{"id":59,"date":"2023-09-17T07:23:14","date_gmt":"2023-09-17T07:23:14","guid":{"rendered":"https:\/\/mathority.org\/pt\/extrair-extrair-exercicios-exemplos-resolvidos-de-fator-comum\/"},"modified":"2023-09-17T07:23:14","modified_gmt":"2023-09-17T07:23:14","slug":"extrair-extrair-exercicios-exemplos-resolvidos-de-fator-comum","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/extrair-extrair-exercicios-exemplos-resolvidos-de-fator-comum\/","title":{"rendered":"Extraia ou remova o fator comum"},"content":{"rendered":"<p>Nesta p\u00e1gina explicamos como obter (ou extrair) o fator comum de um polin\u00f4mio. Aqui voc\u00ea encontrar\u00e1 os diferentes tipos de fator comum e poder\u00e1 ver v\u00e1rios exemplos de como isso \u00e9 conseguido. Al\u00e9m disso, voc\u00ea pode treinar com exerc\u00edcios resolvidos passo a passo. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%C2%BFQue-es-el-factor-comun\"><\/span> Qual \u00e9 o fator comum? <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p class=\"has-background\" style=\"background-color:#ffebee\"> Em matem\u00e1tica, o <strong>fator comum<\/strong> \u00e9 um fator presente em todos os termos de um polin\u00f4mio, ou seja, o fator comum consiste em um n\u00famero ou letra que multiplica cada termo de um polin\u00f4mio.<\/p>\n<p> Como exemplo, identificaremos qual \u00e9 o fator comum do seguinte 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-55851b062b232cfea5dcd03351860f67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4x+4y\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"59\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> No n\u00famero 4 \u00e9 repetido em todos os termos do 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-85078c48d043da6cc82f7ccd94af281c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\color{red} \\bm{4} \\color{black} x + \\color{red} \\bm{4} \\color{black} y\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"191\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> O fator comum deste polin\u00f4mio \u00e9, portanto, igual a 4. <\/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-0f4364038f1bfdff454201e0a039ca86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\text{Factor com\\'un} = 4 {\\phantom{\\frac{1}{2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"139\" style=\"vertical-align: -1px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Como-sacar-o-extraer-factor-comun\"><\/span> Como obter (ou extrair) o fator comum<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de sabermos o significado do fator comum, vamos ver como obter o fator comum de um polin\u00f4mio.<\/p>\n<p class=\"has-background\" style=\"background-color:#ffebee\"> Quando dois ou mais termos de um polin\u00f4mio t\u00eam um fator comum, <strong>o fator comum pode ser tomado (ou extra\u00eddo)<\/strong> para transformar adi\u00e7\u00f5es ou subtra\u00e7\u00f5es do polin\u00f4mio em uma multiplica\u00e7\u00e3o.<\/p>\n<p> Isso pode parecer um pouco dif\u00edcil de escrever, ent\u00e3o vamos ver como extrair o fator comum de um polin\u00f4mio com um exemplo: <\/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\/extraire-ou-supprimer-le-facteur-commun-dun-polynome.png\" alt=\"extrair ou remover o fator comum de um polin\u00f4mio\" class=\"wp-image-1497\" width=\"155\" height=\"156\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Como voc\u00ea pode ver no exemplo, o n\u00famero 5 se repete no mon\u00f4mio 5x e no mon\u00f4mio 5y, ent\u00e3o o fator comum do polin\u00f4mio \u00e9 5. Assim, uma vez identificado o fator comum, podemos converter a soma dos mon\u00f4mios em um produtos.<\/p>\n<p class=\"has-background\" style=\"background-color:#fffde7\"> N\u00e3o esque\u00e7a de colocar os par\u00eanteses na hora de extrair o fator comum, pois o fator comum deve multiplicar todas as adendas.<\/p>\n<p> Retirar o fator comum \u00e9 a opera\u00e7\u00e3o inversa da <strong>propriedade distributiva<\/strong> , ou seja, estamos na verdade aplicando a propriedade distributiva ao contr\u00e1rio. Portanto, podemos sempre verificar se extra\u00edmos corretamente o fator comum realizando o processo inverso:<\/p>\n<ul>\n<li> Se aplicando a propriedade distributiva obtivemos o mesmo polin\u00f4mio desde o in\u00edcio, isso significa que obtivemos o fator comum corretamente.<\/li>\n<li> Por outro lado, quando o resultado da utiliza\u00e7\u00e3o da propriedade distributiva \u00e9 outro polin\u00f4mio diferente do polin\u00f4mio original, implica que cometemos um erro no processo de extra\u00e7\u00e3o do fator comum. <\/li>\n<\/ul>\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\/propriete-distributive-du-facteur-commun.jpg\" alt=\"fator comum e propriedade distributiva\" class=\"wp-image-1502\" width=\"317\" height=\"197\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejemplos-de-sacar-o-extraer-factor-comun\"><\/span> Exemplos de tomada (ou extra\u00e7\u00e3o) de fator comum<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Deixamos mais exemplos para finalizar a compreens\u00e3o do conceito de fator comum:<\/p>\n<h3 class=\"wp-block-heading\"> Exemplo 1<\/h3>\n<p> Como voc\u00ea pode ver neste exemplo, um fator comum pode ser extra\u00eddo de mais de dois termos ao mesmo tempo: <\/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\/calculateur-de-facteur-commun.jpg\" alt=\"calculadora de fator comum\" class=\"wp-image-1509\" width=\"231\" height=\"145\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h3 class=\"wp-block-heading\"> Exemplo 2<\/h3>\n<p> Voc\u00ea tamb\u00e9m pode extrair um fator comum das vari\u00e1veis (ou letras): <\/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\/extrait-du-facteur-commun-dans-les-expressions-algebriques.jpg\" alt=\"extrair fator comum em express\u00f5es alg\u00e9bricas\" class=\"wp-image-1512\" width=\"199\" height=\"145\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Neste caso, a letra x multiplica os dois termos do polin\u00f4mio, ent\u00e3o podemos simplificar a express\u00e3o alg\u00e9brica tomando a vari\u00e1vel x como fator comum.<\/p>\n<h3 class=\"wp-block-heading\"> Exemplo 3<\/h3>\n<p> Neste exemplo, o primeiro termo tem a vari\u00e1vel x elevada \u00e0 pot\u00eancia de 3 e no segundo termo o x \u00e9 elevado \u00e0 pot\u00eancia de 2, portanto ambos os termos t\u00eam dois x&#8217;s. O fator comum n\u00e3o \u00e9, portanto, apenas um x, mas x <sup>2<\/sup> : <\/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\/algebre-facteur-commun-de-x-au-carre.jpg\" alt=\"fator comum de \u00e1lgebra de x ao quadrado\" class=\"wp-image-1515\" width=\"190\" height=\"143\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Por outro lado, observe que <strong>se o fator comum do polin\u00f4mio coincide exatamente com um termo, ao extrairmos o fator comum devemos colocar 1 em seu lugar<\/strong> . Caso contr\u00e1rio, se n\u00e3o coloc\u00e1ssemos nada no seu lugar, n\u00e3o obter\u00edamos uma express\u00e3o equivalente. <\/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\/comment-extraire-ou-extraire-le-facteur-commun.jpg\" alt=\"como extrair ou encontrar um fator comum\" class=\"wp-image-1516\" width=\"284\" height=\"230\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h3 class=\"wp-block-heading\"> Exemplo 4<\/h3>\n<p> \u00c0s vezes, o fator comum n\u00e3o \u00e9 t\u00e3o \u00f3bvio e n\u00e3o \u00e9 diretamente vis\u00edvel, mas \u00e9 um <strong>divisor dos coeficientes<\/strong> dos mon\u00f4mios. Por exemplo, o fator comum no exemplo a seguir \u00e9 3, uma vez que a decomposi\u00e7\u00e3o fatorial de 6 e 9 cont\u00e9m 3: <\/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\/le-plus-grand-facteur-commun.jpg\" alt=\"fator comum m\u00e1ximo\" class=\"wp-image-1521\" width=\"235\" height=\"149\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Este tipo de fator comum \u00e9 denominado em alguns livros de \u00e1lgebra <em>fator comum m\u00e1ximo<\/em> , pois o fator comum \u00e9 ao mesmo tempo o m\u00e1ximo fator comum (MDC) dos coeficientes dos termos polinomiais.<\/p>\n<p> Se voc\u00ea chegou at\u00e9 aqui, significa que provavelmente j\u00e1 sabe como encontrar o fator comum de um polin\u00f4mio, perfeito. Por\u00e9m, voc\u00ea ainda n\u00e3o se perguntou <span style=\"text-decoration: underline;\">para que serve o fator comum?<\/span> Bem, uma aplica\u00e7\u00e3o do fator comum \u00e9 que ele \u00e9 usado para fatorar polin\u00f4mios. Se voc\u00ea ainda n\u00e3o sabe o que \u00e9, neste link voc\u00ea poder\u00e1 ver o que \u00e9 <a href=\"https:\/\/mathority.org\/pt\"><strong><span style=\"text-decoration: underline;\">fatora\u00e7\u00e3o polinomial<\/span><\/strong><\/a> e porque o fator comum \u00e9 t\u00e3o importante para realizar esta opera\u00e7\u00e3o polinomial. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Factor-comun-en-fracciones\"><\/span> Fator comum para fra\u00e7\u00f5es<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> O fator comum tamb\u00e9m \u00e9 muito \u00fatil para simplificar termos em fra\u00e7\u00f5es com polin\u00f4mios no numerador e no denominador.<\/p>\n<p> Para ver como isso \u00e9 feito, vamos simplificar a seguinte fra\u00e7\u00e3o como exemplo:<\/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-926e853d2437425b652f2c47844cf056_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{2x^2+4}{14x+8y}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"68\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> A primeira coisa que precisamos fazer \u00e9 determinar o fator comum do polin\u00f4mio do numerador e do polin\u00f4mio do denominador. Neste caso, o fator comum dos dois polin\u00f4mios \u00e9 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-8dc50e00c20a758293d8d2dc74fb8f7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{\\color{red} \\bm{2} \\color{black}\\cdot x^2+\\color{red} \\bm{2} \\color{black}\\cdot 2 }{\\color{red} \\bm{2} \\color{black}\\cdot 7x+\\color{red} \\bm{2} \\color{black} \\cdot 4y}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"235\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Agora vamos extrair o fator comum dos dois 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-e66d7e737e76389a6294c4ececffbbd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{\\color{red} \\bm{2} \\color{black}\\left(x^2+2\\right)}{\\color{red} \\bm{2} \\color{black}\\left(7x+4y\\right)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"151\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> E uma vez obtido um fator comum para ambos os polin\u00f4mios, <strong>precisamos retirar os fatores que se repetem no numerador e no denominador<\/strong> :<\/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-5bcc4a8a84c802f1510baebfc8f4f126_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{\\cancel{2}\\left(x^2+2\\right)}{ \\cancel{2}\\left(7x+4y\\right)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"85\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Concluindo, a fra\u00e7\u00e3o simplificada \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-f035f7e96fe7afd4c93cef5a93c249f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^2+2}{7x+4y}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"59\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Factor-comun-por-agrupacion\"><\/span> Fator comum por agrupamento<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Uma maneira de reduzir os termos de um polin\u00f4mio \u00e9 usar o <strong>m\u00e9todo do fator comum agrupando os termos<\/strong> , tamb\u00e9m chamado de extra\u00e7\u00e3o de fator comum duplo. Como o pr\u00f3prio nome sugere, este procedimento consiste em simplificar a express\u00e3o de um polin\u00f4mio agrupando seus termos duas vezes.<\/p>\n<p> Este m\u00e9todo \u00e9 um pouco complexo, ent\u00e3o vamos ver como ele \u00e9 aplicado passo a passo com o seguinte 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-fb6a8ed826038a265d2b21c9d086a8ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+2x+5x+10\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"139\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Primeiro precisamos determinar dois poss\u00edveis fatores comuns diferentes, ent\u00e3o separamos o polin\u00f4mio em duas partes:<\/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-d9dfcd08553e61ba0be3791c055bd0ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x^2+2x)+(5x+10)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"164\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Neste caso, os elementos x <sup>2<\/sup> e 2x t\u00eam a letra x como fator comum, e os termos 5x e 10 t\u00eam 5 como fator comum (j\u00e1 que 10 \u00e9 m\u00faltiplo de 5). Portanto, temos esses dois fatores em comum:<\/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-96393c2563ce4dcedade0d979a6ce9c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x\\cdot x+2\\cdot x)+(5\\cdot x+5\\cdot 2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"218\" 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-859b9fdc443c61344efe64ea1c561565_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x(x+2)+5(x+2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E finalmente, como os dois produtos polinomiais restantes t\u00eam o fator (x+2), podemos simplificar o polin\u00f4mio 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-730e68f7964706bfd88cdf23c0c7376f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x+2)\\cdot (x+5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Como voc\u00ea pode ver, esse m\u00e9todo n\u00e3o \u00e9 nada f\u00e1cil. Portanto, n\u00e3o hesite em nos fazer qualquer pergunta nos coment\u00e1rios, e n\u00f3s responderemos o mais breve poss\u00edvel. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejercicios-resueltos-paso-a-paso-de-factor-comun\"><\/span> Exerc\u00edcios de fatores comuns resolvidos passo a passo<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Deixamos v\u00e1rios exerc\u00edcios resolvidos passo a passo para que voc\u00ea possa praticar a extra\u00e7\u00e3o do fator comum de um polin\u00f4mio.<\/p>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Extraia o fator comum dos seguintes 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-d0ebbf8633cd83e917eef6b61be195f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ 6a+6b\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" 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-8e6c228a58b589aea75d96ac06489ce8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ 3x^2+8x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"93\" 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-6b2f716764369b8748180f11c2088405_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ 2x-4\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" 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-4fe0000a9ab02d158e8e5c238614f2a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ -5x^4+x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"113\" 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\"> <strong>A)<\/strong> Todos os termos que comp\u00f5em o primeiro polin\u00f4mio possuem 6, ent\u00e3o o fator comum do polin\u00f4mio \u00e9 6:<\/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-dcfc02b18324ae47e4f6161c02ed10a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6a+6b = 6\\cdot a + 6 \\cdot b = \\bm{6\\cdot (a+b)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"259\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>B)<\/strong> No segundo polin\u00f4mio, todos os seus elementos possuem pelo menos uma letra x. Aqui est\u00e1 o fator comum do 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-d5397f749d2af52e3d25f47ac6763236_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x^2+8x = 3x \\cdot x+8\\cdot x= \\bm{x\\cdot (3x+8)}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"296\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>C)<\/strong> O primeiro mon\u00f4mio do polin\u00f4mio obviamente tem 2, e o segundo mon\u00f4mio \u00e9 um m\u00faltiplo de 2. Portanto, o fator comum do polin\u00f4mio \u00e9 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-403fd331c25d97df08b1a6e265b874f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x-4= 2\\cdot x - 2\\cdot 2 =\\bm{2\\cdot (x-2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"256\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>D)<\/strong> No \u00faltimo polin\u00f4mio todas as vari\u00e1veis s\u00e3o pelo menos elevadas ao quadrado. O fator comum \u00e9, portanto, x <sup>2<\/sup> :<\/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-9f5b04e6c70d455f570dea79a1caf289_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-5x^4+x^2= -5x^2\\cdot x^2 +1 \\cdot x^2 = \\bm{x^2\\cdot (-5x^2+1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"373\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Lembre-se que quando o fator comum \u00e9 id\u00eantico a um termo, deve-se colocar 1 em seu lugar.<\/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> Considere o fator comum dos seguintes 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-6e62899cc63709635736efb76c383ec0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ 8x^2 + 10y^3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"109\" 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-9c467706436dd57e76ccd016cd70110a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ 5x^3-2x^2+4x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"141\" 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-16d9856d9e0d050d484ce1def25a9757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ 25x^5+15x^3-20\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"158\" 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-a03f69699de681226d47227124a68a5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\  9x^4-3x^3-21x^2-6x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"199\" 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\"> <strong>A)<\/strong> Todos os coeficientes dos elementos que comp\u00f5em o primeiro polin\u00f4mio s\u00e3o m\u00faltiplos de 2, portanto, ao extrair o fator comum o polin\u00f4mio fica:<\/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-7862ee57cf098358e43d3b669e0cb54d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l} 8x^2 + 10y^3  = \\\\[2ex] = 2\\cdot 4x^2 +2\\cdot 5y^3 = \\\\[2ex] = \\bm{2\\left(4x^2+5y^3\\right)} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"154\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>B)<\/strong> Em todos os termos do polin\u00f4mio existe pelo menos um x, portanto:<\/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-39551a1ffc0b98f8b9c037fdcc793b01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}5x^3-2x^2+4x = \\\\[2ex] = 5x^2\\cdot x-2x\\cdot x+4\\cdot x= \\\\[2ex] =\\bm{x\\left(5x^2-2x+4\\right)}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"203\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>C)<\/strong> O m\u00e1ximo divisor comum dos coeficientes de todos os termos do polin\u00f4mio \u00e9 5, de modo que o divisor comum do referido polin\u00f4mio \u00e9 5:<\/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-fe8cc019c3511734ff278f68c44066df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}25x^5+15x^3-20 = \\\\[2ex] =5\\cdot 5x^5+5\\cdot 3x^3-5\\cdot 4 = \\\\[2ex] = \\bm{5\\left(5x^5+3x^3-4\\right)}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"207\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>D)<\/strong> Todos os termos do polin\u00f4mio possuem pelo menos um x e, al\u00e9m disso, todos os coeficientes s\u00e3o m\u00faltiplos de 3. Portanto, o fator comum do polin\u00f4mio \u00e9 3x: <\/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-fb70701f39b1050eec730f1591671293_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}9x^4-3x^3-21x^2-6x = \\\\[2ex] = 3x^3\\cdot 3x-x^2\\cdot 3x-7x\\cdot 3x-2\\cdot 3x= \\\\[2ex] = \\bm{3x\\left(3x^3-x^2-7x-2 \\right)}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"301\" 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> Encontre o fator comum de cada um dos seguintes polin\u00f4mios e subtraia-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-1d40bfaef04f667f19afd89bed2c5c6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ 4a^2b^5+7a^4b^3-10a^6b^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"202\" 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-be1557edebbdaa17ea43aa69cf14a8c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ 16x^4y^7z+8x^2y^2z^2+ 24x^3y^5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"243\" 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-d493a7783ca2a21e55e60bcf2af7748d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ 6ab^2c^4-6ab^2c+12a^3b^2c\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"217\" 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-d2cc5c9bbb43ac4387ad806fefc4d643_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\  18x^2y+10xy-5xy^3+4z\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"227\" 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\"> <strong>A)<\/strong> Todos os mon\u00f4mios t\u00eam pelo menos a letra<\/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-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> ao quadrado e a letra<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> ao cubo, ent\u00e3o o fator comum \u00e9<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f369f018d950bc237c46322c9b05d60b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^2b^3:\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"41\" 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-b622844f35f4bd4411c545c352331943_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l} 4a^2b^5+7a^4b^3-10a^6b^4 = \\\\[2ex] = 4b^2\\cdot a^2b^3+7a^2\\cdot a^2b^3-10a^4b\\cdot a^2b^3 = \\\\[2ex] = \\bm{a^2b^3\\left(4b^2+7a^2-10a^4b\\right)} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"308\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>B)<\/strong> Todos os coeficientes do polin\u00f4mio s\u00e3o m\u00faltiplos de 8 e, al\u00e9m disso, possuem pelo menos x <sup>2<\/sup> e y <sup>2<\/sup> como partes literais. O fator comum do polin\u00f4mio \u00e9, portanto, 8x <sup>2<\/sup> y <sup>2<\/sup> .<\/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-2d7018f427d800f6db67335a4d4f238b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}16x^4y^7z+8x^2y^2z^2+ 24x^3y^5 = \\\\[2ex] = 2x^2y^5z \\cdot 8x^2y^2 +z^2\\cdot 8x^2y^2+ 3xy^3\\cdot 8x^2y^2= \\\\[2ex] =\\bm{8x^2y^2\\left(2x^2y^5z+z^2+3xy^3\\right)}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"355\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>C)<\/strong> Neste caso, o fator comum coincide com o valor do mon\u00f4mio intermedi\u00e1rio<\/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-57647d998c69941eadc01ee28cf3e5cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(6ab^2c\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"54\" style=\"vertical-align: -7px;\"><\/p>\n<p> , uma vez que os coeficientes dos outros mon\u00f4mios s\u00e3o m\u00faltiplos de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b5c3e12330dabaeec7413281aba0f134_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> e absolutamente todo mundo tem<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4302e041ad21bfe7a384223fd3d35f1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"ab^2c:\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"41\" 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-76f91563726822a51c7bfe88aa461e6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}6ab^2c^4-6ab^2c+12a^3b^2c = \\\\[2ex] =c^3\\cdot 6ab^2c -1\\cdot 6ab^2c+2a^2 \\cdot 6ab^2c = \\\\[2ex] = \\bm{6ab^2c\\left(c^3-1+2a^2\\right)}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"292\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>D)<\/strong> Neste caso particular, o polin\u00f4mio n\u00e3o possui fator comum, pois nenhum fator se repete em todos os termos do polin\u00f4mio. Portanto, a express\u00e3o polinomial n\u00e3o pode ser simplificada algebricamente. <\/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-3c58c3237bfa4a57d2e87d36b1eebab0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"18x^2y+10xy-5xy^3+4z\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"200\" style=\"vertical-align: -4px;\"><\/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> Simplifique as seguintes fra\u00e7\u00f5es alg\u00e9bricas tomando o fator comum: <\/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-73e11e8ebb8c064d2981506461d693c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\cfrac{10x^2+30}{5x-20}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"103\" 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-401feeca45cc41913c5e9f64d2821535_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\cfrac{16x^2-8}{24x-32}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"95\" 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-b4354346c86ff374869cc9a6029bbcdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\cfrac{49x^3+7x}{35x^2-14}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"105\" 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-4221a799a5650e5afdab728cb2797be9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\  \\cfrac{8x^4+16x^3-4x^2}{12x^2+20x}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"161\" style=\"vertical-align: -14px;\"><\/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 procedimento para simplificar uma fra\u00e7\u00e3o alg\u00e9brica, ou seja, uma fra\u00e7\u00e3o com polin\u00f4mios, consiste em extrair o fator comum do numerador e denominador da fra\u00e7\u00e3o e, em seguida, eliminar os fatores que se repetem acima e abaixo. abaixo da fra\u00e7\u00e3o. ENT\u00c3O: <\/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-6ce28a7207aa3bccea5a8afae54c8ad5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\quad \\begin{array}{l} \\cfrac{10x^2+30}{5x-20}=  \\cfrac{5\\cdot 2x^2 +5\\cdot 6}{5\\cdot x-5\\cdot 4} = \\\\[4ex] = \\cfrac{5(2x^2+6)}{5(x-4)}= \\cfrac{\\cancel{5}(2x^2+6)}{\\cancel{5}(x-4)} = \\\\[4ex] = \\cfrac{\\bm{2x^2+6}}{\\bm{x-4}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"173\" width=\"276\" 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-a6024cf2af6c13d0d27f7e5d3e675362_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\quad \\begin{array}{l} \\cfrac{16x^2-8}{24x-32} = \\cfrac{8 \\cdot 2x^2+8 \\cdot (-1)}{8 \\cdot 3x-8 \\cdot 4} =\\\\[4ex] = \\cfrac{8(2x^2-1)}{8(3x-4)}= \\cfrac{\\cancel{8}(2x^2-1)}{\\cancel{8}(3x-4)} =\\\\[4ex] = \\cfrac{\\bm{2x^2-1}}{\\bm{3x-4}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"173\" width=\"292\" 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-5678b7c6d631a693a86fad40aa79d030_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\quad \\begin{array}{l}\\cfrac{49x^3+7x}{35x^2-14}=\\cfrac{7x\\cdot 7x^2+7x\\cdot 1}{7 \\cdot 5x^2+7\\cdot (-2)} =\\\\[4ex] = \\cfrac{7x(7x^2+1)}{7(5x^2-2)}= \\cfrac{\\cancel{7}x(7x^2+1)}{\\cancel{7}(5x^2-2)}=\\\\[4ex] = \\cfrac{\\bm{x(7x^2+1)}}{\\bm{5x^2-2}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"173\" width=\"301\" 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-82199184b3594c18db6888c58fa0bdfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\quad \\begin{array}{l} \\cfrac{8x^4+16x^3-4x^2}{12x^2+20x}=\\cfrac{4x^2\\cdot 2x^2+4x^2\\cdot 4x+4x^2\\cdot (-1)}{4x\\cdot 3x+4x\\cdot 5}=\\\\[4ex] = \\cfrac{4x^2(2x^2+4x-1)}{4x(3x+5)}= \\cfrac{\\cancel{4}x^{\\cancel{2}}(2x^2+4x-1)}{\\cancel{4}\\cancel{x}(3x+5)}=\\\\[4ex] = \\cfrac{\\bm{x(2x^2+4x-1)}}{\\bm{3x+5}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"175\" width=\"474\" 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 5<\/h3>\n<p> Tome o fator comum das seguintes express\u00f5es polinomiais: <\/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-8e5c91e0867615671956c23daf199f56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\cfrac{7}{3}x^2+ \\cfrac{8}{3}x-\\cfrac{2}{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"130\" 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-861358103d4265a73b6bf2c12a5c315d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\sqrt{12x^3+16}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"115\" 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-592bb1e112ee966a6c8266f8453176d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ x^2+4x-3x-12\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"163\" 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-6ea1c6302c0ee60f46718b42a2741bf3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\  \\cfrac{3}{4}x^2+ \\cfrac{1}{2}x-\\cfrac{5}{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"130\" style=\"vertical-align: -12px;\"><\/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\"> <strong>A)<\/strong> Todos os termos do polin\u00f4mio podem ser decompostos fatorialmente ao terceiro, portanto:<\/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-66174ecc5c7264ae95507f632d1f2630_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l} \\cfrac{7}{3}x^2+ \\cfrac{8}{3}x-\\cfrac{2}{3}= \\\\[3ex] = \\cfrac{1}{3}\\cdot 7x^2+ \\cfrac{1}{3}\\cdot 8x-\\cfrac{1}{3}\\cdot 2 = \\\\[3ex] = \\mathbf{\\cfrac{1}{3}}\\bm{\\left(7x^2+8x-2\\right)} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"156\" width=\"205\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>B)<\/strong> O fator comum do polin\u00f4mio dentro da raiz \u00e9 4, mas, ent\u00e3o, podemos gerar o fator comum calculando sua raiz quadrada:<\/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-3a362644fe5b22fefc962f164db26f5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\sqrt{12x^3+16}= \\\\[2ex] =\\sqrt{4\\cdot 3x^3+4\\cdot 4}= \\\\[2ex]=\\sqrt{4\\left(3x^3+4\\right)}= \\\\[2ex] =\\bm{2\\sqrt{3x^3+4}}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"133\" width=\"152\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>C)<\/strong> Neste polin\u00f4mio podemos aplicar o processo de extra\u00e7\u00e3o do fator comum agrupando:<\/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-fd43d5f40797c39b0401769ff392d9e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}x^2+4x-3x-12= \\\\[2ex] =x(x+4)-3(x+4) = \\\\[2ex] = \\bm{(x+4)(x-3)}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"95\" width=\"186\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>D)<\/strong> Todos os coeficientes fracion\u00e1rios do polin\u00f4mio s\u00e3o m\u00faltiplos de metade, ent\u00e3o o fator comum do polin\u00f4mio \u00e9 \u00bd. <\/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-a35ab3db2e1f1197d6652e6d4d0a8a8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle \\frac{3}{4}x^2+ \\frac{1}{2}x-\\frac{5}{6}= \\\\[4ex] \\displaystyle = \\frac{1}{2}\\cdot\\frac{3}{2}x^2+ \\frac{1}{2}\\cdot x-\\frac{1}{2}\\cdot\\frac{5}{3}= \\\\[4ex] \\displaystyle = \\mathbf{\\frac{1}{2}}\\bm{\\left(}\\mathbf{\\frac{3}{2}}\\bm{x^2+x-}\\mathbf{\\frac{5}{3}} \\bm{\\right)}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"166\" width=\"210\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p> \ud83d\udc47\ud83d\udc47\ud83d\udc47 O que voc\u00ea achou da explica\u00e7\u00e3o? Voc\u00ea gostou? Lembre-se tamb\u00e9m que se voc\u00ea tiver alguma d\u00favida sobre como \u00e9 determinado o fator comum de um polin\u00f4mio ou se n\u00e3o entender algum exerc\u00edcio, pode sempre nos perguntar nos coment\u00e1rios, e n\u00f3s responderemos. \ud83d\udc47\ud83d\udc47\ud83d\udc47<\/p>\n<div id=\"ezoic-pub-ad-placeholder-176\" data-inserter-version=\"-1\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina explicamos como obter (ou extrair) o fator comum de um polin\u00f4mio. Aqui voc\u00ea encontrar\u00e1 os diferentes tipos de fator comum e poder\u00e1 ver v\u00e1rios exemplos de como isso \u00e9 conseguido. Al\u00e9m disso, voc\u00ea pode treinar com exerc\u00edcios resolvidos passo a passo. Qual \u00e9 o fator comum? Em matem\u00e1tica, o fator comum \u00e9 um &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/extrair-extrair-exercicios-exemplos-resolvidos-de-fator-comum\/\"> <span class=\"screen-reader-text\">Extraia ou remova o fator comum<\/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-59","post","type-post","status-publish","format-standard","hentry","category-polinomios"],"yoast_head":"<!-- This site is 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