{"id":56,"date":"2023-09-17T07:25:02","date_gmt":"2023-09-17T07:25:02","guid":{"rendered":"https:\/\/mathority.org\/pt\/exemplos-e-exercicios-do-teorema-do-resto-resolvidos\/"},"modified":"2023-09-17T07:25:02","modified_gmt":"2023-09-17T07:25:02","slug":"exemplos-e-exercicios-do-teorema-do-resto-resolvidos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/exemplos-e-exercicios-do-teorema-do-resto-resolvidos\/","title":{"rendered":"Teorema do resto (ou res\u00edduo)"},"content":{"rendered":"<p>Aqui voc\u00ea encontrar\u00e1 a explica\u00e7\u00e3o do que \u00e9 o teorema do resto (ou teorema do resto) e como ele \u00e9 aplicado aos polin\u00f4mios. Voc\u00ea tamb\u00e9m poder\u00e1 ver exemplos e, al\u00e9m disso, praticar com exerc\u00edcios resolvidos passo a passo sobre o teorema do resto. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%C2%BFQue-es-el-teorema-del-resto\"><\/span> Qual \u00e9 o teorema do resto? <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p class=\"has-background\" style=\"background-color:#ffebee\"> Em matem\u00e1tica, o <strong>teorema do resto<\/strong> diz que o resto da divis\u00e3o de qualquer polin\u00f4mio P(x) por outro polin\u00f4mio da forma (xa) \u00e9 igual ao valor num\u00e9rico do polin\u00f4mio P(x) para o valor x=a, Em em outras palavras, o resto da divis\u00e3o P(x):(xa) \u00e9 equivalente a P(a). <\/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-restes.jpg\" alt=\"teorema do resto\" class=\"wp-image-1041\" width=\"202\" height=\"203\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejemplo-del-teorema-del-resto\"><\/span>Exemplo do teorema do resto<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de vermos o que \u00e9 o teorema do resto, vejamos um exemplo pr\u00e1tico de sua aplica\u00e7\u00e3o:<\/p>\n<ul>\n<li> Calcule o restante da divis\u00e3o entre os dois polin\u00f4mios a seguir:<\/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-eba049cd57efaf6f586d13f45f82f98b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) = x^3+2x^2-4x+3 \\qquad \\qquad Q(x)=x-1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"371\" style=\"vertical-align: -5px;\"><\/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-d4fc4381966c56691db34f3b902a9fec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{P(x)}{Q(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"38\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Para encontrar o resto (ou res\u00edduo) da divis\u00e3o polinomial podemos aproveitar o teorema do resto, pois neste caso o polin\u00f4mio divisor \u00e9 da forma (xa), ou seja, \u00e9 de primeiro grau, o coeficiente de a vari\u00e1vel x \u00e9 1 e tem um termo independente.<\/p>\n<p> Ent\u00e3o aplicamos o teorema do resto, que diz que o resto de uma divis\u00e3o como esta \u00e9 igual ao valor num\u00e9rico do polin\u00f4mio do dividendo avaliado no termo independente do polin\u00f4mio divisor com sinal alterado, ou seja, P(1). <\/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-reste-et-du-facteur-pdf.jpg\" alt=\"teorema dos restos e fatores pdf\" class=\"wp-image-1048\" width=\"476\" height=\"118\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Portanto, para encontrar o resto da divis\u00e3o, precisamos avaliar o polin\u00f4mio 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-ff03f53066d698ee3d76e0024f3b51ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} P(1) &amp;= 1^3+2\\cdot 1^2-4\\cdot 1+3\\\\[2ex] &amp;= 1+2\\cdot 1-4 \\cdot 1+3  \\\\[2ex] &amp; = 1+2-4+3 \\\\[2ex] &amp; =\\bm{2}  \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"219\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> <strong>O restante da divis\u00e3o entre os polin\u00f4mios \u00e9, portanto, 2<\/strong> .<\/p>\n<p> Por outro lado, tamb\u00e9m podemos verificar com <strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/regras-resolvidas-exemplos-exercicios-ruffini\/\">a regra de Ruffini para divis\u00e3o de polin\u00f4mios<\/a><\/span><\/strong> que o resto coincide com o resultado que encontramos: <\/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-reste-de-ruffini.jpg\" alt=\"Teorema do resto de Ruffini\" class=\"wp-image-1043\" width=\"252\" height=\"133\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Como voc\u00ea pode ver, \u00e9 muito mais r\u00e1pido e f\u00e1cil determinar o resto de uma divis\u00e3o de um polin\u00f4mio por um bin\u00f4mio com o teorema do resto do que com a regra de Ruffini, porque s\u00e3o realizados muito menos c\u00e1lculos. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Teorema-del-resto-y-del-factor\"><\/span> Teorema do resto e do fator<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Do teorema do resto e da defini\u00e7\u00e3o da raiz (ou zero) de um polin\u00f4mio podemos deduzir o teorema do fator. Portanto, o teorema do fator implica o seguinte:<\/p>\n<p class=\"has-background\" style=\"background-color:#ffebee\"> 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. E, neste caso, isso significa que a \u00e9 uma raiz ou zero do polin\u00f4mio P(x).<\/p>\n<p> Al\u00e9m disso, de acordo com o teorema do resto, isto significa que se um polin\u00f3mio \u00e9 divis\u00edvel por outro polin\u00f3mio, o resto da referida divis\u00e3o \u00e9 zero, uma vez que P(a)=0.<\/p>\n<p> Por exemplo, se tivermos um certo polin\u00f4mio:<\/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-f3d378a4916cbd7a268ed60de50589d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^2+2x-8\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"151\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Este polin\u00f4mio \u00e9 divis\u00edvel pelo bin\u00f4mio (x-2) porque P(2)=0:<\/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-1e90c14ff06cdfa041299e016051b1dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} P(2) &amp;= 2^2+2\\cdot 2-8\\\\[2ex] &amp;= 4+4-8 \\\\[2ex] &amp; =\\bm{0} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"100\" width=\"159\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Como x=2 cancela o polin\u00f4mio P(x), isso significa que x=2 \u00e9 uma raiz do referido polin\u00f4mio.<\/p>\n<p> E al\u00e9m disso, como P(2)=0, podemos saber gra\u00e7as ao teorema do resto que o resto da divis\u00e3o<\/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-ba6849f9b139a364e36b1e7f461969c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^2+2x-8}{x-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"90\" style=\"vertical-align: -12px;\"><\/p>\n<p> \u00e9 igual a 0. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejercicios-resueltos-del-teorema-del-resto\"><\/span> Exerc\u00edcios resolvidos do teorema do resto<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Para finalizar a compreens\u00e3o do teorema do resto, preparamos alguns exerc\u00edcios resolvidos passo a passo para que voc\u00ea possa praticar. Aconselhamos que voc\u00ea experimente primeiro o exerc\u00edcio e depois verifique se o fez corretamente.<\/p>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Encontre, pelo teorema do resto, o resto da divis\u00e3o polinomial<\/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-d4fc4381966c56691db34f3b902a9fec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{P(x)}{Q(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"38\" style=\"vertical-align: -17px;\"><\/p>\n<p> , sendo os polin\u00f4mios envolvidos na opera\u00e7\u00e3o: <\/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-4201895897fc514d2ec7ef49b43ca580_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) =x^3+4x^2-2x+1\\qquad \\qquad Q(x)=x-2\" 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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> O polin\u00f4mio divisor \u00e9 composto apenas por um termo de primeiro grau e um termo independente e, al\u00e9m disso, o coeficiente do termo de primeiro grau \u00e9 1. Podemos, portanto, utilizar o teorema do resto.<\/p>\n<p class=\"has-text-align-left\"> E para aplicar o teorema do resto, basta avaliar o polin\u00f4mio do dividendo no termo independente do polin\u00f4mio divisor com sinal alterado, ou seja, devemos calcular P(2).<\/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-23790b78a8463a23a7b8202ab544ade9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} P(2) &amp;= 2^3+4\\cdot 2^2-2\\cdot 2+1\\\\[2ex] &amp;=8+4\\cdot 4-2\\cdot 2+1  \\\\[2ex] &amp; = 8+16-4+1 \\\\[2ex] &amp; =\\bm{21}  \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"218\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>O restante da divis\u00e3o entre os dois polin\u00f4mios \u00e9, portanto, 21<\/strong> .<\/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> Dado o polin\u00f4mio<\/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-82d5f84d7abf17919f73133b0624418d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^4-2x^3+5x^2-3x+4 ,\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"251\" style=\"vertical-align: -5px;\"><\/p>\n<p> Encontre o resto obtido dividindo-o por cada um dos seguintes polin\u00f4mios: <\/p>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8d8cd22b173b522307e57fd84bcd5cf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\left(x-1 \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2dd9c5036ff9ff34b1ea263893f64e24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\left(x+1 \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"82\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-65e5cacdc716b7c9a064099e0ee78439_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\left(x+2 \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"82\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e2a7b8e26b769665de5041f278d15686_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\left(x-3 \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Como todos os polin\u00f4mios divis\u00f3rios satisfazem as condi\u00e7\u00f5es do teorema do resto, podemos usar este teorema para determinar o resto de cada divis\u00e3o: <\/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-f8ae7d7c667bf9ca6bd7417356756447_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\mathbf{A}\\bm{)} \\ P(1) &amp;= 1^4-2\\cdot 1^3+5\\cdot 1^2-3\\cdot 1+4\\\\[2ex] &amp;=1-2\\cdot 1+5\\cdot 1 -3 \\cdot 1+4 \\\\[2ex] &amp; = 1-2+5-3+4 \\\\[2ex] &amp; =\\bm{5} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"307\" style=\"vertical-align: 0px;\"><\/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-e2a31e7c1334f8d1a24ba246d0459e4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\mathbf{B}\\bm{)} \\ P(-1) &amp;= (-1)^4-2\\cdot (-1)^3+5\\cdot (-1)^2-3\\cdot (-1)+4\\\\[2ex] &amp;=1-2\\cdot (-1)+5\\cdot 1 -3 \\cdot (-1)+4 \\\\[2ex] &amp; = 1+2+5+3+4 \\\\[2ex] &amp; =\\bm{15} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"431\" style=\"vertical-align: 0px;\"><\/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-2e8191f6ce490a0786515d84efaf45ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\mathbf{C}\\bm{)} \\ P(-2) &amp;= (-2)^4-2\\cdot (-2)^3+5\\cdot (-2)^2-3\\cdot (-2)+4\\\\[2ex] &amp;=16-2\\cdot (-8)+5\\cdot 4 -3 \\cdot (-2)+4 \\\\[2ex] &amp; = 16+16+20+6+4 \\\\[2ex] &amp; =\\bm{62} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"430\" style=\"vertical-align: 0px;\"><\/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-2d2e1e17bbcf91abb36d8cad24cadf0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\mathbf{D}\\bm{)} \\ P(3) &amp;= 3^4-2\\cdot 3^3+5\\cdot 3^2-3\\cdot 3+4\\\\[2ex] &amp;=81-2\\cdot 27+5\\cdot 9 -3 \\cdot 3+4 \\\\[2ex] &amp; = 81-54+45-9+4 \\\\[2ex] &amp; =\\bm{67} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"308\" style=\"vertical-align: 0px;\"><\/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> Calcule quanto o par\u00e2metro deve valer<\/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-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> de modo que o resto da divis\u00e3o dos polin\u00f4mios<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4fc4381966c56691db34f3b902a9fec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{P(x)}{Q(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"38\" style=\"vertical-align: -17px;\"><\/p>\n<p> ser igual a 3, sendo ambos polin\u00f4mios: <\/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-39aa975b852cd190f864491d029e3d58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) =x^3-5x^2-mx+9 \\qquad \\qquad Q(x)=x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"379\" 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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Neste caso particular, o polin\u00f4mio divisor \u00e9 composto por um mon\u00f4mio de primeiro grau e um termo independente e, al\u00e9m disso, o coeficiente do mon\u00f4mio de primeiro grau \u00e9 1. Podemos, portanto, utilizar o teorema do resto.<\/p>\n<p class=\"has-text-align-left\"> E para usar o teorema do resto, basta substituir o termo independente do polin\u00f4mio divisor por uma mudan\u00e7a de sinal onde no polin\u00f4mio dividido h\u00e1 um x, devemos portanto resolver P(-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-693cd65a618f884d0dd1be2f20594229_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} P(-3) &amp;=(-3)^3-5\\cdot (-3)^2-m\\cdot (-3)+9\\\\[2ex] &amp;=-27-5\\cdot 9 -m\\cdot (-3)+9 \\\\[2ex] &amp; = -27-45+3m+9 \\\\[2ex] &amp; =3m-63 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Mas obviamente obtemos um resultado baseado na inc\u00f3gnita<\/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-d4a447f470995624a31e9fc621325af2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"20\" style=\"vertical-align: 0px;\"><\/p>\n<p> No entanto, a defini\u00e7\u00e3o do problema nos diz que o resto deve ser igual a tr\u00eas, ent\u00e3o devemos definir o resto igual a 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-c683d547456ad6fc7d9a9c123be3ce3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3m-63 = 3\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"96\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E finalmente, resolvemos a equa\u00e7\u00e3o: <\/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-dd907c2a7cc446f45435dd543851062a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3m = 3+63\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"97\" style=\"vertical-align: -2px;\"><\/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-89214c7e36a0809ce927b0de14579ff7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3m = 66\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"66\" style=\"vertical-align: 0px;\"><\/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-be9d6df593c2b5cd5333f8450a1e2afa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = \\cfrac{66}{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"59\" style=\"vertical-align: -12px;\"><\/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-8631ca446a606a9511ad9a1bbf07bf1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{m = 22}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 4<\/h3>\n<p> Determine com o teorema do fator e do resto se o polin\u00f4mio<\/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-a80be6e42ac3b3c6528958bbfa21f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p> \u00e9 divis\u00edvel pelo polin\u00f4mio <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-12d5846d8a96763047fb4c9f458420f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"><\/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-16c539cdb7c8de8425a9d943f4ffa4df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) =-2x^3-5x^2-x+2 \\qquad \\qquad Q(x)=x+2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"385\" 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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para que o polin\u00f4mio<\/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-a80be6e42ac3b3c6528958bbfa21f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p> ser divis\u00edvel pelo polin\u00f4mio<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8061e215a5d055a2cf14c44c4febfad5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p> a divis\u00e3o entre esses dois polin\u00f4mios deve ser exata e, portanto, o resto deve ser zero.<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o, como o polin\u00f4mio divisor \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-f8aa3ce3e7e742e73a1b997a758126f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x+2),\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -5px;\"><\/p>\n<p> Pelo teorema do fator e pelo teorema do resto, sabemos que o polin\u00f4mio<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a80be6e42ac3b3c6528958bbfa21f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p> ser\u00e1 divis\u00edvel pelo polin\u00f4mio<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8061e215a5d055a2cf14c44c4febfad5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p> se estiver preenchido<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ba038b936f6d3cecf0aac3be3a9fbcbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-2)=0.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -5px;\"><\/p>\n<p> Devemos, portanto, ver se esta igualdade se verifica: <\/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-921817de49081656ab2dd5a8fc6a97ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-2)=0 \\quad \\color{blue} \\bm{?}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"140\" style=\"vertical-align: -5px;\"><\/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-5e8793835809000092b15eaec3877c18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} P(-2) &amp;=-2\\cdot (-2)^3-5\\cdot (-2)^2-(-2)+2\\\\[2ex] &amp;=-2 \\cdot (-8) -5 \\cdot 4+2 +2\\\\[2ex] &amp; =16-20+2+2 \\\\[2ex] &amp; =0\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"329\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Na verdade, o resto da divis\u00e3o<\/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-d4fc4381966c56691db34f3b902a9fec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{P(x)}{Q(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"38\" style=\"vertical-align: -17px;\"><\/p>\n<p> \u00e9 igual a 0, ent\u00e3o o polin\u00f4mio<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a80be6e42ac3b3c6528958bbfa21f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p> Sim, \u00e9 divis\u00edvel pelo outro polin\u00f4mio<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-12d5846d8a96763047fb4c9f458420f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p> O que voc\u00ea acha da explica\u00e7\u00e3o? Voc\u00ea gostou? Esperemos! N\u00e3o se esque\u00e7a que voc\u00ea pode nos deixar suas sugest\u00f5es ou d\u00favidas nos coment\u00e1rios. \u2b07\u2b07\u2b07 Lemos todos voc\u00eas! \ud83d\ude01\ud83d\ude01<\/p>\n<div id=\"ezoic-pub-ad-placeholder-176\" data-inserter-version=\"-1\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Aqui voc\u00ea encontrar\u00e1 a explica\u00e7\u00e3o do que \u00e9 o teorema do resto (ou teorema do resto) e como ele \u00e9 aplicado aos polin\u00f4mios. Voc\u00ea tamb\u00e9m poder\u00e1 ver exemplos e, al\u00e9m disso, praticar com exerc\u00edcios resolvidos passo a passo sobre o teorema do resto. Qual \u00e9 o teorema do resto? Em matem\u00e1tica, o teorema do resto &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/exemplos-e-exercicios-do-teorema-do-resto-resolvidos\/\"> <span class=\"screen-reader-text\">Teorema do resto (ou res\u00edduo)<\/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-56","post","type-post","status-publish","format-standard","hentry","category-polinomios"],"yoast_head":"<!-- This site is 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