{"id":323,"date":"2023-07-06T08:26:38","date_gmt":"2023-07-06T08:26:38","guid":{"rendered":"https:\/\/mathority.org\/pt\/teorema-do-fator\/"},"modified":"2023-07-06T08:26:38","modified_gmt":"2023-07-06T08:26:38","slug":"teorema-do-fator","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/teorema-do-fator\/","title":{"rendered":"Teorema do fator"},"content":{"rendered":"<p>Nesta p\u00e1gina explicamos o que \u00e9 o teorema do fator. Al\u00e9m disso, mostramos para que serve o teorema do fator: divisibilidade de polin\u00f4mios, encontrar ra\u00edzes, fatorar polin\u00f4mios, etc. Finalmente, voc\u00ea poder\u00e1 praticar exerc\u00edcios passo a passo sobre o teorema do fator. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%C2%BFQue-es-el-teorema-del-factor\"><\/span> Qual \u00e9 o teorema do fator? <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p class=\"has-background\" style=\"background-color:#ffebee\"> Em matem\u00e1tica, o <strong>teorema do fator<\/strong> diz que um polin\u00f4mio P(x) \u00e9 divis\u00edvel por outro polin\u00f4mio da forma (xa) se e somente se P(a)=0. <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/theoreme-des-facteurs.jpg\" alt=\"teorema do fator\" class=\"wp-image-2179\" width=\"247\" height=\"247\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Da mesma forma, como consequ\u00eancia do teorema do fator, segue-se que se um polin\u00f4mio P(x) \u00e9 divis\u00edvel pelo termo (x\u2212a), isso significa que o valor a \u00e9 uma raiz (ou zero) do polin\u00f4mio P( x ).<\/p>\n<p> O fato de um polin\u00f4mio ser divis\u00edvel por outro significa que o resto (ou resto) da divis\u00e3o entre os dois polin\u00f4mios \u00e9 igual a zero. Caso voc\u00ea n\u00e3o se lembre completamente deste conceito, no link a seguir voc\u00ea poder\u00e1 ver <a href=\"https:\/\/mathority.org\/pt\/divisao-de-polinomios-exemplos-exercicios-resolvidos-dividir\/\"><strong><span style=\"text-decoration: underline;\">exemplos de divis\u00e3o de polin\u00f4mios<\/span><\/strong><\/a> , l\u00e1 voc\u00ea tamb\u00e9m encontrar\u00e1 a explica\u00e7\u00e3o de como dividir polin\u00f4mios e exerc\u00edcios resolvidos passo a passo. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejemplos-del-teorema-del-factor\"><\/span> Exemplos de teoremas de fator<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Agora que conhecemos a defini\u00e7\u00e3o matem\u00e1tica do teorema do fator, vejamos v\u00e1rios exemplos para ver como ele \u00e9 aplicado.<\/p>\n<h3 class=\"wp-block-heading\"> Exemplo 1<\/h3>\n<p> Uma aplica\u00e7\u00e3o do teorema do fator \u00e9 descobrir se um determinado polin\u00f4mio \u00e9 divis\u00edvel por um <strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/binomios\/\">bin\u00f4mio<\/a><\/span><\/strong> . Vejamos um exemplo de como isso \u00e9 feito com o teorema do fator:<\/p>\n<ul>\n<li> Determine se o polin\u00f4mio P(x) \u00e9 divis\u00edvel pelo bin\u00f4mio Q(x), sendo ambos:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-67a5b0b8df744da98b4d71433f73c9e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^2-4x+3 \\qquad \\qquad Q(x)=x-1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"323\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Primeiro, o polin\u00f4mio divisor, Q(x), \u00e9 um polin\u00f4mio do tipo (xa), portanto podemos aplicar o teorema do fator para resolver o problema.<\/p>\n<p> Ent\u00e3o, para verificar se P(x) pode ser dividido por Q(x) precisamos calcular o valor num\u00e9rico do polin\u00f4mio P(x) para x=1, j\u00e1 que 1 \u00e9 o termo independente do polin\u00f4mio divisor com seu sinal alterado :<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-00216efc4a2e53b0b38de1175e73a5bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} P(1) &amp; =1^2-4\\cdot 1+3 \\\\[2ex] &amp; = 1-4+3 \\\\[2ex] &amp; = 0 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"100\" width=\"159\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> O valor num\u00e9rico do polin\u00f4mio P(x) em x = 1 d\u00e1 zero, ent\u00e3o de acordo com o teorema do fator P(x) \u00e9 divis\u00edvel por Q(x), ou em outras palavras, o resto da divis\u00e3o por ambos ser\u00e1 nulo.<\/p>\n<p> Podemos verificar que a condi\u00e7\u00e3o de divisibilidade \u00e9 satisfeita dividindo os 2 polin\u00f4mios pelo <strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/regras-resolvidas-exemplos-exercicios-ruffini\/\">teorema de Ruffini<\/a><\/span><\/strong> : <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/theoreme-factoriel-exercices-resolus-pdf.jpg\" alt=\"teorema do fator exerc\u00edcios resolvidos pdf online\" class=\"wp-image-2189\" width=\"172\" height=\"130\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Como voc\u00ea pode ver neste exemplo, o teorema do fator \u00e9 um caso especial do teorema do resto (ou resto). Deixo para voc\u00eas este artigo que explica o que \u00e9 <strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/exemplos-e-exercicios-do-teorema-do-resto-resolvidos\/\">o teorema do resto<\/a><\/span><\/strong> , voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos e exerc\u00edcios resolvidos com ele. E, al\u00e9m do mais, voc\u00ea poder\u00e1 ver qual \u00e9 a diferen\u00e7a entre o teorema do resto e o teorema do fator.<\/p>\n<h3 class=\"wp-block-heading\"> Exemplo 2<\/h3>\n<p> O teorema do fator tamb\u00e9m pode ser usado para encontrar as ra\u00edzes (ou zeros) de um polin\u00f4mio. Mas, obviamente, para entender esse tipo de problema \u00e9 preciso saber <a href=\"https:\/\/mathority.org\/pt\/raizes-de-um-polinomio\/\"><strong><span style=\"text-decoration: underline;\">quais s\u00e3o as ra\u00edzes de um polin\u00f4mio<\/span><\/strong><\/a> . Se voc\u00ea ainda n\u00e3o entende esse conceito, pode dar uma olhada na p\u00e1gina do link, que \u00e9 explicada detalhadamente.<\/p>\n<p> Ent\u00e3o, vamos ver atrav\u00e9s de um exemplo como o teorema do fator \u00e9 aplicado para encontrar a raiz de um polin\u00f4mio:<\/p>\n<ul>\n<li> Dado o polin\u00f4mio P(x), calcule se uma de suas ra\u00edzes \u00e9 x=2:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c3d4052b7ff040dd41473d225569289b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^3-3x^2+5x-6\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"199\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Aplicando o teorema do fator, o termo x=2 s\u00f3 ser\u00e1 raiz do polin\u00f4mio P(x) se o valor num\u00e9rico de P(x) para x=2 for zero. Portanto, precisamos encontrar este valor num\u00e9rico:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-82d50f0361613cb6c540051f8da4bc20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} P(2) &amp; =2^3-3\\cdot 2^2+5\\cdot 2-6 \\\\[2ex] &amp; = 8-3\\cdot 4 +5\\cdot 2 -6\\\\[2ex] &amp; = 8-12+10-6 \\\\[2ex] &amp; = 0\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"219\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Na verdade, o valor num\u00e9rico do polin\u00f4mio P(x) desaparece em x=2, ent\u00e3o gra\u00e7as ao teorema do fator podemos afirmar que x=2 \u00e9 uma raiz do polin\u00f4mio P(x). <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Factorizacion-de-polinomios-utilizando-el-teorema-del-factor\"><\/span> Fatora\u00e7\u00e3o de polin\u00f4mios usando o teorema do fator<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Outra aplica\u00e7\u00e3o do teorema do fator \u00e9 a <strong>fatora\u00e7\u00e3o de polin\u00f4mios<\/strong> . Caso voc\u00ea n\u00e3o saiba o que \u00e9, fatorar um polin\u00f4mio significa transformar a express\u00e3o de um polin\u00f4mio em produto de fatores, ou seja, fatorar um polin\u00f4mio simplifica sua express\u00e3o alg\u00e9brica.<\/p>\n<p class=\"has-background\" style=\"background-color:#ffebee\"> Assim, o teorema fatorial estabelece que se um polin\u00f4mio P(x) satisfaz P(a)=0 para um determinado valor a, ent\u00e3o a express\u00e3o desse polin\u00f4mio pode ser fatorada no produto P(x)=(xa)\u00b7 Q( x), onde Q(x) \u00e9 o polin\u00f4mio resultante da divis\u00e3o do polin\u00f4mio P(x) por (xa). <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/facteur-theoreme-preuve.jpg\" alt=\"prova do teorema do fator\" class=\"wp-image-2199\" width=\"470\" height=\"157\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Como exemplo, fatoraremos o seguinte polin\u00f4mio usando o teorema fatorial:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e7291b669031afd3421168b7662c71c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^3+2x^2+4x+8\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"199\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Do polin\u00f4mio anterior, podemos saber que x=-2 \u00e9 uma de suas ra\u00edzes, pois o valor num\u00e9rico do polin\u00f4mio para x=-2 \u00e9 igual a zero:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1d94a4657a385e672badeabd7458b376_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} P(-2) &amp; =(-2)^3+2\\cdot (-2)^2+4\\cdot (-2)+8 \\\\[2ex] &amp; =-8+2\\cdot 4+4\\cdot (-2)+8 \\\\[2ex] &amp; = -8+8-8+8 \\\\[2ex] &amp; = 0 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"316\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Dividimos portanto com a regra de Ruffini o polin\u00f4mio P(x) entre o bin\u00f4mio formado por x e esta raiz mudou de sinal, ou seja, o fator (x+2): <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/theoreme-du-facteur-zero.jpg\" alt=\"O teorema do fator zero afirma que\" class=\"wp-image-2201\" width=\"205\" height=\"130\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Portanto, o quociente da divis\u00e3o polinomial \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8787f77c00f3813ff7e93f147ae7a8d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{P(x)}{x+2} =x^2+4\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"113\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> E finalmente, a partir do teorema do fator, podemos expressar o polin\u00f4mio P(x) na forma de uma multiplica\u00e7\u00e3o do fator (x+2) pelo quociente obtido na divis\u00e3o anterior:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2a63580effbfd7304133453960e84843_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) = (x+2) \\cdot (x^2+4)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"190\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Assim, fatoramos o polin\u00f4mio P(x), mas apenas parcialmente. Para fatorar completamente um polin\u00f4mio, um procedimento mais longo deve ser aplicado. Fizemos um guia onde ensinamos passo a passo <a href=\"https:\/\/mathority.org\/pt\"><strong><span style=\"text-decoration: underline;\">como fatorar polin\u00f4mios de Ruffini<\/span><\/strong><\/a> , al\u00e9m disso, neste artigo explicamos todos os tipos de fatora\u00e7\u00f5es e voc\u00ea poder\u00e1 praticar com exerc\u00edcios resolvidos. Ent\u00e3o clique no link para descobrir como fatorar um polin\u00f4mio do conjunto. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejercicios-resueltos-del-teorema-del-factor\"><\/span> Problemas resolvidos do teorema do fator<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A seguir, preparamos diversos exerc\u00edcios resolvidos passo a passo sobre o teorema do fator para que voc\u00ea possa praticar, e assim verificar se entendeu este teorema. Recomendamos que voc\u00ea tente fazer isso sozinho e veja se entendeu a solu\u00e7\u00e3o corretamente. N\u00e3o se esque\u00e7a tamb\u00e9m que voc\u00ea pode nos deixar suas d\u00favidas abaixo nos coment\u00e1rios! \u2753\u2753\ud83d\udcac\ud83d\udcac<\/p>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Use o teorema fatorial para descobrir se o polin\u00f4mio P(x) \u00e9 divis\u00edvel pelo bin\u00f4mio Q(x) e, em caso afirmativo, encontre uma raiz do polin\u00f4mio e fatore-a. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d91041d9502129f8feb71f75ec493bab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=2x^3-4x^2+x-7 \\qquad \\qquad Q(x)=x-3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"372\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Neste caso, o divisor polinomial Q(x) \u00e9 um bin\u00f4mio composto apenas por um x e um termo independente. Ent\u00e3o, para mostrar que o polin\u00f4mio P(x) pode ser dividido pelo outro polin\u00f4mio Q(x) com o teorema fatorial, devemos avaliar o valor num\u00e9rico do polin\u00f4mio P(x) no termo independente do polin\u00f4mio divisor de sinal alterado, isto \u00e9, em x = 3:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9e8ac752f8e16fae1d66386e9d2a02a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} P(3) &amp; =2\\cdot 3^3-4\\cdot 3^2+3-7\\\\[2ex] &amp; = 2\\cdot 27-4\\cdot 9+3-7 \\\\[2ex] &amp; = 54-36+3-7\\\\[2ex] &amp; = 14 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"219\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O valor num\u00e9rico do polin\u00f4mio P(x) em x=3 \u00e9 equivalente a 14, ou seja, \u00e9 diferente de zero. Portanto, de acordo com o teorema do fator, P(x) N\u00c3O \u00e9 divis\u00edvel por Q(x) porque o resto da divis\u00e3o n\u00e3o \u00e9 zero.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 2<\/h3>\n<p> Descubra pelo teorema fatorial se o polin\u00f4mio P(x) \u00e9 divis\u00edvel pelo bin\u00f4mio Q(x) e, em caso afirmativo, encontre uma raiz do polin\u00f4mio P(x) e fatore-o. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c66c882f468d9bf67c8ae19b9629a24c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^3+5x^2+3x-1 \\qquad \\qquad Q(x)=x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"371\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Neste caso o divisor polinomial Q(x) \u00e9 um bin\u00f4mio composto apenas por um x e um termo independente, podemos portanto aplicar o teorema fatorial.<\/p>\n<p class=\"has-text-align-left\"> E para verificar se o polin\u00f4mio P(x) pode ser dividido pelo polin\u00f4mio Q(x), devemos encontrar o valor num\u00e9rico do polin\u00f4mio P(x) para o termo independente do polin\u00f4mio Q(x) mudou de sinal, \u00e9 ou seja, em x=-1:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-34b63772a1b44bee2c746d94b6ca4785_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} P(-1) &amp; =(-1)^3+5\\cdot (-1)^2+3\\cdot (-1)-1\\\\[2ex] &amp; = -1+5\\cdot 1+3\\cdot (-1)-1\\\\[2ex] &amp; = -1+5-3-1\\\\[2ex] &amp; = 0 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"315\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Neste problema, o valor num\u00e9rico do polin\u00f4mio em x=-1 \u00e9 zero, ent\u00e3o P(x) \u00e9 divis\u00edvel por Q(x).<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o, podemos deduzir pelo teorema fatorial que x=-1 \u00e9 uma raiz do polin\u00f4mio P(x), pois o valor num\u00e9rico de P(x) em x=-1 desaparece.<\/p>\n<p class=\"has-text-align-left\"> Assim, como x=-1 \u00e9 raiz do polin\u00f4mio P(x), para fator\u00e1-lo, basta dividi-lo por x+1. E, para isso, vamos utilizar o m\u00e9todo Ruffini: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/facteur-et-reste-theoreme-2.jpg\" alt=\"teorema do fator e resto\" class=\"wp-image-2223\" width=\"212\" height=\"130\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Ent\u00e3o o resultado da opera\u00e7\u00e3o \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7d359811cb1941ccb7181216d4eb2667_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+4x-1\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"88\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Podemos, portanto, fatorar o polin\u00f4mio P(x) da seguinte forma: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4793f7847d400b7df22361f6b856a0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) = (x+1) \\cdot (x^2+4x-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"231\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 3<\/h3>\n<p> Descubra com o teorema fatorial se o polin\u00f4mio P(x) \u00e9 divis\u00edvel pelo bin\u00f4mio Q(x) e, em caso afirmativo, encontre tamb\u00e9m uma raiz do polin\u00f4mio P(x) e fatore-a. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e3aead584139f397504eb04454974899_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^3+5x^2+4x-6 \\qquad \\qquad Q(x)=x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"372\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Neste caso, o polin\u00f4mio que divide Q(x) \u00e9 um bin\u00f4mio formado apenas por um x e um termo independente, portanto podemos utilizar o teorema do fator.<\/p>\n<p class=\"has-text-align-left\"> E para verificar se o polin\u00f4mio P(x) \u00e9 divis\u00edvel pelo polin\u00f4mio Q(x), devemos determinar o valor num\u00e9rico do polin\u00f4mio P(x) para o termo independente do polin\u00f4mio Q(x) mudou de sinal, ou seja- ou seja, em x = -3:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef8bb895fe041193d71351ffadb94f2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} P(-3) &amp; =(-3)^3+5\\cdot (-3)^2+4\\cdot (-3)-6\\\\[2ex] &amp; = -27+5\\cdot 9+4\\cdot (-3)-6\\\\[2ex] &amp; = -27+45-12-6\\\\[2ex] &amp; = 0 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"316\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Neste caso, o valor num\u00e9rico do polin\u00f4mio em x=-3 \u00e9 zero, ent\u00e3o de fato P(x) \u00e9 divis\u00edvel por Q(x).<\/p>\n<p class=\"has-text-align-left\"> Por isso, deduzimos do teorema fatorial que x=-3 \u00e9 raiz do polin\u00f4mio P(x), pois P(-3) \u00e9 igual a zero.<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o, como x=-3 \u00e9 raiz do polin\u00f4mio P(x), para fator\u00e1-lo devemos dividi-lo por x+3. E, para isso, utilizaremos a regra de Ruffini: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/facteur-theoreme-factorisation-par-ruffini.jpg\" alt=\"fator teorema fator por ruffini\" class=\"wp-image-2226\" width=\"216\" height=\"130\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Ent\u00e3o o resultado da divis\u00e3o \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-990e12f6ae2d0b9effdb52dfaea8edbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+2x-2\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"88\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E, portanto, podemos fatorar o polin\u00f4mio P(x) da seguinte maneira: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4929cce053644c03cdafec2fbfd77008_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) = (x+3) \\cdot (x^2+2x-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"231\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p class=\"has-text-align-center\"> O que voc\u00ea acha do teorema do fator? Voc\u00ea acha que \u00e9 \u00fatil em \u00e1lgebra? N\u00f3s lemos voc\u00ea nos coment\u00e1rios!<br \/> \ud83d\udc40\u2b07\u2b07\u2b07\ud83d\udc40<\/p>\n<div id=\"ezoic-pub-ad-placeholder-176\" data-inserter-version=\"-1\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina explicamos o que \u00e9 o teorema do fator. Al\u00e9m disso, mostramos para que serve o teorema do fator: divisibilidade de polin\u00f4mios, encontrar ra\u00edzes, fatorar polin\u00f4mios, etc. Finalmente, voc\u00ea poder\u00e1 praticar exerc\u00edcios passo a passo sobre o teorema do fator. Qual \u00e9 o teorema do fator? Em matem\u00e1tica, o teorema do fator diz que &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/teorema-do-fator\/\"> <span class=\"screen-reader-text\">Teorema do fator<\/span> Leia mais &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[21],"tags":[],"class_list":["post-323","post","type-post","status-publish","format-standard","hentry","category-polinomios"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Teorema do fator - Matoridade<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/pt\/teorema-do-fator\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Teorema do fator - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina explicamos o que \u00e9 o teorema do fator. 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Em matem\u00e1tica, o teorema do fator diz que &hellip; Teorema do fator Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/teorema-do-fator\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T08:26:38+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/theoreme-des-facteurs.jpg\" \/>\n<meta name=\"author\" content=\"Equipe Mathoridade\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Equipe Mathoridade\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"8 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/teorema-do-fator\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/teorema-do-fator\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Teorema do fator\",\"datePublished\":\"2023-07-06T08:26:38+00:00\",\"dateModified\":\"2023-07-06T08:26:38+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/teorema-do-fator\/\"},\"wordCount\":1551,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Polin\u00f4mios\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/teorema-do-fator\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/teorema-do-fator\/\",\"url\":\"https:\/\/mathority.org\/pt\/teorema-do-fator\/\",\"name\":\"Teorema do fator - Matoridade\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/#website\"},\"datePublished\":\"2023-07-06T08:26:38+00:00\",\"dateModified\":\"2023-07-06T08:26:38+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/pt\/teorema-do-fator\/#breadcrumb\"},\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/pt\/teorema-do-fator\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/pt\/teorema-do-fator\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/pt\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Teorema do fator\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathority.org\/pt\/#website\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"name\":\"Mathority\",\"description\":\"Onde a curiosidade encontra o c\u00e1lculo!\",\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathority.org\/pt\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"pt-BR\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathority.org\/pt\/#organization\",\"name\":\"Mathority\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"contentUrl\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"width\":703,\"height\":151,\"caption\":\"Mathority\"},\"image\":{\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\",\"name\":\"Equipe Mathoridade\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"caption\":\"Equipe Mathoridade\"},\"sameAs\":[\"http:\/\/mathority.org\/pt\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Teorema do fator - Matoridade","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/pt\/teorema-do-fator\/","og_locale":"pt_BR","og_type":"article","og_title":"Teorema do fator - Matoridade","og_description":"Nesta p\u00e1gina explicamos o que \u00e9 o teorema do fator. Al\u00e9m disso, mostramos para que serve o teorema do fator: divisibilidade de polin\u00f4mios, encontrar ra\u00edzes, fatorar polin\u00f4mios, etc. Finalmente, voc\u00ea poder\u00e1 praticar exerc\u00edcios passo a passo sobre o teorema do fator. Qual \u00e9 o teorema do fator? 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