{"id":60,"date":"2023-09-17T07:22:32","date_gmt":"2023-09-17T07:22:32","guid":{"rendered":"https:\/\/mathority.org\/pt\/fracoes-algebricas-operacoes-simplificadas-adicao-subtracao-multiplicacao-divisao-exercicios-resolvidos\/"},"modified":"2023-09-17T07:22:32","modified_gmt":"2023-09-17T07:22:32","slug":"fracoes-algebricas-operacoes-simplificadas-adicao-subtracao-multiplicacao-divisao-exercicios-resolvidos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/fracoes-algebricas-operacoes-simplificadas-adicao-subtracao-multiplicacao-divisao-exercicios-resolvidos\/","title":{"rendered":"Fra\u00e7\u00f5es alg\u00e9bricas: simplifica\u00e7\u00e3o, opera\u00e7\u00f5es, exerc\u00edcios resolvidos,\u2026"},"content":{"rendered":"<p>Nesta p\u00e1gina explicamos o que s\u00e3o fra\u00e7\u00f5es alg\u00e9bricas, quando s\u00e3o equivalentes, como simplific\u00e1-las e como realizar opera\u00e7\u00f5es com fra\u00e7\u00f5es alg\u00e9bricas (adi\u00e7\u00e3o, subtra\u00e7\u00e3o, multiplica\u00e7\u00e3o e divis\u00e3o). Al\u00e9m disso, voc\u00ea poder\u00e1 ver passo a passo exerc\u00edcios resolvidos para fra\u00e7\u00f5es alg\u00e9bricas. Resumindo, aqui voc\u00ea encontrar\u00e1 tudo sobre fra\u00e7\u00f5es alg\u00e9bricas. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%C2%BFQue-son-las-fracciones-algebraicas\"><\/span> O que s\u00e3o fra\u00e7\u00f5es alg\u00e9bricas? <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p class=\"has-background\" style=\"background-color:#ffebee\"> Em matem\u00e1tica, uma <strong>fra\u00e7\u00e3o alg\u00e9brica<\/strong> \u00e9 uma fra\u00e7\u00e3o que possui um polin\u00f4mio no numerador e outro polin\u00f4mio no denominador. <\/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\/fractions-algebriques.jpg\" alt=\"fra\u00e7\u00f5es alg\u00e9bricas resolvidas\" class=\"wp-image-1594\" width=\"135\" height=\"135\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Por exemplo, a express\u00e3o fracion\u00e1ria acima consiste em uma fra\u00e7\u00e3o alg\u00e9brica porque seu numerador e denominador s\u00e3o compostos de polin\u00f4mios. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Fracciones-algebraicas-equivalentes\"><\/span> Equivalente de fra\u00e7\u00e3o alg\u00e9brica<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de sabermos a defini\u00e7\u00e3o de fra\u00e7\u00f5es alg\u00e9bricas, vamos ver quando duas dessas fra\u00e7\u00f5es s\u00e3o iguais.<\/p>\n<p> Matematicamente, duas <strong>fra\u00e7\u00f5es alg\u00e9bricas s\u00e3o equivalentes<\/strong> se a seguinte condi\u00e7\u00e3o for atendida: <\/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\/exemples-de-fractions-equivalentes-algebriques.jpg\" alt=\"exemplos de fra\u00e7\u00f5es alg\u00e9bricas equivalentes\" class=\"wp-image-1599\" width=\"454\" height=\"55\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Como exemplo, verificaremos se as 2 fra\u00e7\u00f5es alg\u00e9bricas a seguir s\u00e3o equivalentes:<\/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-6ba3a266ca555e28002e4c27378731ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x+3}{x^2+5x+6} \\qquad \\cfrac{1}{x+2}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"167\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Para determinar se as fra\u00e7\u00f5es s\u00e3o algebricamente iguais, multiplicamos seus termos transversalmente:<\/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-358c219d5ec5e8ba791c0f3f807d7f1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x+3)\\cdot (x+2) = (x^2+5x+6)\\cdot 1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"268\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Agora vamos calcular as multiplica\u00e7\u00f5es de 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-106b457477b09880c099437a8aaebd46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+2x+3x+6 = x^2+5x+6\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"242\" 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-48884f070eddcdf12f0f4389e4a4b44d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+5x+6 = x^2+5x+6\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"202\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Obtivemos a mesma express\u00e3o em ambos os lados da equa\u00e7\u00e3o, portanto s\u00e3o efetivamente duas fra\u00e7\u00f5es alg\u00e9bricas equivalentes. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Simplificar-fracciones-algebraicas\"><\/span> Simplifique fra\u00e7\u00f5es alg\u00e9bricas <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p class=\"has-background\" style=\"background-color:#ffebee\"> Para simplificar uma fra\u00e7\u00e3o alg\u00e9brica, primeiro voc\u00ea deve fatorar os polin\u00f4mios no numerador e no denominador e, em seguida, eliminar os fatores que eles t\u00eam em comum.<\/p>\n<p> Obviamente, para simplificar fra\u00e7\u00f5es alg\u00e9bricas, \u00e9 fundamental que voc\u00ea saiba <strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\">o que \u00e9 fatora\u00e7\u00e3o polinomial<\/a><\/span><\/strong> e como ela \u00e9 feita. Se voc\u00ea ainda n\u00e3o sabe como os polin\u00f4mios s\u00e3o fatorados ou n\u00e3o se lembra completamente, recomendo ir \u00e0 p\u00e1gina vinculada antes de continuar, caso contr\u00e1rio voc\u00ea mal entender\u00e1 o procedimento. Explica passo a passo como fatorar polin\u00f4mios e, al\u00e9m disso, voc\u00ea poder\u00e1 ver diversos exemplos e praticar com exerc\u00edcios resolvidos.<\/p>\n<p> Agora vamos ver como uma fra\u00e7\u00e3o alg\u00e9brica \u00e9 simplificada aplicando o m\u00e9todo de fatora\u00e7\u00e3o de polin\u00f4mios usando um exemplo:<\/p>\n<ul>\n<li> Simplifique a seguinte fra\u00e7\u00e3o alg\u00e9brica:<\/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-9d4a5d7aff1a98650ab53864b88f40f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^3+2x^2-x-2}{x^2-2x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"129\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Primeiro, fatoramos os polin\u00f4mios do numerador e denominador da fra\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-44fc5a57c780655bb62672a6bf0cf283_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{(x-1)(x+1)(x+2)}{(x-1)(x-1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"164\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> \u2b06(Se voc\u00ea n\u00e3o sabe como os polin\u00f4mios foram fatorados, d\u00ea uma olhada no link acima)\u2b06<\/p>\n<p> E uma vez fatorados os polin\u00f4mios, eliminamos os fatores comuns entre o numerador e o denominador, ou seja, removemos todos os termos que se repetem:<\/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-efa638e7cb20b509ce8671d761fb8e7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{\\cancel{(x-1)}(x+1)(x+2)}{\\cancel{(x-1)}(x-1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"164\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> A fra\u00e7\u00e3o alg\u00e9brica simplificada, portanto, fica assim:<\/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-ae61b98cdf3f19dc83782b9bfc4a6809_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{(x+1)(x+2)}{x-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"109\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Neste problema, os polin\u00f4mios da fra\u00e7\u00e3o alg\u00e9brica foram fatorados encontrando suas ra\u00edzes; entretanto, \u00e0s vezes um polin\u00f4mio pode ser fatorado diretamente usando o fator comum (m\u00e9todo muito mais r\u00e1pido). Neste link voc\u00ea ver\u00e1 o que significa <a href=\"https:\/\/mathority.org\/pt\/extrair-extrair-exercicios-exemplos-resolvidos-de-fator-comum\/\"><strong><span style=\"text-decoration: underline;\">extrair um fator comum<\/span><\/strong><\/a> de um polin\u00f4mio e descobrir\u00e1 <span style=\"text-decoration: underline;\">como simplificar uma fra\u00e7\u00e3o alg\u00e9brica<\/span> usando um fator comum. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Operaciones-con-fracciones-algebraicas\"><\/span> Opera\u00e7\u00f5es com fra\u00e7\u00f5es alg\u00e9bricas<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Como qualquer tipo de fra\u00e7\u00e3o, as opera\u00e7\u00f5es tamb\u00e9m podem ser realizadas com fra\u00e7\u00f5es alg\u00e9bricas. Especificamente, as fra\u00e7\u00f5es alg\u00e9bricas podem ser adicionadas, subtra\u00eddas, multiplicadas e divididas. Abaixo explicamos passo a passo com exemplos como cada tipo de opera\u00e7\u00e3o \u00e9 calculado. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Suma-y-resta-de-fracciones-algebraicas\"><\/span> Adi\u00e7\u00e3o e subtra\u00e7\u00e3o de fra\u00e7\u00f5es alg\u00e9bricas<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> O procedimento de adi\u00e7\u00e3o e subtra\u00e7\u00e3o de fra\u00e7\u00f5es alg\u00e9bricas \u00e9 praticamente id\u00eantico, por isso iremos analis\u00e1-los em conjunto. Primeiro veremos um exemplo de adi\u00e7\u00e3o de duas fra\u00e7\u00f5es alg\u00e9bricas, e a seguir estudaremos a diferen\u00e7a entre o m\u00e9todo de subtra\u00e7\u00e3o de fra\u00e7\u00f5es alg\u00e9bricas.<\/p>\n<h4 class=\"wp-block-heading\"> Adicionando fra\u00e7\u00f5es alg\u00e9bricas <\/h4>\n<p class=\"has-background\" id=\"block-1594a5f3-1254-4c9e-bbd8-f79bfe778fd8\" style=\"background-color:#ffebee\"> <strong>A adi\u00e7\u00e3o de fra\u00e7\u00f5es alg\u00e9bricas<\/strong> \u00e9 feita da mesma maneira que com fra\u00e7\u00f5es normais: primeiro reduza as fra\u00e7\u00f5es a um denominador comum e depois adicione os numeradores.<\/p>\n<p id=\"block-1395699d-4152-4cd4-8029-52152aa81ba2\"> Vamos ver como as fra\u00e7\u00f5es alg\u00e9bricas s\u00e3o adicionadas usando um 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-27852f0123fcecbe8be07c1cbe492d1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x}{x^2+2x+1} + \\cfrac{3x}{x^2+x}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"163\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p id=\"block-3672173b-c7e6-44a1-8522-63b08b168ddb\"> Primeiro fatoramos os denominadores das fra\u00e7\u00f5es: <\/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-d76a27785a43860878e20f9f19f8d85f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x}{(x+1)(x+1)} + \\cfrac{3x}{x(x+1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"197\" style=\"vertical-align: -17px;\"><\/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-ae0b117b67b96ab97cc34f0395452809_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x}{(x+1)^2} + \\cfrac{3x}{x(x+1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"151\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p id=\"block-816f8d98-19a3-4b75-8c9f-21fc82303588\"> Agora precisamos encontrar o <strong>lcm<\/strong> (m\u00ednimo m\u00faltiplo comum) <strong>dos denominadores<\/strong> para reduzir as fra\u00e7\u00f5es a um denominador comum. <\/p>\n<p class=\"has-background\" id=\"block-42b361c5-dc91-4928-bf6a-cbd2d278158d\" style=\"background-color:#fffde7\"> <strong>Dica:<\/strong> o lcm dos denominadores \u00e9 sempre formado a partir do produto dos <strong>fatores que eles t\u00eam em comum elevado ao maior expoente<\/strong> multiplicado pelos <strong>fatores n\u00e3o comuns<\/strong> .<\/p>\n<p> Por exemplo, no nosso caso<\/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-ae0b117b67b96ab97cc34f0395452809_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x}{(x+1)^2} + \\cfrac{3x}{x(x+1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"151\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> O divisor comum entre os denominadores elevados ao maior expoente \u00e9<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-431f3a0fe905f53a3bba14fdcd2184c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x+1)^2.\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"65\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E o fator n\u00e3o comum entre os denominadores \u00e9<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a9cc293b28f198c32e0356b52e2e23bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Portanto, o lcm dos denominadores neste caso \u00e9:<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8d532daf990ee1d21f27629c90976a48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x+1)^2 \\cdot x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"84\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p id=\"block-436a726a-70a7-40da-a208-7769b5622363\"> O lcm dos denominadores \u00e9, 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-61abd769b2c8924619fc1de9990437b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x+1)^2 \\cdot x,\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"88\" style=\"vertical-align: -5px;\"><\/p>\n<p> este ser\u00e1, portanto, o novo denominador das 2 fra\u00e7\u00f5es.<\/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-4ac9ff4f5390125209474030fdbc2a87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x}{(x+1)^2} + \\cfrac{3x}{x(x+1)} \\ \\longrightarrow \\ \\cfrac{}{x(x+1)^2} + \\cfrac{}{x(x+1)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"371\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p id=\"block-3640c36e-b28c-4faa-8152-67d85f2974c6\"> Depois de encontrar o denominador comum, devemos modificar os numeradores. Para fazer isso, seguimos o mesmo processo da adi\u00e7\u00e3o de fra\u00e7\u00f5es normais: para cada fra\u00e7\u00e3o dividimos o lcm<\/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-657731178831ef2525fd245d9ca550b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bigl( \\ x(x+1)^2 \\ \\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"96\" style=\"vertical-align: -7px;\"><\/p>\n<p> entre o denominador original e multiplique o resultado pelo numerador: <\/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-4f7789228cada95f0a43a23b00ee3e1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x(x+1)^2}{(x+1)^2} = \\cfrac{x\\cancel{(x+1)^2}}{\\cancel{(x+1)^2}} = \\color{red}\\bm{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"230\" style=\"vertical-align: -17px;\"><\/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-85ce335cfa9eeb59efcc2ea66bb7fc0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x(x+1)^2}{x(x+1)}= \\cfrac{\\cancel{x}(x+1)^\\cancel{2}}{\\cancel{x}\\cancel{(x+1)}}=\\color{blue} \\bm{x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"265\" style=\"vertical-align: -17px;\"><\/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-64f36b1034879522e19c809ed07216aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x}{(x+1)^2} + \\cfrac{3x}{x(x+1)} \\ \\longrightarrow \\ \\cfrac{x \\cdot \\color{red}\\bm{x} \\color{black} }{x(x+1)^2} + \\cfrac{3x \\cdot \\color{blue} \\bm{(x+1)} \\color{black}}{x(x+1)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"483\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p id=\"block-eb2ac5b7-1160-4287-8f77-09fa899a0782\"> Ent\u00e3o agora podemos juntar as duas fra\u00e7\u00f5es porque elas t\u00eam o mesmo denominador:<\/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-f2cb949a7a18557fb88da6579adfa641_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^2+3x(x+1)}{x(x+1)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"113\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p id=\"block-bda62c37-98b4-4d75-94f5-69cd7d6b2a00\"> Finalmente, operamos no numerador. Primeiro fazemos o produto do mon\u00f4mio e 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-3572c0c661f3766c98fec0ea7bef241a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^2 +3x\\cdot x+ 3x\\cdot 1 }{x(x+1)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"144\" style=\"vertical-align: -17px;\"><\/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-da3f5de85cee840ea70185242cd4e56d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^2 +3x^2 + 3x }{x(x+1)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"107\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> A seguir, adicionamos os termos semelhantes ao numerador:<\/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-0569c5469be3e30f70b75ca5cb648e19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{4x^2 + 3x }{x(x+1)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"73\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Normalmente j\u00e1 estar\u00edamos l\u00e1, mas se olharmos este problema de perto, podemos simplificar ainda mais a fra\u00e7\u00e3o alg\u00e9brica removendo um fator comum do numerador. Ainda: <\/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-c3a36483296f8ccd582c06b3d6f656df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x(4x + 3)}{x(x+1)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"74\" style=\"vertical-align: -17px;\"><\/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-fe08e4b3b69fc920878413f5f35724e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{\\cancel{x}(4x + 3)}{\\cancel{x}(x+1)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"74\" style=\"vertical-align: -17px;\"><\/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-d320996e579c62a49d8a244591243d16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{4x + 3}{(x+1)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"63\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> E assim j\u00e1 completamos a soma das duas fra\u00e7\u00f5es alg\u00e9bricas.<\/p>\n<h4 class=\"wp-block-heading\"> Subtra\u00e7\u00e3o de fra\u00e7\u00f5es alg\u00e9bricas <\/h4>\n<p class=\"has-background\" id=\"block-1594a5f3-1254-4c9e-bbd8-f79bfe778fd8\" style=\"background-color:#ffebee\"> Para <strong>subtrair fra\u00e7\u00f5es alg\u00e9bricas,<\/strong> devemos seguir um procedimento semelhante \u00e0 adi\u00e7\u00e3o de fra\u00e7\u00f5es alg\u00e9bricas: primeiro reduza as fra\u00e7\u00f5es a um denominador comum e depois subtraia os numeradores.<\/p>\n<p id=\"block-1395699d-4152-4cd4-8029-52152aa81ba2\"> Vamos ver como as fra\u00e7\u00f5es alg\u00e9bricas s\u00e3o subtra\u00eddas com um exemplo: <\/p>\n<p class=\"has-text-align-center\" id=\"block-626dfe43-b064-405b-a4e5-64f8f9d2f3b8\">\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-230f5d12513bd0aa34943bf0bd1bc662_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{2x}{x^2-x-6} - \\cfrac{4x-3}{(x+2)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"167\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p id=\"block-3672173b-c7e6-44a1-8522-63b08b168ddb\"> Primeiro, precisamos fatorar os denominadores das duas fra\u00e7\u00f5es:<\/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-a7203a6355aa93738a6e4da82bc98b82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{2x}{(x+2)(x-3)} - \\cfrac{4x-3}{(x+2)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"195\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p id=\"block-816f8d98-19a3-4b75-8c9f-21fc82303588\"> Tal como acontece com a subtra\u00e7\u00e3o de fra\u00e7\u00f5es normais, devemos agora calcular o <strong>lcm<\/strong> (m\u00ednimo m\u00faltiplo comum) <strong>dos denominadores<\/strong> para reduzir as fra\u00e7\u00f5es a um denominador comum. Neste caso, o lcm dos denominadores \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-7ef9edba310262ed53634436be5c90ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x+2)^2(x-3) ,\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"120\" style=\"vertical-align: -5px;\"><\/p>\n<p> este ser\u00e1, portanto, o novo denominador das 2 fra\u00e7\u00f5es.<\/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-79f2882bb4ddf2035540641da2bd5db9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{2x}{(x+2)(x-3)} - \\cfrac{4x-3}{(x+2)^2} \\ \\longrightarrow \\ \\cfrac{}{(x+2)^2(x-3)} + \\cfrac{}{(x+2)^2(x-3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"504\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p id=\"block-3640c36e-b28c-4faa-8152-67d85f2974c6\"> Agora aplicamos o mesmo processo de subtra\u00e7\u00e3o de fra\u00e7\u00f5es normais: para cada fra\u00e7\u00e3o dividimos o lcm<\/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-657731178831ef2525fd245d9ca550b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bigl( \\ x(x+1)^2 \\ \\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"96\" style=\"vertical-align: -7px;\"><\/p>\n<p> entre o denominador original e multiplique o resultado pelo numerador: <\/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-c9d52f52e2bff5eace93764529daef0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{(x+2)^2(x-3)}{(x+2)(x-3)} = \\cfrac{(x+2)^{\\cancel{2}}\\cancel{(x-3)}}{\\canel{(x+2)}\\cancel{(x-3)}} = \\color{red}\\bm{x+2}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"348\" style=\"vertical-align: -17px;\"><\/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-f2846ba14540fe74dd89606e3a527840_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{(x+2)^2(x-3)}{(x+2)^2}= \\cfrac{\\cancel{(x+2)^2}(x-3)}{\\cancel{(x+2)^2}}=\\color{blue} \\bm{x-3}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"355\" style=\"vertical-align: -17px;\"><\/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-d0cf478a0cf48972d8610efeebc10cd2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{2x}{(x+2)(x-3)} - \\cfrac{4x-3}{(x+2)^2} \\ \\longrightarrow \\ \\cfrac{2x\\cdot \\color{red}\\bm{(x+2)} \\color{black}}{(x+2)^2(x-3)} + \\cfrac{(4x-3)\\cdot \\color{blue} \\bm{(x-3)} \\color{black}}{(x+2)^2(x-3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"92\" width=\"580\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p id=\"block-eb2ac5b7-1160-4287-8f77-09fa899a0782\"> Agora juntamos as duas fra\u00e7\u00f5es alg\u00e9bricas, pois elas t\u00eam o mesmo denominador:<\/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-76d9503723192264d7c0525f1dc5a762_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{2x(x+2)-(4x-3)(x-3)}{(x+2)^2(x-3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"213\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p id=\"block-bda62c37-98b4-4d75-94f5-69cd7d6b2a00\"> E operamos no numerador. Primeiro resolvemos as multiplica\u00e7\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-9a0a7d0ab0b62fced99150257d2646d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{2x^2+4x-\\bigl[4x^2-12x-3x+9\\bigr]}{(x+2)^2(x-3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"252\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-background\" style=\"background-color:#fffde7\"> Um erro muito comum ao subtrair fra\u00e7\u00f5es alg\u00e9bricas \u00e9 esquecer de colocar par\u00eanteses ap\u00f3s realizar esta multiplica\u00e7\u00e3o. Isto seria um erro, pois o sinal negativo afeta todos os elementos resultantes do produto, e n\u00e3o apenas o primeiro termo.<\/p>\n<p> Realizamos as opera\u00e7\u00f5es entre par\u00eanteses:<\/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-afa68a0ebcbad358f5454b2d98a1b813_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{2x^2+4x-\\bigl[4x^2-15x+9\\bigr]}{(x+2)^2(x-3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"211\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Assim, gra\u00e7as ao sinal negativo, mudamos o sinal de todos os termos entre par\u00eanteses:<\/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-dedff86768aa41641373644dbd64e9c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{2x^2+4x-4x^2+15x-9}{(x+2)^2(x-3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"196\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> E, por fim, agrupamos mon\u00f4mios semelhantes: <\/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-c622a1e2620a264cb373c3b0e37b164d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{-2x^2+19x-9}{(x+2)^2(x-3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"129\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Multiplicacion-de-fracciones-algebraicas\"><\/span> Multiplica\u00e7\u00e3o de fra\u00e7\u00f5es alg\u00e9bricas <span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p class=\"has-background\" style=\"background-color:#ffebee\"> Para <strong>multiplicar fra\u00e7\u00f5es alg\u00e9bricas,<\/strong> primeiro fatoramos todos os polin\u00f4mios dessas fra\u00e7\u00f5es, depois multiplicamos os numeradores entre si e os denominadores entre si e, por fim, simplificamos a fra\u00e7\u00e3o obtida.<\/p>\n<p> Portanto, o produto das fra\u00e7\u00f5es alg\u00e9bricas \u00e9 calculado da mesma forma que o produto das fra\u00e7\u00f5es normais.<\/p>\n<p> A seguir, vamos ver como multiplicar duas fra\u00e7\u00f5es alg\u00e9bricas com um 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-b3c8c26c3bbb6d88c38bf4c8ef88da96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{3x}{x^2+x-2} \\cdot \\cfrac{x^2-6x+5}{x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"185\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Primeiro de tudo voc\u00ea tem que fatorar todos os polin\u00f4mios das fra\u00e7\u00f5es, tanto os numeradores quanto os denominadores:<\/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-05d90fbe961a4a9cb043923b03dad497_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{3x}{(x-1)(x+2)} \\cdot \\cfrac{(x-1)(x-5)}{x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"233\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Agora vamos multiplicar fra\u00e7\u00f5es. Para fazer isso, multiplicamos os numeradores e denominadores:<\/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-722813021488c38da6ce48b4a52e8d57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{3x \\cdot (x-1)(x-5)}{(x-1)(x+2)\\cdot (x+1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"177\" style=\"vertical-align: -17px;\"><\/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-184ba5c9f9ac4c307273d787e8448d83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{3x(x-1)(x-5)}{(x-1)(x+2)(x+1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"164\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> E por fim, simplificamos os fatores que se repetem no denominador e no numerador:<\/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-9e62697ec3544aa9fd4d668c8e6e8594_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{3x\\cancel{(x-1)}(x-5)}{\\cancel{(x-1)}(x+2)(x+1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"164\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> O resultado da multiplica\u00e7\u00e3o \u00e9 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-768cd7cd7f16a3c0481bfff2b1a56943_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{3x(x-5)}{(x+2)(x+1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"109\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> A fra\u00e7\u00e3o n\u00e3o pode ser mais simplificada. Ent\u00e3o j\u00e1 terminamos de multiplicar fra\u00e7\u00f5es alg\u00e9bricas. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Division-de-fracciones-algebraicas\"><\/span> Divis\u00e3o de fra\u00e7\u00f5es alg\u00e9bricas <span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p class=\"has-background\" style=\"background-color:#ffebee\"> Para calcular uma <strong>divis\u00e3o de fra\u00e7\u00f5es alg\u00e9bricas<\/strong> , primeiro fatoramos todos os polin\u00f4mios, depois multiplicamos as fra\u00e7\u00f5es transversalmente (o primeiro numerador pelo segundo denominador e o primeiro denominador pelo segundo numerador) e por fim, simplificamos a fra\u00e7\u00e3o alg\u00e9brica.<\/p>\n<p> Ent\u00e3o vamos ver melhor como duas fra\u00e7\u00f5es alg\u00e9bricas s\u00e3o divididas usando um 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-b6d69f56a5f8e07ece76cd2cc7af7758_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^3-7x-6}{2x^2-8} : \\cfrac{x^2+2x+1}{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"196\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> O primeiro passo para dividir duas fra\u00e7\u00f5es alg\u00e9bricas \u00e9 fatorar todos 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-545baef30b02392b2ff48771efc53723_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{(x+1)(x+2)(x-3)}{2(x-2)(x+2)} : \\cfrac{(x+1)^2}{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"243\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Agora precisamos dividir as fra\u00e7\u00f5es. Para isso, multiplicamos as fra\u00e7\u00f5es transversalmente, ou seja, multiplica-se o primeiro numerador pelo segundo denominador e o resultado ser\u00e1 o numerador da nova fra\u00e7\u00e3o, e, da mesma forma, multiplica-se o primeiro denominador pelo segundo numerador e o resultado ser\u00e1 o denominador da nova fra\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-b4acdd6cd4da76cb20c3cadc6a5c1d43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{(x+1)(x+2)(x-3)\\cdot 6}{2(x-2)(x+2)\\cdot (x+1)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"193\" style=\"vertical-align: -17px;\"><\/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-b7212e50bbac42806205c19de22a0551_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{6(x+1)(x+2)(x-3)}{2(x-2)(x+2)(x+1)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"180\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Simplificamos os fatores que se repetem no denominador e no numerador:<\/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-dae6e61f3285d1b55fb0aaac1b46137a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{6\\cancel{(x+1)}\\cancel{(x+2)}(x-3)}{2(x-2)\\cancel{(x+2)}(x+1)^\\cancel{2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"180\" style=\"vertical-align: -17px;\"><\/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-601bd6bee2dbd9229822ee5ed1200709_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{6(x-3)}{2(x-2)(x+1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"118\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> E podemos simplificar ainda mais a fra\u00e7\u00e3o, j\u00e1 que<\/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-1672db950e496c9affb548df70351d16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6:2 = 3.\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"69\" 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-44d5a5ff05275fd0222ff8c0569fdcba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{3(x-3)}{(x-2)(x+1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"109\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> A fra\u00e7\u00e3o n\u00e3o pode ser mais simplificada. Portanto, j\u00e1 dividimos as fra\u00e7\u00f5es alg\u00e9bricas. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejercicios-resueltos-de-fracciones-algebraicas\"><\/span> Exerc\u00edcios resolvidos sobre fra\u00e7\u00f5es alg\u00e9bricas<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A seguir oferecemos diversos exerc\u00edcios resolvidos passo a passo sobre fra\u00e7\u00f5es alg\u00e9bricas, para que voc\u00ea possa praticar e assim finalizar a compreens\u00e3o do conceito. N\u00e3o se esque\u00e7a que voc\u00ea pode nos perguntar qualquer d\u00favida abaixo nos coment\u00e1rios! \ud83d\udcac\ud83d\udcac\ud83d\udcac<\/p>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Determine se as seguintes fra\u00e7\u00f5es alg\u00e9bricas s\u00e3o equivalentes ou n\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-f740b5268b9119666bd42176a4f86842_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x+3}{x^2-9} \\qquad \\cfrac{1}{x-3} \\qquad \\cfrac{x-3}{x^2+2x-3}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"254\" 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\"> Para verificar se duas fra\u00e7\u00f5es alg\u00e9bricas s\u00e3o equivalentes, voc\u00ea deve multiplic\u00e1-las transversalmente e ver se obt\u00e9m uma igualdade. Portanto, primeiro verificaremos a primeira e a segunda fra\u00e7\u00f5es: <\/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-9318b0a40cd808d3abda878dc008bbd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x+3}{x^2-9}= \\cfrac{1}{x-3} \\quad ?\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"140\" 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-c81610b6d1d22e9611f533266d69b825_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x+3)\\cdot (x-3)=(x^2-9)\\cdot 1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"227\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Resolvemos a identidade not\u00e1vel \u00e0 esquerda da 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-b2818cb76662004522c70eea92249fad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-9=x^2-9\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"120\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u2705<\/p>\n<p class=\"has-text-align-left\"> Neste caso, obtivemos uma igualdade, portanto a primeira e a segunda fra\u00e7\u00f5es s\u00e3o algebricamente iguais.<\/p>\n<p class=\"has-text-align-left\"> Aplicamos agora o mesmo procedimento com a primeira e a terceira fra\u00e7\u00f5es alg\u00e9bricas: <\/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-ac874c5337fab5f612e80957bc114410_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x+3}{x^2-9}= \\cfrac{x-3}{x^2+2x-3} \\quad ?\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"189\" style=\"vertical-align: -14px;\"><\/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-919a4415bd1fdb158fa2d85693fdc234_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x+3)\\cdot (x^2+2x-3)=(x^2-9)\\cdot(x-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"321\" 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-d03f74b88827f2678256a955e86489de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^3+2x^2-3x+3x^2+6x-9=x^3-3x^2-9x+27\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"397\" 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-405dbd6c82dedfdad2ffafa7e39caff0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^3+5x^2+3x-9=x^3-3x^2-9x+27\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"308\" style=\"vertical-align: -2px;\"><\/p>\n<p> \u274c<\/p>\n<p class=\"has-text-align-left\"> Por\u00e9m, desta vez as fra\u00e7\u00f5es alg\u00e9bricas n\u00e3o satisfazem a equa\u00e7\u00e3o, ent\u00e3o a primeira e a terceira fra\u00e7\u00f5es s\u00e3o matematicamente diferentes.<\/p>\n<p class=\"has-text-align-left\"> Concluindo, a terceira fra\u00e7\u00e3o \u00e9 diferente da primeira fra\u00e7\u00e3o e, portanto, tamb\u00e9m \u00e9 desigual \u00e0 segunda fra\u00e7\u00e3o, uma vez que a primeira e a segunda fra\u00e7\u00f5es s\u00e3o equivalentes. <\/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-081129bba98116bf2c3236379c5fe973_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x+3}{x^2-9} = \\cfrac{1}{x-3} \\neq \\cfrac{x-3}{x^2+2x-3}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"230\" style=\"vertical-align: -14px;\"><\/p>\n<\/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> Simplifique as seguintes fra\u00e7\u00f5es alg\u00e9bricas: <\/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-fa89ac8c0a9e92441f00f58652927ad2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\cfrac{5x^2+10x}{11x}\" 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-82ba94360c06faf6048680b225f02cc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\cfrac{x^2-4}{x^2+2x-8}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"117\" style=\"vertical-align: -14px;\"><\/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-cb654d3fdf52ba7fce13cd0a69acc692_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\cfrac{x^3-2x^2-3x}{x^2-3x}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"135\" 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-abecf1c451ad906e117a46855f5cd7cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\cfrac{x^3-3x+2}{x^3+4x^2+x-6}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"157\" 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\"> Para simplificar uma fra\u00e7\u00e3o alg\u00e9brica, precisamos fatorar os polin\u00f4mios no numerador e no denominador e, em seguida, eliminar os fatores repetidos. Ainda: <\/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-3f06c8f3d861d237ca41232418bd3e17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\begin{array}{l} \\cfrac{5x^2+10x}{11x} =\\cfrac{5x(x+2)}{11x} = \\\\[4ex] =\\cfrac{5\\cancel{x}(x+2)}{11\\cancel{x}}= \\cfrac{\\bm{5(x+2)}}{\\bm{11}}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"235\" 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-f9577181669de9b9760dfe7ed8425e17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\begin{array}{l} \\cfrac{x^2-4}{x^2+2x-8} = \\cfrac{(x-2)(x+2)}{(x-2)(x+4)}= \\\\[4ex] = \\cfrac{\\cancel{(x-2)}(x+2)}{\\cancel{(x-2)}(x+4)}=\\cfrac{\\bm{x+2}}{\\bm{x+4}}}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"112\" width=\"283\" 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-04505e35cce382f2905db108961c6718_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\begin{array}{l} \\cfrac{x^3-2x^2-3x}{x^2-3x} =  \\cfrac{x(x+1)(x-3)}{x(x-3)}}= \\\\[4ex] = \\cfrac{\\cancel{x} (x+1) \\cancel{x-3}}{\\cancel{x}\\cancel{(x-3)}} = \\cfrac{x+1}{1} = \\\\[4ex] = \\bm{x+1}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"149\" width=\"311\" 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-68ca63836b70d9aa6731e3271247d681_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\begin{array}{l} \\cfrac{x^3-3x+2}{x^3+4x^2+x-6}=\\cfrac{(x-1)^2(x+2)}{(x-1)(x+3)(x+2)}= \\\\[4ex] = \\cfrac{(x-1)^{\\cancel{2}}\\cancel{(x+2)}}{\\cancel{(x-1)}(x+3)\\cancel{(x+2)}}=\\cfrac{\\bm{x-1}}{\\bm{x+3}}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"112\" width=\"378\" 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 as seguintes adi\u00e7\u00f5es e subtra\u00e7\u00f5es de fra\u00e7\u00f5es alg\u00e9bricas: <\/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-a96e5be1d4a8e7b216abe3f5a2bc0ddc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\cfrac{4}{x^2+2x} + \\cfrac{3x-2}{x^2-x-6}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"191\" style=\"vertical-align: -14px;\"><\/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-25e21ca9e99469748c58da61755e32ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\cfrac{4x}{x^3+2x^2+x} - \\cfrac{2}{x^2-3x-4}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"239\" style=\"vertical-align: -14px;\"><\/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-125740a48e020b23010873f17905c6ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\cfrac{7x}{x^2-4x+4} + \\cfrac{-5}{x-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"181\" style=\"vertical-align: -14px;\"><\/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-59ba454283d08de8fcc2e15d4967b00f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ x +\\cfrac{-3x}{x^2-4}  -  \\cfrac{2x^3-1}{2x^2+6x+4}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"230\" 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\"> Para somar (ou subtrair) fra\u00e7\u00f5es alg\u00e9bricas, devemos primeiro reduzir as fra\u00e7\u00f5es a um denominador comum e depois somar (ou subtrair) os numeradores. 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-6524d97070ae44570c7bbd75df0b6bb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\begin{array}{l} \\cfrac{4}{x^2+2x} + \\cfrac{3x-2}{x^2-x-6} = \\cfrac{4}{x(x+2)} + \\cfrac{3x-2}{(x+2)(x-3)} = \\\\[4ex] =\\cfrac{4\\cdot(x-3)}{x(x+2)\\cdot (x-3)} + \\cfrac{(3x-2)\\cdot x}{(x+2)(x-3)\\cdot x} = \\cfrac{4\\cdot(x-3) + (3x-2)\\cdot x}{x(x+2)(x-3)} = \\\\[4ex] = \\cfrac{4x-12 + 3x^2-2x}{x(x+2)(x-3)} = \\cfrac{  \\bm{3x^2+2x-12}}{\\bm{x(x+2)(x-3)}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"175\" width=\"572\" 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-b228a6d7ced30d4dfdca7fa7653cec0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\begin{array}{l} \\cfrac{4x}{x^3+2x^2+x} - \\cfrac{2}{x^2-3x-4} = \\cfrac{4x}{x(x+1)^2} - \\cfrac{2}{(x+1)(x-4)}= \\\\[4ex] = \\cfrac{4x \\cdot (x-4)}{x(x+1)^2 \\cdot (x-4)} - \\cfrac{2 \\cdot (x+1) \\cdot x}{(x+1)^2(x-4)\\cdot x}= \\cfrac{4x \\cdot (x-4) - 2 \\cdot (x+1) \\cdot x }{x(x+1)^2 (x-4) }= \\\\[4ex] = \\cfrac{4x^2 -16x - 2 \\cdot (x^2+x) }{x(x+1)^2 (x-4) }= \\cfrac{4x^2 -16x - 2x^2 - 2x }{x(x+1)^2  (x-4) } =\\\\[4ex] =\\cfrac{2x^2 -18x}{x(x+1)^2 (x-4)}=\\cfrac{x(2x -18)}{x(x+1)^2 (x-4)}= \\\\[4ex] = \\cfrac{\\bm{2x -18}}{\\bm{(x+1)^2 (x-4)}}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"307\" width=\"609\" 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-541ca3698314f502dae6b4144ff2180e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\begin{array}{l}\\cfrac{7x}{x^2-4x+4} + \\cfrac{-5}{x-2}=\\cfrac{7x}{(x-2)^2} + \\cfrac{-5}{x-2}} = \\\\[4ex] = \\cfrac{7x}{(x-2)^2} + \\cfrac{-5\\cdot (x-2)}{(x-2)\\cdot (x-2)}=\\cfrac{7x}{(x-2)^2} + \\cfrac{-5\\cdot (x-2)}{(x-2)^2}= \\\\[4ex] = \\cfrac{7x + [-5\\cdot (x-2)] }{(x-2)^2}  =\\cfrac{7x -5\\cdot (x-2) }{(x-2)^2} = \\\\[4ex] = \\cfrac{7x -5x+10 }{(x-2)^2} = \\cfrac{ \\bm{2x+10}}{\\bm{(x-2)^2 } } \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"242\" width=\"495\" 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-eba4fb225a87d253ea56ae18460f89a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\  \\begin{array}{l}  x +\\cfrac{-3x}{x^2-4}  -  \\cfrac{2x^3-1}{2x^2+6x+4}=\\cfrac{x}{1} +\\cfrac{-3x}{x^2-4}  -  \\cfrac{2x^3-1}{2x^2+6x+4}= \\\\[4ex] =x +\\cfrac{-3x}{(x-2)(x+2)}  -  \\cfrac{2x^3-1}{2(x+1)(x+2)}= \\\\[4ex] = \\cfrac{x\\cdot 2(x-2)(x+2)(x+1)}{1\\cdot 2(x-2)(x+2)(x+1)} \\ + \\ \\cfrac{-3x\\cdot 2(x+1)}{(x-2)(x+2)\\cdot 2(x+1)} \\ - \\  \\cfrac{(2x^3-1)\\cdot(x-2)}{2(x+1)(x+2)\\cdot (x+1)}= \\\\[4ex] = \\cfrac{ 2x(x-2)(x+2)(x+1)}{2(x-2)(x+2)(x+1)} \\ + \\ \\cfrac{-6x(x+1)}{2(x-2)(x+2)(x+1)} \\ - \\ \\cfrac{(2x^3-1)\\cdot(x-2)}{2(x-2)(x+2)(x+1)}= \\\\[4ex]= \\cfrac{ 2x^4+2x^3-8x^2-8x}{2(x-2)(x+2)(x+1)} \\ + \\ \\cfrac{-6x^2-6x}{2(x-2)(x+2)(x+1)} \\ - \\  \\cfrac{2x^4-4x^3-x+2}{2(x-2)(x+2)(x+1)} = \\\\[4ex] = \\cfrac{ 2x^4+2x^3-8x^2-8x -6x^2-6x  -  (2x^4-4x^3-x+2)}{2(x-2)(x+2)(x+1)} = \\\\[4ex] = \\cfrac{ 2x^4+2x^3-8x^2-8x -6x^2-6x  - 2x^4+4x^3+x-2}{2(x-2)(x+2)(x+1)} = \\\\[4ex] = \\cfrac{ \\bm{6x^3-14x^2-13x-2}}{\\bm{2(x-2)(x+2)(x+1)}}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"508\" width=\"711\" 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> Resolva as seguintes multiplica\u00e7\u00f5es e divis\u00f5es de fra\u00e7\u00f5es alg\u00e9bricas: <\/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-6d07300122444585669fcc2bf1b8d1e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\cfrac{x^2+5x+4}{7}\\cdot \\cfrac{x-1}{x^2-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"181\" 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-fa1e3c21cf05605c6a408df67189bd41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\cfrac{3x^2+15x+18}{3x}\\cdot \\cfrac{x^2+x-2}{x^3+3x^2-x-3}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"287\" style=\"vertical-align: -14px;\"><\/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-e9ff7ee82098d3295e3f3911f2e1ff8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\cfrac{3x}{x^2+10x+25}:\\cfrac{2x}{x^2-25}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"209\" style=\"vertical-align: -14px;\"><\/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-6534e92d7d90ee190d64290189008587_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\cfrac{x^2+8x+15}{4x}:\\cfrac{x^2+4x-5}{2x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"233\" 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\"> Para multiplicar fra\u00e7\u00f5es alg\u00e9bricas, devemos primeiro fatorar todos os polin\u00f4mios, depois multiplicar os numeradores e denominadores e, finalmente, simplificar a fra\u00e7\u00e3o resultante. <\/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-cc9600c8e95d957e9004296306ea25fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\begin{array}{l} \\cfrac{x^2+5x+4}{7}\\cdot \\cfrac{x-1}{x^2-1} = \\cfrac{(x+1)(x+4)}{7}\\cdot \\cfrac{x-1}{(x-1)(x+1)}\\\\[4ex] =\\cfrac{(x+1)(x+4)\\cdot (x-1)}{7 \\cdot (x-1)(x+1)}=\\cfrac{(x+1)(x+4) (x-1)}{7(x-1)(x+1)} = \\\\[4ex] = \\cfrac{\\cancel{(x+1)}(x+4)\\cancel{ (x-1)}}{7\\cancel{(x-1)}\\cancel{(x+1)}} = \\cfrac{\\bm{x+4}}{\\bm{7}}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"178\" width=\"452\" 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-71554d3bb6d51cfd8c3202606ca1e6e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\begin{array}{l}\\cfrac{3x^2+15x+18}{3x}\\cdot \\cfrac{x^2+x-2}{x^3+3x^2-x-3} = \\cfrac{3(x+2)(x+3)}{3x}\\cdot \\cfrac{(x-1)(x+2)}{(x-1)(x+1)(x+3)}= \\\\[4ex] =\\cfrac{3(x+2)(x+3)\\cdot (x-1)(x+2)}{3x\\cdot (x-1)(x+1)(x+3)}=\\cfrac{3(x+2)(x+3) (x-1)(x+2)}{3x (x-1)(x+1)(x+3)} = \\\\[4ex] = \\cfrac{\\cancel{3}(x+2)\\cancel{(x+3)} \\cancel{(x-1)}(x+2)}{\\cancel{3}x \\cancel{(x-1)}(x+1)\\cancel{(x+3)}} = \\cfrac{(x+2)(x+2)}{x (x+1)} = \\\\[4ex] = \\cfrac{\\bm{(x+2)^2}}{\\bm{x (x+1)}}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"244\" width=\"636\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Por outro lado, para dividir fra\u00e7\u00f5es alg\u00e9bricas primeiro fatoramos todos os polin\u00f4mios, depois multiplicamos as fra\u00e7\u00f5es transversalmente (o primeiro numerador pelo segundo denominador e o primeiro denominador pelo segundo numerador) e, por fim, simplificamos a fra\u00e7\u00e3o alg\u00e9brica encontrada. <\/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-8994adaa1df1f24822c8102c0d1e69c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\begin{array}{l} \\cfrac{3x}{x^2+10x+25}:\\cfrac{2x}{x^2-25}= \\cfrac{3x}{(x+5)^2}:\\cfrac{2x}{(x-5)(x+5)}=\\\\[4ex] = \\cfrac{3x\\cdot (x-5)(x+5)}{(x+5)^2\\cdot 2x}=\\cfrac{3x(x-5)(x+5)}{2x(x+5)^2 }= \\\\[4ex] =\\cfrac{3\\cancel{x}(x-5)\\cancel{(x+5)}}{2\\cancel{x}(x+5)^\\cancel{2}} = \\cfrac{\\bm{3(x-5)}}{\\bm{2(x+5)}}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"175\" width=\"453\" 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-961a9787bca20a2482c010586614793d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\begin{array}{l} \\cfrac{x^2+8x+15}{4x}:\\cfrac{x^2+4x-5}{2x^2} = \\cfrac{(x+3)(x+5)}{4x}:\\cfrac{(x-1)(x+5)}{2x^2}= \\\\[4ex] = \\cfrac{(x+3)(x+5)\\cdot 2x^2 }{4x \\cdot (x-1)(x+5)} = \\cfrac{2x^2 (x+3)(x+5)}{4x (x-1)(x+5)} = \\\\[4ex] = \\cfrac{2x^{\\cancel{2}}(x+3)\\cancel{ (x+5)}}{4\\cancel{x} (x-1)\\cancel{ (x+5)}} =\\cfrac{2x(x+3)}{4(x-1)} =  \\cfrac{\\bm{x(x+3)}}{\\bm{2(x-1)}}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"178\" width=\"524\" style=\"vertical-align: 0px;\"><\/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? Ou voc\u00ea tem alguma sugest\u00e3o? \ud83d\udcac Conte-nos o que voc\u00ea achou desta p\u00e1gina nos coment\u00e1rios! N\u00f3s lemos todos voc\u00eas! \ud83d\udc40 E n\u00e3o se esque\u00e7a que voc\u00ea tamb\u00e9m pode nos tirar todas as suas d\u00favidas! \u2754\ud83d\udc47\u2754\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 o que s\u00e3o fra\u00e7\u00f5es alg\u00e9bricas, quando s\u00e3o equivalentes, como simplific\u00e1-las e como realizar opera\u00e7\u00f5es com fra\u00e7\u00f5es alg\u00e9bricas (adi\u00e7\u00e3o, subtra\u00e7\u00e3o, multiplica\u00e7\u00e3o e divis\u00e3o). Al\u00e9m disso, voc\u00ea poder\u00e1 ver passo a passo exerc\u00edcios resolvidos para fra\u00e7\u00f5es alg\u00e9bricas. Resumindo, aqui voc\u00ea encontrar\u00e1 tudo sobre fra\u00e7\u00f5es alg\u00e9bricas. O que s\u00e3o fra\u00e7\u00f5es alg\u00e9bricas? Em matem\u00e1tica, uma fra\u00e7\u00e3o &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/fracoes-algebricas-operacoes-simplificadas-adicao-subtracao-multiplicacao-divisao-exercicios-resolvidos\/\"> <span class=\"screen-reader-text\">Fra\u00e7\u00f5es alg\u00e9bricas: simplifica\u00e7\u00e3o, opera\u00e7\u00f5es, exerc\u00edcios resolvidos,\u2026<\/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":[22],"tags":[],"class_list":["post-60","post","type-post","status-publish","format-standard","hentry","category-representacao-de-funcao"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Fra\u00e7\u00f5es alg\u00e9bricas: simplifica\u00e7\u00e3o, opera\u00e7\u00f5es, exerc\u00edcios resolvidos, etc. -<\/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\/fracoes-algebricas-operacoes-simplificadas-adicao-subtracao-multiplicacao-divisao-exercicios-resolvidos\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Fra\u00e7\u00f5es alg\u00e9bricas: simplifica\u00e7\u00e3o, opera\u00e7\u00f5es, exerc\u00edcios resolvidos, etc. -\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina explicamos o que s\u00e3o fra\u00e7\u00f5es alg\u00e9bricas, quando s\u00e3o equivalentes, como simplific\u00e1-las e como realizar opera\u00e7\u00f5es com fra\u00e7\u00f5es alg\u00e9bricas (adi\u00e7\u00e3o, subtra\u00e7\u00e3o, multiplica\u00e7\u00e3o e divis\u00e3o). Al\u00e9m disso, voc\u00ea poder\u00e1 ver passo a passo exerc\u00edcios resolvidos para fra\u00e7\u00f5es alg\u00e9bricas. Resumindo, aqui voc\u00ea encontrar\u00e1 tudo sobre fra\u00e7\u00f5es alg\u00e9bricas. O que s\u00e3o fra\u00e7\u00f5es alg\u00e9bricas? 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