{"id":132737,"date":"2024-12-30T10:18:51","date_gmt":"2024-12-30T13:18:51","guid":{"rendered":"https:\/\/vestibulares.estrategia.com\/portal\/?p=132737"},"modified":"2025-02-19T14:21:03","modified_gmt":"2025-02-19T17:21:03","slug":"polinomios-e-relacoes-de-girard-definicoes-aplicacoes-e-mais","status":"publish","type":"post","link":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/polinomios-e-relacoes-de-girard-definicoes-aplicacoes-e-mais\/","title":{"rendered":"Polin\u00f4mios e rela\u00e7\u00f5es de Girard: defini\u00e7\u00f5es, aplica\u00e7\u00f5es e mais"},"content":{"rendered":"<p class=\"wp-block-paragraph\">Os <span id=\"urn:enhancement-42118607-4c1b-4aa1-8569-1a0c0286ec96\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mios<\/span> desempenham um papel <span id=\"urn:enhancement-e5d38377-b88d-4894-b147-7ec85d6db296\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/ensino-fundamental\">fundamental<\/span> no estudo da <span id=\"urn:enhancement-bf2fa655-4653-4cb0-8284-6a00a085fc25\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;tica<\/span>. Tendo em vista sua variada aplicabilidade, esse assunto costuma ser cobrado no <span id=\"urn:enhancement-1a3e2b7e-7375-415d-8ced-fe56b1503a2d\" class=\"textannotation disambiguated wl-event\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/enem\">Enem<\/span> e vestibulares em diversos contextos, como na determina&ccedil;&atilde;o de &aacute;reas e volumes, na an&aacute;lise de padr&otilde;es em progress&otilde;es num&eacute;ricas e na interpreta&ccedil;&atilde;o gr&aacute;fica.<p class=\"wp-block-paragraph\">Para explorar essas aplica&ccedil;&otilde;es, &eacute; essencial dominar conceitos b&aacute;sicos, como o <span id=\"urn:enhancement-07be29bb-815d-4fdd-8698-922c8fc69e32\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">Teorema<\/span> do Resto, rela&ccedil;&otilde;es de Girard e <span id=\"urn:enhancement-ace84ecd-2009-40e1-8527-9b555497f9d4\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/calculo\">c&aacute;lculo<\/span> de ra&iacute;zes de fun&ccedil;&otilde;es polinomiais. Preparado para aprender mais? Vamos nessa!<\/p><p class=\"wp-block-paragraph\"><div class=\"cta-medio\" style=\"background-image: url(https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/08\/curso-enem-1-estrategia-vestibulares-ctamedio.jpg);\"><h3>Pacote Enem TOP <\/h3> <p>Estude com o Estrat\u00e9gia Vestibulares<\/p><div class=\"cta-botao\"><a id=\"cta-medio\" href=\"https:\/\/vestibulares.estrategia.com\/curso\/enem-top\/\" target=\"_blank\" style=\"background:rgb(255,150,0)\">Saiba mais<\/a><\/div><\/div>\n\n\n\n<\/p><div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 counter-hierarchy ez-toc-counter ez-toc-transparent ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\" style=\"cursor:inherit\">Navegue pelo conte\u00fado<\/p>\n<\/div><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:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/polinomios-e-relacoes-de-girard-definicoes-aplicacoes-e-mais\/#O-que-sao-polinomios\" >O que s&atilde;o polin&ocirc;mios?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/polinomios-e-relacoes-de-girard-definicoes-aplicacoes-e-mais\/#Grau-de-um-polinomio\" >Grau de um polin&ocirc;mio<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/polinomios-e-relacoes-de-girard-definicoes-aplicacoes-e-mais\/#Classificacao-dos-polinomios\" >Classifica&ccedil;&atilde;o dos polin&ocirc;mios<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/polinomios-e-relacoes-de-girard-definicoes-aplicacoes-e-mais\/#Raizes-de-um-polinomio\" >Ra&iacute;zes de um polin&ocirc;mio<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/polinomios-e-relacoes-de-girard-definicoes-aplicacoes-e-mais\/#Numero-de-raizes-complexas-de-um-polinomio\" >N&uacute;mero de ra&iacute;zes complexas de um polin&ocirc;mio<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/polinomios-e-relacoes-de-girard-definicoes-aplicacoes-e-mais\/#Relacoes-de-Girard\" >Rela&ccedil;&otilde;es de Girard<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/polinomios-e-relacoes-de-girard-definicoes-aplicacoes-e-mais\/#Questao-sobre-polinomios-e-relacoes-de-Girard\" >Quest&atilde;o sobre polin&ocirc;mios e rela&ccedil;&otilde;es de Girard<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/polinomios-e-relacoes-de-girard-definicoes-aplicacoes-e-mais\/#Estude-com-o-Estrategia-Vestibulares\" >Estude com o Estrat&eacute;gia Vestibulares<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"h-o-que-sao-polinomios\"><span class=\"ez-toc-section\" id=\"O-que-sao-polinomios\"><\/span>O que s&atilde;o polin&ocirc;mios?<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Em s&iacute;ntese, os <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/polinomios\/\" target=\"_blank\">polin&ocirc;mios<\/a> podem ser definidos como equa&ccedil;&otilde;es em que h&aacute; mon&ocirc;mios separados por <span id=\"urn:enhancement-e881cb1d-e72f-481f-9d56-00a8fedd1f95\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/operacoes\">opera&ccedil;&otilde;es<\/span> de <span id=\"urn:enhancement-ba93a2a5-1805-4776-a460-4b3eeca2b506\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">adi&ccedil;&atilde;o<\/span> ou <span id=\"urn:enhancement-e6400264-6698-49cc-928f-05a7a55787bc\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/subtracao\">subtra&ccedil;&atilde;o<\/span>. Nessas equa&ccedil;&otilde;es, encontramos letras, denominadas <strong>vari&aacute;veis<\/strong>, e n&uacute;meros que multiplicam-nas, denominados <strong><span id=\"urn:enhancement-56086247-5922-4ea6-8bad-93ffbd3ce196\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficientes<\/span><\/strong>. Al&eacute;m disso, h&aacute; os <strong>expoentes<\/strong>, que representam as pot&ecirc;ncias da vari&aacute;vel.<\/p><p class=\"wp-block-paragraph\">Dessa maneira, a vari&aacute;vel elevada a um expoente e multiplicada por um <span id=\"urn:enhancement-d1b10790-4309-42fd-bfad-56476cdceb26\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficiente<\/span>, <span id=\"urn:enhancement-d82da5c1-ebb2-4c9f-84ac-b1f09c3d7b1e\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span> um <strong>termo <\/strong>do <span id=\"urn:enhancement-c424e8cd-c15c-4ec2-8336-5a68371cdd87\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span>.<\/p><ul class=\"wp-block-list\">\n<li>Para um polin&ocirc;mio gen&eacute;rico: P(x) = a<sub>n<\/sub>&#8203;x<sup>n<\/sup> + a<sub>n&minus;1<\/sub>&#8203;x<sup>n&minus;1<\/sup> + a<sub>n&minus;2&#8203;<\/sub>x<sup>n&minus;2<\/sup> + &hellip; + a<sub>1<\/sub>&#8203;x + a<sub>0<\/sub>&#8203;\n<ul class=\"wp-block-list\">\n<li>Coeficientes: a<sub>n<\/sub>,<sub> <\/sub>a<sub>n-1<\/sub>, a<sub>n-2<\/sub>, &hellip;, a<sub>1<\/sub>, a<sub>0<\/sub><\/li>\n\n\n\n<li>Vari&aacute;vel: &#8203;x<\/li>\n\n\n\n<li>Termos: a<sub>n<\/sub>&#8203;x<sup>n<\/sup>, a<sub>n&minus;1<\/sub>&#8203;x<sup>n&minus;1<\/sup>, a<sub>n&minus;2&#8203;<\/sub>x<sup>n&minus;2<\/sup>, &hellip; , a<sub>1<\/sub>&#8203;x, a<sub>0<\/sub>&#8203;<\/li>\n<\/ul>\n<\/li>\n<\/ul><h2 class=\"wp-block-heading\" id=\"h-grau-de-um-polinomio\"><span class=\"ez-toc-section\" id=\"Grau-de-um-polinomio\"><\/span>Grau de um polin&ocirc;mio<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">&Eacute; poss&iacute;vel identificar expoentes distintos em cada termo do <span id=\"urn:enhancement-e36df764-a015-4317-9d20-cbfde8ed5353\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span>. Nesse sentido, &eacute; necess&aacute;rio verificar o termo cuja vari&aacute;vel apresenta maior expoente para, ent&atilde;o, determinarmos o grau do <span id=\"urn:enhancement-bedf429d-7be7-460b-97d3-3b1ed732c7a2\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> em quest&atilde;o. Assim, o grau do <span id=\"urn:enhancement-711038f4-f78b-46d3-8c74-6af518e229e3\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> ser&aacute; de mesmo valor do maior expoente encontrado.<\/p><p class=\"wp-block-paragraph\"><strong>Exemplos:<\/strong><\/p><ul class=\"wp-block-list\">\n<li>P(x)= 2x<sup>3<\/sup> &minus; x<sup>2<\/sup> + 4x &minus; 5\n<ul class=\"wp-block-list\">\n<li>O maior expoente &eacute; 3 (encontrado no termo 2x<sup>3<\/sup>), o que indica que o grau desse polin&ocirc;mio &eacute; igual a 3.<\/li>\n<\/ul>\n<\/li>\n<\/ul><ul class=\"wp-block-list\">\n<li>P(x)= 5x<sup>4 <\/sup>&ndash; 7x<sup>3<\/sup> + 2x<sup>2<\/sup> + 4x &minus; 1\n<ul class=\"wp-block-list\">\n<li>O maior expoente &eacute; 4 (encontrado no termo 5x<sup>4<\/sup>), o que indica que o grau desse polin&ocirc;mio &eacute; igual a 4.<\/li>\n<\/ul>\n<\/li>\n<\/ul><h2 class=\"wp-block-heading\" id=\"h-classificacao-dos-polinomios\"><span class=\"ez-toc-section\" id=\"Classificacao-dos-polinomios\"><\/span>Classifica&ccedil;&atilde;o dos polin&ocirc;mios<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Os <span id=\"urn:enhancement-f0b1ba38-f7e0-47c9-964a-adbd663e0d2a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mios<\/span> s&atilde;o classificados de acordo com a quantidade de termos encontrados, da seguinte maneira:&nbsp;<\/p><ul class=\"wp-block-list\">\n<li><strong>Mon&ocirc;mio<\/strong>: polin&ocirc;mio com apenas um termo (3x<sup>2<\/sup>)<\/li>\n\n\n\n<li><strong>Bin&ocirc;mio<\/strong>: polin&ocirc;mio formado por dois termos (x + 5)<\/li>\n\n\n\n<li><strong>Trin&ocirc;mio<\/strong>: polin&ocirc;mio composto por tr&ecirc;s termos (x<sup>2<\/sup> + 5x &ndash; 1)<\/li>\n\n\n\n<li><strong>Polin&ocirc;mio geral<\/strong>: quando h&aacute; quatro ou mais termos, n&atilde;o h&aacute; uma nomenclatura especial (6x<sup>4<\/sup> + x<sup>3<\/sup> + 2x<sup>2<\/sup> + x &minus; 4)<\/li>\n<\/ul><h2 class=\"wp-block-heading\" id=\"h-raizes-de-um-polinomio\"><span class=\"ez-toc-section\" id=\"Raizes-de-um-polinomio\"><\/span>Ra&iacute;zes de um polin&ocirc;mio<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">As ra&iacute;zes de um <span id=\"urn:enhancement-ba65d57b-bc76-4263-8d02-d75496783712\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> s&atilde;o os valores que tornam o <span id=\"urn:enhancement-2d62838e-61be-4182-b98c-a63963aa2634\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> igual a zero quando o substitu&iacute;mos na vari&aacute;vel. Dessa <span id=\"urn:enhancement-e9f11729-93fd-4ea7-af7d-ca98fc633265\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span>, sendo P(x) um <span id=\"urn:enhancement-37653b66-d741-4945-baf5-ffb7618e4495\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span>, ent&atilde;o r ser&aacute; raiz se P(r) = 0. Para encontrarmos as ra&iacute;zes, podemos utilizar alguns m&eacute;todos, como <span id=\"urn:enhancement-0eea6144-eccd-42bb-906e-84acd62aa610\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">Teorema<\/span> do Resto, <span id=\"urn:enhancement-c31429b7-df99-4e33-bbdb-67a1090960da\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">Teorema<\/span> de D&rsquo;Alembert.<\/p><h3 class=\"wp-block-heading\" id=\"h-teorema-do-resto-nbsp\">Teorema do resto&nbsp;<\/h3><p class=\"wp-block-paragraph\">Segundo esse <span id=\"urn:enhancement-464dcbf2-8d66-4f41-9687-abb3d2d9194f\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">teorema<\/span>, o resto (R) da <span id=\"urn:enhancement-5bd64927-294b-48d8-ad11-cd655483e727\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/divisao\">divis&atilde;o<\/span> um <span id=\"urn:enhancement-b1aaf43d-432b-4764-9766-ac550367c184\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> P(x) por um bin&ocirc;mio ax + b &eacute; igual ao valor desse <span id=\"urn:enhancement-3038f760-ef5f-4085-b09a-8bb128836d02\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> para x = -b\/a. Portanto, o resto (R) dessa <span id=\"urn:enhancement-4af13f7f-58a3-41a8-9da2-a508a18b8aea\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/divisao\">divis&atilde;o<\/span> ser&aacute; calculado da seguinte maneira: P(-b\/a) = R.<\/p><p class=\"wp-block-paragraph\"><strong>Exemplo:<\/strong><\/p><p class=\"wp-block-paragraph\">Verifique se <span id=\"urn:enhancement-6202bef1-b8e6-4775-808f-e734fb899f09\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> P(x) = x<sup>3<\/sup> &minus; 4x<sup>2<\/sup> + 5x &minus; 2 &eacute; divis&iacute;vel por x &ndash; 2.<\/p><ul class=\"wp-block-list\">\n<li>Devemos calcular o resto da divis&atilde;o de P(x) por x &ndash; 2. Para isso, calcularemos o valor desse polin&ocirc;mio para x = 2.<\/li>\n\n\n\n<li>Calculamos o valor de P(x) para x = 2: <br>P(2) = 2<sup>3<\/sup> &minus; 4.(2<sup>2<\/sup>) + 5.2 &minus; 2<br>P(2) = 8 &ndash; 4.4 + 10 &ndash; 2<br>P(2) = 8 &ndash; 16 + 10 &ndash; 2<br>P(2) = 0<\/li>\n\n\n\n<li>Como P(2) = 0, o resto da divis&atilde;o &eacute; igual a 0.<\/li>\n<\/ul><h3 class=\"wp-block-heading\" id=\"h-teorema-de-d-alembert\">Teorema de D&rsquo;Alembert<\/h3><p class=\"wp-block-paragraph\">Esse <span id=\"urn:enhancement-0777e583-1ea1-4e36-bafa-598bd1a702ce\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">teorema<\/span> &eacute; uma consequ&ecirc;ncia do <span id=\"urn:enhancement-337f33ad-e69f-4acc-9fb1-824709771a32\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">teorema<\/span> do resto. De acordo com esse <span id=\"urn:enhancement-93d00886-5d5b-41cc-be22-01e1006afb6a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">teorema<\/span>, em uma <span id=\"urn:enhancement-f4173661-ea0f-4cb3-a50d-b24360614d05\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/divisao\">divis&atilde;o<\/span> de um <span id=\"urn:enhancement-e91ee5c4-e8af-4a87-bc18-6dc6b5dd0edf\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> P(x) por um bin&ocirc;mio do tipo x &ndash; a, o resto ser&aacute; igual a zero se e somente se a constante a for raiz do <span id=\"urn:enhancement-357bd6f5-ce16-4985-962f-2c9c8f722e20\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> P(x).&nbsp;<\/p><p class=\"wp-block-paragraph\">Portanto, levando-se em considera&ccedil;&atilde;o tamb&eacute;m o <span id=\"urn:enhancement-1fecaba8-b5e0-4de8-83d2-5dd86abf6639\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">Teorema<\/span> do Resto, pode-se afirmar que P(a) = 0, se e somente se, &ldquo;a&rdquo; for raiz do <span id=\"urn:enhancement-c76cc54f-9725-4eaf-b501-4d0f288523b3\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span>.<\/p><p class=\"wp-block-paragraph\"><strong>Exemplo:<\/strong>&nbsp;<\/p><p class=\"wp-block-paragraph\">Confira, atrav&eacute;s do <span id=\"urn:enhancement-466c8014-afb5-48f8-9362-3ecdcb218efe\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">Teorema<\/span> de D&rsquo;Alembert, se o <span id=\"urn:enhancement-0ff48893-bc78-4ed2-a4a8-1a430e646851\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> P(x) = x<sup>2<\/sup> &ndash; 5x + 6 &eacute; divis&iacute;vel por x-2.<\/p><ul class=\"wp-block-list\">\n<li>Para o divisor x &ndash; 2, a constante <strong>a<\/strong> &eacute; igual a 2.<\/li>\n\n\n\n<li>Devemos conferir se 2 &eacute; raiz do polin&ocirc;mio. Para isso, P(2) deve ser igual a 0.<br><br>P(2) = 2<sup>2<\/sup> &ndash; 5.2 + 6<br>P(2) = 4 &ndash; 10 + 6<br>P(2) = -6 + 6<br>P(2) = 0, ent&atilde;o chegamos &agrave; conclus&atilde;o de que P(x) &eacute; divis&iacute;vel por x &ndash; 2.<\/li>\n<\/ul><h2 class=\"wp-block-heading\" id=\"h-numero-de-raizes-complexas-de-um-polinomio\"><span class=\"ez-toc-section\" id=\"Numero-de-raizes-complexas-de-um-polinomio\"><\/span>N&uacute;mero de ra&iacute;zes complexas de um polin&ocirc;mio<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Um <span id=\"urn:enhancement-9eea3b4c-94e4-43af-9cb6-b41292012095\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> de grau n possui exatamente n ra&iacute;zes no conjunto dos <span id=\"urn:enhancement-091de74f-8ca8-4215-af49-ce8d48e5ac6b\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/numeros-complexos\">n&uacute;meros complexos<\/span>, contando as multiplicidades. Por exemplo, o <span id=\"urn:enhancement-42daffc6-167b-4474-ae63-97ed782946b6\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> P(x)=(x&minus;2)<sup>2<\/sup>(x+3) &eacute; de grau 3 e possui as ra&iacute;zes x=2 (multiplicidade 2) e x=&minus;3 (multiplicidade 1).<\/p><p class=\"wp-block-paragraph\">Veja mais:<br><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/teorema-fundamental-da-algebra\/\" target=\"_blank\">+ Teorema fundamental da &aacute;lgebra<\/a><\/p><h2 class=\"wp-block-heading\" id=\"h-relacoes-de-girard\"><span class=\"ez-toc-section\" id=\"Relacoes-de-Girard\"><\/span>Rela&ccedil;&otilde;es de Girard<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">As rela&ccedil;&otilde;es de Girard s&atilde;o f&oacute;rmulas que conectam os <span id=\"urn:enhancement-5029889d-6da2-4db6-9d72-8674cb695268\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficientes<\/span> de um <span id=\"urn:enhancement-8e39fa17-ed89-44d2-83b7-696950e635d6\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> com as ra&iacute;zes desse <span id=\"urn:enhancement-d4ec7bdd-7917-4703-b16b-fae9fdf7a31c\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span>. Elas s&atilde;o &uacute;teis porque nos permitem calcular somas e produtos das ra&iacute;zes sem precisar encontr&aacute;-las explicitamente.<\/p><h3 class=\"wp-block-heading\" id=\"h-relacoes-de-girard-para-polinomios-de-2&ordm;-grau\">Rela&ccedil;&otilde;es de Girard para polin&ocirc;mios de 2&ordm; grau<\/h3><p class=\"wp-block-paragraph\">Considerando um <span id=\"urn:enhancement-e3b931c4-acab-4365-9b25-aaf8a851c718\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> P(x)=ax<sup>2<\/sup> + bx + c, com ra&iacute;zes x<sub>1<\/sub>&#8203; e x<sub>2<\/sub>&#8203;.<\/p><ul class=\"wp-block-list\">\n<li>Soma das ra&iacute;zes:<br>x<sub>1<\/sub>&#8203; + x<sub>2<\/sub>&#8203;= &minus;b\/a&#8203;.<\/li>\n\n\n\n<li>Produto das ra&iacute;zes:<br>x<sub>1<\/sub>&#8203;.x<sub>2<\/sub>&#8203; = c\/a<\/li>\n\n\n\n<li>Demonstra&ccedil;&atilde;o da rela&ccedil;&atilde;o de Girard para polin&ocirc;mios de 2&ordm; grau: \n<ul class=\"wp-block-list\">\n<li>As ra&iacute;zes x<sub>1<\/sub>&#8203; e x<sub>2<\/sub>&#8203; podem ser encontradas pela f&oacute;rmula de Bhaskara: <br><img decoding=\"async\" width=\"150\" height=\"126\" class=\"wp-image-132738\" style=\"width: 150px;\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/Captura-de-tela-2024-12-30-090457.png\" alt=\"formula-de-Bhaskara\"><\/li>\n\n\n\n<li>Ent&atilde;o, a soma de x<sub>1<\/sub>&#8203; e x<sub>2<\/sub>&#8203; ser&aacute;:<br><img decoding=\"async\" width=\"1000\" height=\"240\" class=\"wp-image-132739\" style=\"width: 1000px;\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/Captura-de-tela-2024-12-30-090845.png\" alt=\"soma-das-raizes\"><\/li>\n\n\n\n<li>E o produto de x<sub>1<\/sub>&#8203; e x<sub>2<\/sub> ser&aacute;:<br><img decoding=\"async\" width=\"1000\" height=\"191\" class=\"wp-image-132740\" style=\"width: 1000px;\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/Captura-de-tela-2024-12-30-090901.png\" alt=\"produto-das-raizes\"><\/li>\n<\/ul>\n<\/li>\n<\/ul><h3 class=\"wp-block-heading\" id=\"h-relacoes-de-girard-para-polinomios-de-3&ordm;-grau\">Rela&ccedil;&otilde;es de Girard para polin&ocirc;mios de 3&ordm; grau<\/h3><p class=\"wp-block-paragraph\">Considerando, agora, P(x) = ax<sup>3 <\/sup>+ bx<sup>2&nbsp; <\/sup>+ cx + d = 0, com ra&iacute;zes x<sub>1<\/sub>, x<sub>2<\/sub> e x<sub>3<\/sub>:<\/p><ul class=\"wp-block-list\">\n<li>Soma das ra&iacute;zes:<br>x<sub>1<\/sub> + x<sub>2<\/sub> + x<sub>3<\/sub> = &ndash; b\/a<\/li>\n\n\n\n<li>Soma do produto das ra&iacute;zes tomadas duas a duas:<br>x<sub>1<\/sub>&sdot;x<sub>2<\/sub> + x<sub>1<\/sub>&sdot;x<sub>3<\/sub> + x<sub>2<\/sub>&sdot;x<sub>3<\/sub> = c\/a<\/li>\n\n\n\n<li>Produto de todas as ra&iacute;zes:<br>x<sub>1<\/sub>&sdot;x<sub>2<\/sub>&sdot;x<sub>3<\/sub> = &minus;d\/a<\/li>\n<\/ul><h3 class=\"wp-block-heading\" id=\"h-relacoes-de-girard-para-polinomios-de-grau-n\">Rela&ccedil;&otilde;es de Girard para polin&ocirc;mios de grau n<\/h3><p class=\"wp-block-paragraph\">Para com um <span id=\"urn:enhancement-69ea44b9-2a57-4e06-85f2-e62266991f42\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> P(x) = a<sub>n<\/sub>&#8203;x<sup>n<\/sup> + a<sub>n&minus;1<\/sub>&#8203;x<sup>n&minus;1<\/sup> + a<sub>n&minus;2&#8203;<\/sub>x<sup>n&minus;2<\/sup> + &hellip; + a<sub>1<\/sub>&#8203;x + a<sub>0<\/sub>, com ra&iacute;zes x<sub>1<\/sub>, x<sub>2<\/sub>, x<sub>3<\/sub>, &hellip; , x<sub>n<\/sub>.<\/p><ul class=\"wp-block-list\">\n<li>Soma das ra&iacute;zes:<br>x<sub>1<\/sub> + x<sub>2<\/sub> + x<sub>3<\/sub> + &hellip; + x<sub>n<\/sub> = &ndash; a<sub>n&minus;1<\/sub>\/a<sub>n<\/sub><\/li>\n\n\n\n<li>Soma dos produtos das ra&iacute;zes tomadas duas a duas:<br>x<sub>1<\/sub>x<sub>2<\/sub> + x<sub>1<\/sub>x<sub>3<\/sub> + x<sub>1<\/sub>x<sub>4<\/sub> + &hellip; + x<sub>n-1<\/sub>x<sub>n<\/sub> = a<sub>n&minus;2<\/sub>\/a<sub>n<\/sub><\/li>\n\n\n\n<li>Produto das ra&iacute;zes (todas):<br>x<sub>1<\/sub>x<sub>2<\/sub>&hellip;x<sub>n-1<\/sub>x<sub>n<\/sub> = (-1)<sup>n&nbsp;<\/sup> a<sub>0<\/sub>\/a<sub>n<\/sub><\/li>\n<\/ul><h3 class=\"wp-block-heading\" id=\"h-exercicios-envolvendo-relacoes-de-girard\">Exerc&iacute;cios envolvendo rela&ccedil;&otilde;es de Girard<\/h3><ol class=\"wp-block-list\">\n<li>Considere o polin&ocirc;mio P(x) = &minus; 4x<sup>2<\/sup> + 5x &minus; 2. Determine:<br>a) A soma das ra&iacute;zes.<br>b) O produto das ra&iacute;zes.<\/li>\n\n\n\n<li>Considere o polin&ocirc;mio P(x) = 2x<sup>3<\/sup> &minus; 3x<sup>2<\/sup> &minus; 5x + 6. Determine:<br>a) A soma das ra&iacute;zes.<br>b) A soma do produto das ra&iacute;zes tomadas duas a duas.<br>c) O produto das ra&iacute;zes.<\/li>\n<\/ol><details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary><strong>Respostas<\/strong><\/summary>\n<ol class=\"wp-block-list\">\n<li><br>a) x<sub>1<\/sub> + x<sub>2<\/sub> = -b\/a <br>x<sub>1<\/sub> + x<sub>2<\/sub> = -(-5\/4) <br><strong>x<sub>1<\/sub> + x<sub>2<\/sub> = 5\/4<\/strong><br><br>b) x<sub>1<\/sub>x<sub>2<\/sub> = c\/a<br>x<sub>1<\/sub>x<sub>2<\/sub> = 2\/4<br><strong>x<sub>1<\/sub>x<sub>2<\/sub> = 1\/2<\/strong><\/li>\n\n\n\n<li><br>a) x<sub>1<\/sub> + x<sub>2<\/sub> + x<sub>3&nbsp; <\/sub>= -b\/a<br>x<sub>1<\/sub> + x<sub>2<\/sub> + x<sub>3<\/sub> = -(-3\/2)<br><strong>x<sub>1<\/sub> + x<sub>2<\/sub> + x<sub>3<\/sub> = 3\/2<\/strong><br><br>b) x<sub>1<\/sub>x<sub>2<\/sub> + x<sub>1<\/sub>x<sub>3<\/sub> + x<sub>2<\/sub>x<sub>3 <\/sub>= c\/a<br><strong>x<sub>1<\/sub>x<sub>2<\/sub> + x<sub>1<\/sub>x<sub>3<\/sub> + x<sub>2<\/sub>x<sub>3<\/sub> = -5\/2<br><\/strong><br>c) x<sub>1<\/sub>x<sub>2<\/sub>x<sub>3<\/sub> = &ndash; d\/a <br>x<sub>1<\/sub>x<sub>2<\/sub>x<sub>3<\/sub> = -6\/2<br><strong>x<sub>1<\/sub>x<sub>2<\/sub>x<sub>3<\/sub> = -3<\/strong><\/li>\n<\/ol>\n<\/details><h2 class=\"wp-block-heading\" id=\"h-questao-sobre-polinomios-e-relacoes-de-girard\"><span class=\"ez-toc-section\" id=\"Questao-sobre-polinomios-e-relacoes-de-Girard\"><\/span>Quest&atilde;o sobre polin&ocirc;mios e rela&ccedil;&otilde;es de Girard<span class=\"ez-toc-section-end\"><\/span><\/h2><h3 class=\"wp-block-heading\" id=\"h-uea-2022\">UEA (2022)<\/h3><p class=\"wp-block-paragraph\">Na <span id=\"urn:enhancement-14efbda9-5be5-4db8-874c-544950f9ff0b\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao-polinomial\">equa&ccedil;&atilde;o polinomial<\/span> x<sup>3<\/sup> &ndash; kx<sup>2<\/sup> + 5x &ndash; p = 0, em que k e p s&atilde;o n&uacute;meros reais n&atilde;o nulos, a <span id=\"urn:enhancement-d17c6f3e-821c-4ba7-8b68-f5d6ea971af8\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> das ra&iacute;zes &eacute; o dobro do produto delas. Sabendo que uma das ra&iacute;zes &eacute; 2, os valores de k e p s&atilde;o iguais a<\/p><p class=\"wp-block-paragraph\"><strong>A)<\/strong> k = &minus; 2 e p = 4.<br><strong>B)<\/strong> k = 4 e p = &minus;2.<br><strong>C)<\/strong> k = &minus; 4 e p = 2.<br><strong>D)<\/strong> k = 4 e p = 2.<br><strong>E)<\/strong> k = 2 e p = 4.<\/p><details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary><strong>Resolu&ccedil;&atilde;o<\/strong><\/summary>\n<p class=\"wp-block-paragraph\"><strong>Passo 1<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Definimos as ra&iacute;zes da equa&ccedil;&atilde;o x<sup>3<\/sup> &ndash; kx<sup>2<\/sup> + 5x &ndash; p = 0 como x<sub>1<\/sub>, x<sub>2<\/sub> e x<sub>3<\/sub>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sabemos que:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>x<sub>1<\/sub> = 2<\/li>\n\n\n\n<li>A soma das ra&iacute;zes &eacute; o dobro do produto delas: 2 (x<sub>1<\/sub>.x<sub>2<\/sub>.x<sub>3<\/sub>) = x<sub>1<\/sub> + x<sub>2<\/sub> + x<sub>3<\/sub><\/li>\n\n\n\n<li>Soma das ra&iacute;zes = x<sub>1<\/sub> + x<sub>2<\/sub> + x<sub>3<\/sub> = -(-k\/1) = k<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Ent&atilde;o:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">2 (x<sub>1<\/sub>.x<sub>2<\/sub>.x<sub>3<\/sub>) = x<sub>1<\/sub> + x<sub>2<\/sub> + x<sub>3<\/sub><br>2 (2.x<sub>2<\/sub>.x<sub>3<\/sub>) = k<br>4x<sub>2<\/sub>x<sub>3<\/sub> = k&nbsp;<br><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">x<sub>2<\/sub>x<sub>3<\/sub> = k\/4<br><\/mark><br>x<sub>1<\/sub> + x<sub>2<\/sub> + x<sub>3<\/sub> =k<br>2+ x<sub>2<\/sub> + x<sub>3<\/sub> = k<br><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">x<sub>2<\/sub> + x<sub>3<\/sub> = k &ndash; 2<br><\/mark><br><strong>Passo 2<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sabemos que:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>x<sub>1<\/sub> = 2<\/li>\n\n\n\n<li>x<sub>2<\/sub>x<sub>3<\/sub> = k\/4<\/li>\n\n\n\n<li>x<sub>2<\/sub> + x<sub>1<\/sub> = k-2<\/li>\n\n\n\n<li>Soma dos produtos das ra&iacute;zes duas a duas = x<sub>1<\/sub>x<sub>2<\/sub> + x<sub>1<\/sub>x<sub>3<\/sub> + x<sub>2<\/sub>x<sub>3<\/sub> = 5<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Ent&atilde;o:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">x<sub>1<\/sub>x<sub>2<\/sub> + x<sub>1<\/sub>x<sub>3<\/sub> + x<sub>2<\/sub>x<sub>3<\/sub> = 5<br>2x<sub>2<\/sub> + 2x<sub>3<\/sub> + x<sub>2<\/sub>x<sub>3<\/sub> = 5<br>2 (x<sub>2<\/sub> + x<sub>3<\/sub>) + x<sub>2<\/sub>x<sub>3<\/sub> = 5<br>2(k-2) + k\/4 = 5<br>2(k-2) + k\/4 = 5<br>2k &ndash; 4 + k\/4 = 5<br>2k + k\/4 = 5 + 4<br>2k + k\/4 = 9<br>8k + k = 36<br>9k = 36<br><strong>k = 4<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Passo 3<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sabemos que:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>2x<sub>1<\/sub>x<sub>2<\/sub>x<sub>3<\/sub> = x<sub>1<\/sub> + x<sub>2<\/sub> + x<sub>3<\/sub><\/li>\n\n\n\n<li>x<sub>1<\/sub> + x<sub>2<\/sub> + x<sub>3<\/sub> = k<\/li>\n\n\n\n<li>k = 4<\/li>\n\n\n\n<li>Produto de todas as ra&iacute;zes = x<sub>1<\/sub>x<sub>2<\/sub>x<sub>3<\/sub> = -(-p\/1) = p<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Ent&atilde;o:<br><br>2x<sub>1<\/sub>x<sub>2<\/sub>x<sub>3<\/sub> = x<sub>1<\/sub> + x<sub>2<\/sub> + x<sub>3<\/sub><br>2x<sub>1<\/sub>x<sub>2<\/sub>x<sub>3<\/sub> = k<br>2p = k<br>2p = 4<br><strong>p = 2<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Gabarito: <strong>D)<\/strong> k = 4 e p = 2.<\/p>\n<\/details><h2 class=\"wp-block-heading\" id=\"h-estude-com-o-estrategia-vestibulares\"><span class=\"ez-toc-section\" id=\"Estude-com-o-Estrategia-Vestibulares\"><\/span>Estude com o Estrat&eacute;gia Vestibulares<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Quer aprender <span id=\"urn:enhancement-1c697498-2fd6-4079-a146-b2a70ead9ba4\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;tica<\/span> para o <span id=\"urn:enhancement-e2af60a5-1bd4-4fc4-98bc-fb664b6c57e5\" class=\"textannotation disambiguated wl-event\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/enem\">Enem<\/span> e vestibulares? 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Clique no banner abaixo e prepare-se com a Coruja!<\/p><p class=\"wp-block-paragraph\"><a id=\"cta\" class=\"cta-imagem\" href=\"https:\/\/vestibulares.estrategia.com\/cursos\/\" target=\"blank\">\n                <img decoding=\"async\" width=\"100%\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/07\/ESTRATEGIA.jpg\" alt=\"\" title=\"ESTRAT\u00c9GIA\">\n        <\/a>\n\n\n<\/p><\/p>\n","protected":false},"excerpt":{"rendered":"Aprenda a classificar, encontrar ra\u00edzes e determinar o grau dos polin\u00f4mios; entenda, tamb\u00e9m, o teorema do resto e as rela\u00e7\u00f5es de Girard\n","protected":false},"author":58,"featured_media":132748,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29],"tags":[],"class_list":["post-132737","post","type-post","status-publish","format-standard","has-post-thumbnail","category-matematica"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v25.9 (Yoast SEO v28.2) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Polin\u00f4mios e rela\u00e7\u00f5es de Girard: defini\u00e7\u00f5es, aplica\u00e7\u00f5es e mais<\/title>\n<meta name=\"description\" content=\"Aprenda a classificar, encontrar ra\u00edzes e determinar o grau dos polin\u00f4mios. 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