_histoire_du_bonhomme\"><\/span>Formule de Vi\u00e8te: histoire du bonhomme<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Fran\u00e7ois Vi\u00e8te est un math\u00e9maticien fran\u00e7ais du XVI\u00e8me si\u00e8cle. Il est n\u00e9 en 1540 alors que Fran\u00e7ois 1er \u00e9tait encore sur le tr\u00f4ne (no comment). Ce dernier quitta le tr\u00f4ne \u00e0 sa mort, en 1547, pour laisser la place \u00e0 Henri II. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Le petit Henri ne resta pas longtemps car il mourut en 1560, laissant sa place \u00e0 Charles IX. Pas pour tr\u00e8s longtemps car peu de temps apr\u00e8s, ce fut au tour d&rsquo;Henri III, puis d&rsquo;Henri IV : voir la <a href=\"http:\/\/www.thucydide.com\/realisations\/utiliser\/chronos\/rois_france.htm\" target=\"_blank\" rel=\"noreferrer noopener\">chronologie des rois de France<\/a>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pourquoi diable vous parle-je des rois ? Et bien parce que Fran\u00e7ois Vi\u00e8te fut tr\u00e8s proche de la royaut\u00e9. En effet, il fut :<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>conseiller de Charles IX;<\/li><li>ma\u00eetre des requ\u00eates ordinaires de l&rsquo;h\u00f4tel du roi sous Henri III;<\/li><li>ma\u00eetre des requ\u00eates et d\u00e9chiffreur de Henri IV.<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Parall\u00e8lement (sans jeu de mot) \u00e0 sa carri\u00e8re politique, Fran\u00e7ois Vi\u00e8te fut math\u00e9maticien amateur&#8230; Mais pas si amateur que \u00e7a!<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En effet, entre autre, il annonce l&rsquo;incommensurabilit\u00e9 de \\(\\pi\\) (qui ne fut prouv\u00e9e qu&rsquo;au XVIII\u00e8me si\u00e8cle) et qu&rsquo;il avance la formule:$$\\pi=2\\times\\frac{2}{\\sqrt2}\\times\\frac{2}{\\sqrt{2+\\sqrt2}}\\times\\frac{2}{\\sqrt{2+\\sqrt{2+\\sqrt2}}}\\times\\cdots$$Balaise pour un politicard non ?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Mais bon ! Ce n&rsquo;est pas la formule qui nous int\u00e9resse ici&#8230; Et oui ! Fran\u00e7ois a plus d&rsquo;une corde \u00e0 son arc (de cercle)&#8230;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"en-route-vers-la-formule-de-viete-sur-les-polynomes\"><span class=\"ez-toc-section\" id=\"En_route_vers_la_formule_de_Viete_sur_les_polynomes\"><\/span>En route vers la formule de Vi\u00e8te sur les polyn\u00f4mes<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Un polyn\u00f4me de degr\u00e9 <em>n<\/em> est une expression de la forme:$$P(x)=\\sum_{k=0}^n a_kx^k=a_nx^n+a_{n-1}x^{n-1}+\\cdots+a_1x+a_0.$$Par exemple, $$P(x)=5x^3-3x^2+2x-1$$est un polyn\u00f4me de degr\u00e9 3.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Une <em>racine<\/em> d&rsquo;un polyn\u00f4me est une valeur <em>r<\/em> telle que P(<em>r<\/em>)=0. Par exemple, <em>r<\/em> = 1 est une racine du polyn\u00f4me P(<em>x<\/em>) = <em>x<\/em>\u00b2 &#8211; 2<em>x<\/em> + 1 = (<em>x<\/em> &#8211; 1)\u00b2.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Vous savez ce que sont les nombres complexes ? Ce sont des nombres qui s&rsquo;\u00e9crivent sous la forme <em>a + <\/em>i<em>b<\/em>, o\u00f9 i\u00b2 = -1. Ce sont des nombres <em>imaginaires<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">La formule de Vi\u00e8te nous dit que la somme des racines complexes du polyn\u00f4me <em>P<\/em> est \u00e9gale \u00e0 \\(-\\frac{a_{n-1}}{a_n}\\).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"demonstration\"><span class=\"ez-toc-section\" id=\"Demonstration\"><\/span>D\u00e9monstration<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">La d\u00e9monstration de cette formule est assez simple si l&rsquo;on conna\u00eet le th\u00e9or\u00e8me de Gauss stipulant que tout polyn\u00f4me de degr\u00e9 <em>n<\/em> admet exactement <em>n<\/em> racines complexes. Ainsi, tout polyn\u00f4me de degr\u00e9 <em>n<\/em> peut se factoriser sous la forme : $$P(x)=a_n(x-r_1)(x-r_2)(x-r_3)\\cdots(x-r_{n-1})(x-r_n)$$ o\u00f9 \\(r_1,\\ r_2,\\ \\ldots,\\ r_n\\) repr\u00e9sentent les <em>n<\/em> racines complexes du polyn\u00f4me.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En d\u00e9veloppant <em>partiellement<\/em> la forme factoris\u00e9e, on obtient:$$P(x)=a_nx^n-a_n(r_1+r_2+\\cdots+r_n)x^{n-1}+\\cdots+(-1)^na_nr_1r_2\\cdots r_n.$$Par identification avec la forme d\u00e9velopp\u00e9e:$$P(x)=a_nx^n+a_{n-1}x^{n-1}+\\cdots+a_1x+a_0,$$les coefficients des \\(x^{n-1}\\) doivent \u00eatre \u00e9gaux, et donc:$$a_{n-1}=-a_n(r_1+r_2+\\cdots+r_n)$$ce qui donne:$$r_1+r_2+\\cdots+r_n=-\\frac{a_{n-1}}{a_n}.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On peut m\u00eame affirmer de la m\u00eame fa\u00e7on que:$$a_0=(-1)^na_nr_1r_2\\cdots r_n$$soit:$$r_1r_2\\cdots r_n=(-1)^n\\frac{a_0}{a_n}.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Voil\u00e0 ! Ce n&rsquo;\u00e9tait pas si compliqu\u00e9 que \u00e7a au final&#8230;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"un-exemple-pour-la-route\"><span class=\"ez-toc-section\" id=\"Un_exemple_pour_la_route%E2%80%A6\"><\/span>Un exemple pour la route&#8230;<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Consid\u00e9rons le polyn\u00f4me:$$P(x)=x^4+x^3-x^2+x-2.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ici,$$-\\frac{a_{n-1}}{a_n}=-1$$donc la somme des quatre racines complexes de P est \u00e9gale \u00e0 -1.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On peut le v\u00e9rifier en constatant que <em>x<\/em> = 1 est une racine \u00e9vidente de P. On peut ainsi factoriser P(<em>x<\/em>) par (<em>x<\/em>&#8211;  1), ce qui donne:$$P(x)=(x-1)(x^3+2x^2+x+2).$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Un \u0153il aiguis\u00e9 peut aussi voir que <em>x<\/em> = i est aussi une racine \u00e9vidente. En effet, $$\\text{i}^3+2\\text{i}^2+\\text{i}+2=-\\text{i}-2+\\text{i}+2=0.$$Ainsi, on peut factoriser par (<em>x<\/em> &#8211; i), ce qui donne:$$P(x)=(x-1)(x-\\text{i})(x^2+(2+\\text{i})x+2\\text{i}).$$Comme \u00ab\u00a0i\u00a0\u00bb est une racine de P, son conjugu\u00e9 aussi: <em>x<\/em> = -i est donc une racine de P. En factorisant par (<em>x<\/em> + i), on obtient finalement:$$P(x)=(x-\\text{i})(x+\\text{i})(x-1)(x+2).$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">La somme des racines est donc:$$\\text{i}+(-\\text{i})+1+(-2)=-1.$$C&rsquo;est bien ce que l&rsquo;on avait annonc\u00e9 \u00e0 l&rsquo;aide de la formule de Vi\u00e8te.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"a-quoi-sert-la-formule-de-viete\"><span class=\"ez-toc-section\" id=\"A_quoi_sert_la_formule_de_Viete\"><\/span>\u00c0 quoi sert la formule de Vi\u00e8te ?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"polynome-de-degre-2\"><span class=\"ez-toc-section\" id=\"Polynome_de_degre_2\"><\/span>Polyn\u00f4me de degr\u00e9 2<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Dans le cas particulier d&rsquo;un polyn\u00f4me de degr\u00e9 2:$$P(x)=ax^2+bx+c,$$la formule de Vi\u00e8te stipule que:$$\\begin{cases}x_1+x_2 &amp; =-\\frac{b}{a}\\\\x_1 x_2 &amp; = \\frac{c}{a}\\end{cases}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi, si nous avons la chance de conna\u00eetre une racine, nous pouvons d\u00e9terminer l&rsquo;autre \u00e0 l&rsquo;aide de l&rsquo;une de ces deux \u00e9galit\u00e9s.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Par exemple, le polyn\u00f4me:$$P(x)=3x^2-5x+2$$admet pour racine \u00e9vidente <em>x<\/em> = 1 (car la somme des coefficients est nulle). Ainsi, en utilisant la deuxi\u00e8me \u00e9galit\u00e9 (celle du produit), on obtient pour seconde racine :\\(x=\\frac{2}{3}\\). Inutile de sortir le bazooka pour tuer la mouche ! Ici, inutile de calculer le discriminant de P.<\/p>\n\n\n\n<p class=\"has-text-align-center has-black-color has-light-green-cyan-background-color has-text-color has-background has-medium-font-size wp-block-paragraph\"><strong>Retrouvez des exercices corrig\u00e9s sur le second degr\u00e9, et plus encore, dans le recueil d&rsquo;exercices de 1\u00e8re sur <a href=\"https:\/\/www.mathweb.fr\/euclide\/exercices-corriges-maths-1ere\/\" target=\"_blank\" rel=\"noreferrer noopener\">cette page<\/a>.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"polynome-de-degre-3\"><span class=\"ez-toc-section\" id=\"Polynome_de_degre_3\"><\/span>Polyn\u00f4me de degr\u00e9 3<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">On peut imaginer un sc\u00e9nario identique pour le degr\u00e9 3. Consid\u00e9rons le polyn\u00f4me:$$P(x)=3x^3-2x^2-3x+2.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">La somme des coefficients \u00e9tant nulle, <em>x<\/em> = 1 est une racine \u00e9vidente.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Les coefficients des termes de degr\u00e9s impairs \u00e9tant oppos\u00e9s (3 et -2 d&rsquo;une part, -2 et 2 d&rsquo;autre part), <em>x<\/em> = -1 est aussi une racine \u00e9vidente.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D&rsquo;apr\u00e8s la formule de Vi\u00e8te, la somme des racines est \u00e9gale \u00e0 \\(\\frac{2}{3}\\), et comme la somme des deux premi\u00e8res racines est nulle, cela signifie que la troisi\u00e8me racine est \\(\\frac{2}{3}\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">C&rsquo;est pas top-moumoute tout \u00e7a ? \ud83d\ude42<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"vers-la-theorie-de-galois\"><span class=\"ez-toc-section\" id=\"Vers_la_theorie_de_Galois\"><\/span>Vers la th\u00e9orie de Galois<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">La formule de Vi\u00e8te peut \u00eatre g\u00e9n\u00e9ralis\u00e9e, en fonction des r\u00e9sultats obtenus dans la d\u00e9monstration pr\u00e9c\u00e9dente, \u00e0:$$\\sum_{1\\leqslant i_1 &lt; i_2 &lt; \\cdots &lt; i_p\\leqslant n}x_{i_1}x_{i_2}\\cdots x_{i_p} = (-1)^p\\frac{a_{n-p}}{a_n}$$o\u00f9 \\(x_1\\) , &#8230; , \\(x_n\\) repr\u00e9sentent les <em>n<\/em> racines du polyn\u00f4me.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Par exemple, pour un polyn\u00f4me de degr\u00e9 3:$$P(x)=ax^3+bx^2+cx+d,$$on a :$$\\begin{cases}x_1+x_2+x_3 &amp; = -\\frac{b}{a}\\\\x_1x_2 + x_1x_3 + x_2x_3 &amp; = \\frac{c}{a}\\\\ x_1x_2x_3 &amp; = -\\frac{d}{a} \\end{cases}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ces \u00e9galit\u00e9s jouent un r\u00f4le important dans la <a href=\"https:\/\/www.mathweb.fr\/euclide\/vers-la-theorie-de-galois\/\" target=\"_blank\" rel=\"noreferrer noopener\">th\u00e9orie de Galois<\/a>. Mais \u00e7a, c&rsquo;est une autre histoire&#8230;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>La formule de Vi\u00e8te sur les polyn\u00f4mes nous donne la valeur de la somme des racines complexes d&rsquo;un polyn\u00f4me. Cette somme est l&rsquo;oppos\u00e9e du quotient de deux coefficients cons\u00e9cutifs du polyn\u00f4me; elle est donc r\u00e9elle. Regardons cela de plus pr\u00e8s&#8230; Cet article est accessible aux \u00e9l\u00e8ves de lyc\u00e9e d\u00e8s la [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,6],"tags":[],"class_list":["post-1084","post","type-post","status-publish","format-standard","hentry","category-enseignement","category-mathematiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.8 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>La formule de Vi\u00e8te sur les polyn\u00f4mes - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"La formule de Vi\u00e8te sur les polyn\u00f4mes nous donne la valeur de la somme des racines complexes d&#039;un polyn\u00f4me. 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