{"id":283,"date":"2023-07-06T21:11:26","date_gmt":"2023-07-06T21:11:26","guid":{"rendered":"https:\/\/mathority.org\/pt\/subtracao-de-polinomios-exemplos-exercicios-resolvidos-subtracao\/"},"modified":"2023-07-06T21:11:26","modified_gmt":"2023-07-06T21:11:26","slug":"subtracao-de-polinomios-exemplos-exercicios-resolvidos-subtracao","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/subtracao-de-polinomios-exemplos-exercicios-resolvidos-subtracao\/","title":{"rendered":"Subtra\u00e7\u00e3o de polin\u00f4mios"},"content":{"rendered":"<p>Nesta p\u00e1gina explicamos como subtrair polin\u00f4mios. Al\u00e9m disso, voc\u00ea encontrar\u00e1 diversos exemplos e exerc\u00edcios resolvidos passo a passo para subtra\u00e7\u00e3o de polin\u00f4mios. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%C2%BFComo-se-hace-la-resta-de-polinomios\"><\/span> Como subtrair polin\u00f4mios?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Para subtrair dois polin\u00f4mios, voc\u00ea deve subtrair os termos dos polin\u00f4mios que s\u00e3o semelhantes. Em outras palavras, subtrair polin\u00f4mios consiste em subtrair termos que possuem a mesma parte literal (mesmas vari\u00e1veis e mesmos expoentes).<\/strong><\/p>\n<p> Em matem\u00e1tica, voc\u00ea pode calcular a subtra\u00e7\u00e3o de polin\u00f4mios de duas maneiras diferentes: com o m\u00e9todo vertical ou com o m\u00e9todo horizontal. Abaixo est\u00e1 a explica\u00e7\u00e3o de ambos os procedimentos, mas aconselhamos que voc\u00ea aprenda primeiro como subtrair polin\u00f4mios verticalmente e depois passe para o m\u00e9todo horizontal. Obviamente fique com o que voc\u00ea preferir. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Resta-de-polinomios-vertical\"><\/span> Subtra\u00e7\u00e3o de polin\u00f4mios verticais<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A seguir, veremos como dois polin\u00f4mios s\u00e3o subtra\u00eddos verticalmente usando um exemplo:<\/p>\n<ul>\n<li> fa\u00e7a a subtra\u00e7\u00e3o\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fb1a50d4312d4058c8eb09943f235f0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) - Q(x),\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"101\" style=\"vertical-align: -5px;\"><\/p>\n<p> sendo os dois polin\u00f4mios:<\/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-a6c08cd58c0d1b7efc11c82847970390_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) = 7x^4+2x^3+5x-4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"208\" 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-064d41c149c568d4f3094a2b6ece595c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x) = 4x^4-3x^3+8x^2-2x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"255\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A primeira coisa que precisamos fazer para encontrar uma subtra\u00e7\u00e3o polinomial \u00e9 colocar um polin\u00f4mio abaixo do outro, de modo que os termos semelhantes dos dois polin\u00f4mios fiquem alinhados em colunas: <\/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\/soustraction-de-polynomes-exemples.jpg\" alt=\"exemplos de subtra\u00e7\u00e3o de polin\u00f4mios\" class=\"wp-image-509\" width=\"342\" height=\"83\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-background\" style=\"background-color:#fffde7\"> <strong>Aten\u00e7\u00e3o:<\/strong> Se um polin\u00f4mio n\u00e3o possuir termo de determinado grau, o espa\u00e7o deve ser deixado vazio. Por exemplo, o polin\u00f4mio<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8156aa8394d88c61cfead0f1bb502f23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=7x^4+2x^3+5x-4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"208\" style=\"vertical-align: -5px;\"><\/p>\n<p> n\u00e3o possui um mon\u00f4mio quadr\u00e1tico, ent\u00e3o h\u00e1 um espa\u00e7o vazio em seu lugar. <\/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\/soustraction-de-polynomes-vides.jpg\" alt=\"subtra\u00e7\u00e3o de polin\u00f4mios passo a passo\" class=\"wp-image-510\" width=\"342\" height=\"118\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Embora agora possamos subtrair polin\u00f4mios diretamente, \u00e9 muito f\u00e1cil errar o sinal se fizermos desta forma. Portanto, para subtrair polin\u00f4mios, \u00e9 melhor <strong>alterar o sinal de todos os termos do polin\u00f4mio subtrator<\/strong> (o polin\u00f4mio subtrativo) e depois fazer a adi\u00e7\u00e3o. J\u00e1 que subtrair um polin\u00f4mio \u00e9 o mesmo que adicionar seu polin\u00f4mio oposto. <\/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\/soustraire-polynomes-changer-de-signe.jpg\" alt=\"subtra\u00e7\u00e3o de polin\u00f4mios com expoentes diferentes\" class=\"wp-image-512\" width=\"342\" height=\"83\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> E uma vez organizados todos os termos em ordem do grau mais alto para o mais baixo e negados os termos do polin\u00f4mio abaixo, somamos os coeficientes de cada coluna mantendo as partes literais iguais: <\/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\/soustraction-de-polynomes.jpg\" alt=\"subtra\u00e7\u00e3o de polin\u00f4mios\" class=\"wp-image-514\" width=\"353\" height=\"131\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Portanto, o resultado obtido da subtra\u00e7\u00e3o dos 2 polin\u00f4mios \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-b4bd78f9b776ec568554534c61cbb463_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{P(x)-Q(x) = 3x^4+5x^3-8x^2+7x-5}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"315\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Caso o \u00faltimo passo n\u00e3o esteja claro para voc\u00ea, deixo-vos com a explica\u00e7\u00e3o de como \u00e9 feita a <a href=\"https:\/\/mathority.org\/pt\/soma-de-polinomios-exemplos-adicao-de-exercicios-resolvidos\/\"><strong><span style=\"text-decoration: underline;\">adi\u00e7\u00e3o de polin\u00f4mios<\/span><\/strong><\/a> , na verdade \u00e9 essencial que voc\u00ea domine a adi\u00e7\u00e3o de polin\u00f4mios para poder subtrair polin\u00f4mios com sucesso. Na p\u00e1gina vinculada voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos e exerc\u00edcios resolvidos de adi\u00e7\u00e3o de polin\u00f4mios e, al\u00e9m disso, poder\u00e1 ver as diferen\u00e7as entre adicionar e subtrair polin\u00f4mios. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Resta-de-polinomios-horizontal\"><\/span> Subtra\u00e7\u00e3o de polin\u00f4mios horizontais<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Acabamos de ver como subtrair polin\u00f4mios verticalmente, mas agora veremos o outro m\u00e9todo que existe para subtrair polin\u00f4mios: subtrair polin\u00f4mios horizontalmente. Este procedimento \u00e9 certamente mais r\u00e1pido que o anterior, por\u00e9m \u00e9 necess\u00e1rio ter um dom\u00ednio superior dos conceitos de polin\u00f4mios.<\/p>\n<p> Ent\u00e3o vamos ver em que consiste esse m\u00e9todo de subtra\u00e7\u00e3o de polin\u00f4mios atrav\u00e9s de um exemplo. E para que voc\u00ea possa ver as diferen\u00e7as entre os dois m\u00e9todos, subtrairemos os mesmos polin\u00f4mios do exemplo anterior:<\/p>\n<ul>\n<li> Calcule o restante\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fb1a50d4312d4058c8eb09943f235f0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) - Q(x),\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"101\" style=\"vertical-align: -5px;\"><\/p>\n<p> sendo os dois polin\u00f4mios:<\/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-a6c08cd58c0d1b7efc11c82847970390_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) = 7x^4+2x^3+5x-4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"208\" 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-064d41c149c568d4f3094a2b6ece595c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x) = 4x^4-3x^3+8x^2-2x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"255\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Devemos primeiro colocar os dois polin\u00f4mios na forma de uma opera\u00e7\u00e3o alg\u00e9brica, ou seja, um ap\u00f3s o outro: <\/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\/exemple-de-soustraction-polynomiale.jpg\" alt=\"\" class=\"wp-image-522\" width=\"533\" height=\"40\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Os mon\u00f4mios do primeiro par\u00eantese permanecem os mesmos, por\u00e9m os termos do segundo par\u00eantese devem mudar de sinal porque possuem um negativo na frente: <\/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-soustraire-deux-polynomes.jpg\" alt=\"como subtrair dois polin\u00f4mios\" class=\"wp-image-4009\" width=\"520\" height=\"42\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> E agora agrupamos termos que possuem partes literais id\u00eanticas, ou seja, termos com as mesmas vari\u00e1veis (letras) e expoentes. Termos que n\u00e3o s\u00e3o semelhantes n\u00e3o podem ser adicionados ou subtra\u00eddos. <\/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\/soustraire-des-polynomes.jpg\" alt=\"subtra\u00e7\u00e3o de polin\u00f4mios m\u00e9todo horizontal\" class=\"wp-image-534\" width=\"510\" height=\"265\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Portanto, o polin\u00f4mio resultante da subtra\u00e7\u00e3o \u00e9: <\/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\/solution-soustraire-des-polynomes.jpg\" alt=\"resultado da subtra\u00e7\u00e3o de polin\u00f4mios\" class=\"wp-image-536\" width=\"314\" height=\"41\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Como voc\u00ea pode ver, obtivemos o mesmo resultado com os dois m\u00e9todos, ent\u00e3o voc\u00ea pode usar o que melhor lhe convier.<\/p>\n<p> Agora que voc\u00ea viu os 2 m\u00e9todos para resolver a subtra\u00e7\u00e3o polinomial, <span style=\"text-decoration: underline;\">voc\u00ea sabia que tamb\u00e9m pode subtrair fra\u00e7\u00f5es com polin\u00f4mios?<\/span> E n\u00e3o apenas subtra\u00e7\u00f5es, mas todos os tipos de opera\u00e7\u00f5es. Descubra como <strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/fracoes-algebricas-operacoes-simplificadas-adicao-subtracao-multiplicacao-divisao-exercicios-resolvidos\/\">s\u00e3o feitas as opera\u00e7\u00f5es com fra\u00e7\u00f5es alg\u00e9bricas<\/a><\/span><\/strong> clicando neste link. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejercicios-resueltos-de-la-resta-de-polinomios\"><\/span> Problemas resolvidos de subtra\u00e7\u00e3o de polin\u00f4mios<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Para que voc\u00ea possa praticar, deixamos v\u00e1rios exerc\u00edcios resolvidos de subtra\u00e7\u00e3o polinomial. Se voc\u00ea tiver d\u00favidas sobre algum exerc\u00edcio pode perguntar nos coment\u00e1rios da p\u00e1gina, responderemos o mais breve poss\u00edvel.<\/p>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Subtraia o polin\u00f4mio<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a80be6e42ac3b3c6528958bbfa21f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p> menos o polin\u00f4mio <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4d38083076c97f0893079e8fed89adb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" 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-ddd25ba5aeb117e4adea646cdc4cd301_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) = 3x^3-6x^2-2x+4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"208\" 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-fd08d7de922ca14fcfe0d3fbbe8c5f0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x) = -5x^3-8x^2+4x+7\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"222\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Neste caso, subtrairemos os dois polin\u00f4mios verticalmente. Para isso, primeiro ordenamos os polin\u00f4mios por grau, depois mudamos o sinal dos demais termos polinomiais e, por fim, somamos os mon\u00f4mios localizados na mesma coluna: <\/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\/exercices-de-soustraction-de-polynomes-resolus-pas-a-pas.jpg\" alt=\"exerc\u00edcios de subtra\u00e7\u00e3o polinomial resolvidos passo a passo\" class=\"wp-image-545\" width=\"258\" height=\"122\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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> Resolva o polin\u00f4mio de subtra\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-a80be6e42ac3b3c6528958bbfa21f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p> menos o polin\u00f4mio <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4d38083076c97f0893079e8fed89adb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" 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-e0c8d8bdcae533c7a94c1cee08bdd8b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) = 9x^4+4x^3+3x^2-6x-1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"255\" 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-733c47af69965682e421c39533ebe0b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x) = 7x^4+8x^2-7x+5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"207\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Subtrair dois polin\u00f4mios equivale a adicionar o oposto do polin\u00f4mio subtra\u00eddo ao polin\u00f4mio diminuto. Portanto, alteramos o sinal dos termos do polin\u00f4mio de subcontrata\u00e7\u00e3o (o restante) e somamos os polin\u00f4mios: <\/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\/quest-ce-que-la-soustraction-de-polynomes.jpg\" alt=\"qual \u00e9 a subtra\u00e7\u00e3o de polin\u00f4mios\" class=\"wp-image-4018\" width=\"349\" height=\"130\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Observe que neste caso particular deve ser deixado um espa\u00e7o vazio na coluna de grau 3 do segundo polin\u00f4mio, pois n\u00e3o possui termo de terceiro grau.<\/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 resultado da subtra\u00e7\u00e3o 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-a80be6e42ac3b3c6528958bbfa21f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p> menos o polin\u00f4mio <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4d38083076c97f0893079e8fed89adb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" 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-f588c27f4669758cb7c4ca71436670da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) = -4x^5-2x^4+6x^3-9x^2-5x+7\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"318\" 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-0ba234e1c81c35d1dac12700205128f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x) = -7x^5+3x^4+9x^3-4x^2+2x+4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"318\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Neste caso, resolveremos a subtra\u00e7\u00e3o dos dois polin\u00f4mios verticalmente. Ent\u00e3o, primeiro colocamos os polin\u00f4mios em ordem do maior para o menor grau, depois trocamos o sinal dos termos do polin\u00f4mio restante e, por fim, adicionamos os termos semelhantes: <\/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-resoudre-une-soustraction-de-polynomes.jpg\" alt=\"como resolver uma subtra\u00e7\u00e3o de polin\u00f4mios\" class=\"wp-image-4020\" width=\"417\" height=\"119\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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> Calcule a seguinte opera\u00e7\u00e3o com 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-727af8d0a8192ade6534b54715dd5158_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) -Q(x)-R(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"156\" 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-f870cf5e45e0e7245b35c2091b753365_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) = 5x^4+2x^3-7x^2-3x-4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"256\" 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-08ef8fe5837b793cbbf7515128823b97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x) = 8x^4+5x^3-3x^2+6x-9\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"256\" 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-dcebf43abca7d3b85c8ec8e5219cad16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"R(x) = 2x^4-9x^3+4x^2-4x+7\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"256\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Neste caso, 3 polin\u00f4mios est\u00e3o envolvidos na opera\u00e7\u00e3o, 2 dos quais s\u00e3o subtra\u00e7\u00f5es. Portanto, para resolver a opera\u00e7\u00e3o, mudaremos o sinal de todos os termos dos dois polin\u00f4mios restantes e depois somaremos os polin\u00f4mios. <\/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\/soustraction-de-trois-polynomes.jpg\" alt=\"subtra\u00e7\u00e3o de tr\u00eas polin\u00f4mios\" class=\"wp-image-567\" width=\"356\" height=\"160\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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 achou \u00fatil? Qual m\u00e9todo de subtra\u00e7\u00e3o de polin\u00f4mios voc\u00ea prefere, vertical ou horizontal? N\u00f3s lemos voc\u00ea nos coment\u00e1rios! \ud83d\udc40<\/p>\n<div id=\"ezoic-pub-ad-placeholder-176\" data-inserter-version=\"-1\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina explicamos como subtrair polin\u00f4mios. Al\u00e9m disso, voc\u00ea encontrar\u00e1 diversos exemplos e exerc\u00edcios resolvidos passo a passo para subtra\u00e7\u00e3o de polin\u00f4mios. Como subtrair polin\u00f4mios? Para subtrair dois polin\u00f4mios, voc\u00ea deve subtrair os termos dos polin\u00f4mios que s\u00e3o semelhantes. Em outras palavras, subtrair polin\u00f4mios consiste em subtrair termos que possuem a mesma parte literal (mesmas &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/subtracao-de-polinomios-exemplos-exercicios-resolvidos-subtracao\/\"> <span class=\"screen-reader-text\">Subtra\u00e7\u00e3o de polin\u00f4mios<\/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-283","post","type-post","status-publish","format-standard","hentry","category-polinomios"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Subtra\u00e7\u00e3o de polin\u00f4mios - Matoridade<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/pt\/subtracao-de-polinomios-exemplos-exercicios-resolvidos-subtracao\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Subtra\u00e7\u00e3o de polin\u00f4mios - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina explicamos como subtrair polin\u00f4mios. Al\u00e9m disso, voc\u00ea encontrar\u00e1 diversos exemplos e exerc\u00edcios resolvidos passo a passo para subtra\u00e7\u00e3o de polin\u00f4mios. Como subtrair polin\u00f4mios? Para subtrair dois polin\u00f4mios, voc\u00ea deve subtrair os termos dos polin\u00f4mios que s\u00e3o semelhantes. Em outras palavras, subtrair polin\u00f4mios consiste em subtrair termos que possuem a mesma parte literal (mesmas &hellip; Subtra\u00e7\u00e3o de polin\u00f4mios Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/subtracao-de-polinomios-exemplos-exercicios-resolvidos-subtracao\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T21:11:26+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fb1a50d4312d4058c8eb09943f235f0f_l3.png\" \/>\n<meta name=\"author\" content=\"Equipe Mathoridade\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Equipe Mathoridade\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/subtracao-de-polinomios-exemplos-exercicios-resolvidos-subtracao\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/subtracao-de-polinomios-exemplos-exercicios-resolvidos-subtracao\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Subtra\u00e7\u00e3o de polin\u00f4mios\",\"datePublished\":\"2023-07-06T21:11:26+00:00\",\"dateModified\":\"2023-07-06T21:11:26+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/subtracao-de-polinomios-exemplos-exercicios-resolvidos-subtracao\/\"},\"wordCount\":1023,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Polin\u00f4mios\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/subtracao-de-polinomios-exemplos-exercicios-resolvidos-subtracao\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/subtracao-de-polinomios-exemplos-exercicios-resolvidos-subtracao\/\",\"url\":\"https:\/\/mathority.org\/pt\/subtracao-de-polinomios-exemplos-exercicios-resolvidos-subtracao\/\",\"name\":\"Subtra\u00e7\u00e3o de polin\u00f4mios - 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Al\u00e9m disso, voc\u00ea encontrar\u00e1 diversos exemplos e exerc\u00edcios resolvidos passo a passo para subtra\u00e7\u00e3o de polin\u00f4mios. Como subtrair polin\u00f4mios? Para subtrair dois polin\u00f4mios, voc\u00ea deve subtrair os termos dos polin\u00f4mios que s\u00e3o semelhantes. 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