{"id":62025,"date":"2022-01-26T15:47:25","date_gmt":"2022-01-26T18:47:25","guid":{"rendered":"https:\/\/vestibulares.estrategia.com\/portal\/?p=62025"},"modified":"2024-02-21T16:05:07","modified_gmt":"2024-02-21T19:05:07","slug":"polinomios","status":"publish","type":"post","link":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/polinomios\/","title":{"rendered":"Polin\u00f4mios: o que s\u00e3o e como resolver"},"content":{"rendered":"<p class=\"wp-block-paragraph\">Quando <span id=\"urn:enhancement-f7de8092-2154-4336-91e6-4e124a830da5\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> possuem inc&oacute;gnitas e n&uacute;meros conhecidos s&atilde;o chamadas de <span id=\"urn:enhancement-5632f68e-7a8e-47ff-87b1-77202ca549ee\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mios<\/span>. As <span id=\"urn:enhancement-fee053a2-90fb-41d0-855b-e3f9265ef4fa\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&otilde;es<\/span> polinomiais al&eacute;m de fornecerem resultados, podem ser somadas, subtra&iacute;das, multiplicadas e divididas, como voc&ecirc; aprender&aacute; no artigo a seguir. Acompanhe agora!<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 counter-hierarchy ez-toc-counter ez-toc-transparent ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\" style=\"cursor:inherit\">Navegue pelo conte\u00fado<\/p>\n<\/div><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/polinomios\/#Qual-a-definicao-de-polinomios\" >Qual a defini&ccedil;&atilde;o de polin&ocirc;mios?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/polinomios\/#Operacoes-simples-com-polinomios\" >Opera&ccedil;&otilde;es simples com polin&ocirc;mios<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/polinomios\/#Aprenda-mais-de-polinomios-com-o-Estrategia-Vestibulares\" >Aprenda mais de polin&ocirc;mios com o Estrat&eacute;gia Vestibulares<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"h-qual-a-definicao-de-polinomios\"><span class=\"ez-toc-section\" id=\"Qual-a-definicao-de-polinomios\"><\/span>Qual a defini&ccedil;&atilde;o de polin&ocirc;mios?<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Os <span id=\"urn:enhancement-daa516d2-56f9-43a8-ae8f-5134e985e0f2\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mios<\/span> s&atilde;o formados por <strong>termos<\/strong>: por&ccedil;&otilde;es de <span id=\"urn:enhancement-61168d6f-d4bb-45b3-aa37-e23f007440ed\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&otilde;es alg&eacute;bricas<\/span> em que a &uacute;nica opera&ccedil;&atilde;o existente &eacute; a <span id=\"urn:enhancement-d9f03eea-c65a-4e3e-9027-e935ecc25161\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/multiplicacao\">multiplica&ccedil;&atilde;o<\/span>. Por exemplo, 4.x &eacute; um termo, assim como 5.x.y.<\/p><h3 class=\"wp-block-heading\" id=\"h-monomios\">Mon&ocirc;mios<\/h3><p class=\"wp-block-paragraph\">Quando a <span id=\"urn:enhancement-8c764c01-9427-4484-a107-1de9ffdc220f\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> &eacute; formada por apenas um termo, &eacute; denominada de mon&ocirc;mio, por exemplo:<\/p><ul class=\"wp-block-list\">\n<li>10.x<\/li>\n\n\n\n<li>23.x.y.z<\/li>\n\n\n\n<li>a.b<\/li>\n<\/ul><p class=\"wp-block-paragraph\">Como padr&atilde;o, foi determinado que o n&uacute;mero conhecido do termo &eacute; o <strong><span id=\"urn:enhancement-d68b14ec-b145-4bca-92b5-aaac694f1d45\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficiente<\/span><\/strong>. As inc&oacute;gnitas, geralmente representadas por letras, ser&atilde;o chamadas de <strong>parte literal <\/strong>da <span id=\"urn:enhancement-be4ebd87-d4b4-497d-a066-1566dae87ba8\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span>.&nbsp;<\/p><h3 class=\"wp-block-heading\" id=\"h-binomios\">Bin&ocirc;mios<\/h3><p class=\"wp-block-paragraph\">As opera&ccedil;&otilde;es matem&aacute;ticas entre termos s&atilde;o a soma e <span id=\"urn:enhancement-1cf2a7fd-37f8-4d19-873d-1d00da7383d4\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/subtracao\">subtra&ccedil;&atilde;o<\/span> entre eles, no caso em que existem dois termos, encontram-se os bin&ocirc;mios:<\/p><ul class=\"wp-block-list\">\n<li>13x + 25y &rarr; coeficientes +13 e +25;<\/li>\n\n\n\n<li>8x &ndash; 14w &rarr; coeficientes +8 e -14; e<\/li>\n\n\n\n<li>&ndash; x + 9z &rarr; coeficientes -1 e +9.<\/li>\n<\/ul><p class=\"wp-block-paragraph\">em que a parte literal s&atilde;o os valores desconhecidos w, x, y, e z.&nbsp;<\/p><h3 class=\"wp-block-heading\" id=\"h-trinomios\">Trin&ocirc;mios<\/h3><p class=\"wp-block-paragraph\">Quando a <span id=\"urn:enhancement-85ffb232-0267-437e-8a2a-28b9782d14d7\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o<\/span> apresenta tr&ecirc;s termos, ser&aacute; chamada de trin&ocirc;mio. Como voc&ecirc; pode ver a seguir:<\/p><ul class=\"wp-block-list\">\n<li>10.x + 23.x.y<sup>2<\/sup>.z + x<sup>2<\/sup>.y<\/li>\n\n\n\n<li>&ndash; 13a.b<sup>3<\/sup> &ndash; 11a<sup>2<\/sup> + 4.b<\/li>\n<\/ul><p class=\"wp-block-paragraph\">Com os exemplos acima, podemos introduzir o conceito de <strong>grau<\/strong>. Essa grandeza ser&aacute; definida pela <strong>soma dos <span id=\"urn:enhancement-a09948c2-1d08-4cf7-b05b-523bbece343b\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/potencias\">expoentes<\/span><\/strong> da <strong>parte literal em cada termo<\/strong>. O maior resultado ser&aacute; o grau do <span id=\"urn:enhancement-d8e911f4-e28e-4726-8932-78bd72c88daa\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span>, compreenda melhor no t&oacute;pico abaixo.<\/p><h3 class=\"wp-block-heading\" id=\"h-polinomios\">Polin&ocirc;mios<\/h3><p class=\"wp-block-paragraph\">Os <span id=\"urn:enhancement-1dfa60b1-be48-4cb3-ab6a-d8bd750c0e16\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mios<\/span>, por fim, s&atilde;o <span id=\"urn:enhancement-bcb83f02-8abf-44eb-905a-5e7cd8f65ae1\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&otilde;es alg&eacute;bricas<\/span> em que se encontram quatro ou mais termos. Veja a descri&ccedil;&atilde;o de um <span id=\"urn:enhancement-199c47e4-21d2-47e8-853f-710e5dc06c85\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> com o esquema abaixo:<\/p><p class=\"wp-block-paragraph\"><strong>P = -3.a.b + 21.b<\/strong><strong><sup>2<\/sup><\/strong><strong>.a<\/strong><strong><sup>3<\/sup><\/strong><strong> + 7.b &ndash; 9.a<\/strong><strong><sup>4<\/sup><\/strong><strong>.b<\/strong><strong><sup>2<\/sup><\/strong><\/p><ul class=\"wp-block-list\">\n<li>Para o termo de coeficiente<strong> -3,<\/strong> temos uma parte literal com duas inc&oacute;gnitas. &ldquo;a&rdquo; tem expoente 1, assim como b: <strong>1+1=2<\/strong>;<\/li>\n\n\n\n<li>No caso do coeficiente <strong>21<\/strong>, a parte literal possui expoentes 2 e 3: <strong>2+3=5<\/strong>;<\/li>\n\n\n\n<li>No termo que tem <strong>+7<\/strong> como coeficiente, a &uacute;nica inc&oacute;gnita &eacute; o &ldquo;b&rdquo;, de expoente <strong>1<\/strong>; e<\/li>\n\n\n\n<li>No &uacute;ltimo termo, que leva <strong>-9<\/strong> como coeficiente, os expoentes da parte literal, quando somados, ser&atilde;o <strong>4+2=6<\/strong>.<\/li>\n<\/ul><p class=\"wp-block-paragraph\">Com essas informa&ccedil;&otilde;es, observa-se que a maior soma entre os <span id=\"urn:enhancement-16391880-5a55-42b4-a5cc-a8ed75224193\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/potencias\">expoentes<\/span> da parte literal de cada termo &eacute; equivalente a 6. Ent&atilde;o, o <span id=\"urn:enhancement-0bc12f62-068a-4f9c-ae44-b35b4852fbe0\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> P &eacute; do sexto grau, ou grau seis.&nbsp;<\/p><p class=\"wp-block-paragraph\">Os <span id=\"urn:enhancement-f5593107-f562-4e22-b89a-f0a454bca274\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mios<\/span> podem admitir um resultado preciso, quando se determina um valor n&uacute;mero para as inc&oacute;gnitas. Por exemplo, no caso de P, se a=1 e b=2, teremos que:<\/p><p class=\"wp-block-paragraph\">P = -3.1.2 + 21.2<sup>2<\/sup>.1<sup>3<\/sup> + 7.2 &ndash; 9.1<sup>4<\/sup>.2<sup>2<\/sup><br>P = -3.2 + 21.4.1 + 14 &ndash; 9.1.4<br>P = -6 + 84 + 14 &ndash; 36<br>P = 108 &ndash; 42<br><strong>P = 66<\/strong><\/p><p class=\"wp-block-paragraph\">Perceba que, para valores diferentes de a e b, o resultado da <span id=\"urn:enhancement-df0711f0-ffbf-423f-9468-32aa9e7df74f\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o<\/span> P ser&aacute; alterado.<\/p><figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<div class=\"cs-embed cs-embed-responsive\"><iframe title=\"Matem&aacute;tica para ENEM\" width=\"1200\" height=\"675\" src=\"https:\/\/www.youtube.com\/embed\/5ebNpJrkarA?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/div>\n<\/div><\/figure><h2 class=\"wp-block-heading\" id=\"h-operacoes-simples-com-polinomios\"><span class=\"ez-toc-section\" id=\"Operacoes-simples-com-polinomios\"><\/span>Opera&ccedil;&otilde;es simples com polin&ocirc;mios<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Pensando na intera&ccedil;&atilde;o entre as diferentes inc&oacute;gnitas e <span id=\"urn:enhancement-4b8e6244-a4b0-4da0-8d86-0ebe07300e01\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&otilde;es<\/span> polinomiais, &eacute; poss&iacute;vel realizar opera&ccedil;&otilde;es matem&aacute;ticas entre dois ou mais <span id=\"urn:enhancement-10b55454-c78d-42b5-b3f1-ecf87521755c\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mios<\/span>.&nbsp;<\/p><h3 class=\"wp-block-heading\" id=\"h-soma-de-polinomios\">Soma de polin&ocirc;mios<\/h3><p class=\"wp-block-paragraph\">A principal <span id=\"urn:enhancement-e0202836-c83f-4cba-b558-57f3845a2992\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">regra<\/span> da soma de <span id=\"urn:enhancement-03f43c3e-7885-4c4e-8397-3b4ddb24a945\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mios<\/span> &eacute; a realiza&ccedil;&atilde;o da opera&ccedil;&atilde;o somente para termos que possuem a parte literal igual. Para isso, o ideal &eacute; agrupar esses termos semelhantes na escrita da <span id=\"urn:enhancement-5edbfe74-0d79-4e7d-8a3e-99005a55d7bc\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o<\/span>, facilitando a <span id=\"urn:enhancement-e2c9d622-70cc-4904-821e-4cae7e5c526f\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/compreensao\">compreens&atilde;o<\/span> e os c&aacute;lculos:<\/p><p class=\"wp-block-paragraph\"><strong>P = -5x + 8x<\/strong><strong><sup>2<\/sup><\/strong><strong>y + 15x<\/strong><strong><sup>2<\/sup><\/strong><strong>&nbsp;<\/strong><\/p><p class=\"wp-block-paragraph\"><strong>Q = 7y<\/strong><strong><sup>2<\/sup><\/strong><strong>x &ndash; 12yx<\/strong><strong><sup>2 <\/sup><\/strong><strong>+ x &ndash; 5x<\/strong><strong><sup>2 <\/sup><\/strong><strong>&ndash; 9.y<\/strong><strong><sup>2<\/sup><\/strong><strong>.x<\/strong><\/p><p class=\"wp-block-paragraph\">P + Q = S&nbsp;<\/p><p class=\"wp-block-paragraph\">S = ?<\/p><ol class=\"wp-block-list\">\n<li>O primeiro passo ser&aacute; agrupar termos de parte literal igual:<\/li>\n<\/ol><p class=\"wp-block-paragraph\"><br>S = -5x + x + 15x<sup>2 <\/sup>&ndash; 5x<sup>2 <\/sup>+ 8x<sup>2<\/sup>y &ndash; 12yx<sup>2<\/sup> + 7y<sup>2<\/sup>x <sup>&nbsp;&nbsp;<\/sup>&ndash; 9.y<sup>2<\/sup>.x<br><br>Veja como a invers&atilde;o dos fatores em -12yx<sup>2<\/sup> e +8x<sup>2<\/sup>y n&atilde;o altera o fato de as partes literais serem iguais, j&aacute; que na <span id=\"urn:enhancement-aa681b61-fa2e-4e3f-9dde-980fb58f2ebc\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/multiplicacao\">multiplica&ccedil;&atilde;o<\/span> &ldquo;a ordem dos fatores n&atilde;o altera o produto&rdquo;, ou seja yx<sup>2<\/sup> = x<sup>2<\/sup>y.<br><\/p><ol class=\"wp-block-list\" start=\"2\">\n<li>Com essa organiza&ccedil;&atilde;o feita, pode ser feita a opera&ccedil;&atilde;o simples entre os coeficientes dos termos semelhantes:<br><strong>S = -4x + 10x<\/strong><strong><sup>2&nbsp; <\/sup><\/strong><strong>&ndash; 4x<\/strong><strong><sup>2<\/sup><\/strong><strong>y &ndash; 2.y<\/strong><strong><sup>2<\/sup><\/strong><strong>.x<\/strong><\/li>\n<\/ol><p class=\"wp-block-paragraph\">Observe que n&atilde;o h&aacute; mais como agrupar os termos dos <span id=\"urn:enhancement-17d7fc5c-ca2a-4335-b590-8f7bef045022\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mios<\/span> P e Q, de forma que essa &eacute; a forma mais simplificada da opera&ccedil;&atilde;o.<\/p><h3 class=\"wp-block-heading\" id=\"h-subtracao-de-polinomios\">Subtra&ccedil;&atilde;o de polin&ocirc;mios<\/h3><p class=\"wp-block-paragraph\">No caso da <span id=\"urn:enhancement-b7dce6d2-0daf-4545-847f-6c0a11933d3a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/subtracao\">subtra&ccedil;&atilde;o<\/span>, antes de iniciarmos com o agrupamento dos termos, &eacute; necess&aacute;rio aplicar a propriedade distributiva da Matem&aacute;tica &mdash; o <span id=\"urn:enhancement-e0c63795-f2d1-442c-bb99-556f0388154a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/pontos\">ponto<\/span> principal &eacute; inverter o sinal dos <span id=\"urn:enhancement-0258dc33-a152-42d7-9939-4249c088114d\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficientes<\/span> do subtraendo. Com as mesmas <span id=\"urn:enhancement-f2af6d48-c42f-4d6f-8036-c0a7b0e7ffc7\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&otilde;es<\/span> polinomiais P e Q:<\/p><p class=\"wp-block-paragraph\"><strong>P = -5x + 8x<sup>2<\/sup>y + 15x<sup>2<\/sup>&nbsp;<\/strong><br><strong>Q = 7y<sup>2<\/sup>x &ndash; 12yx<sup>2 <\/sup>+ x &ndash; 5x<sup>2 <\/sup>&ndash; 9.y<sup>2<\/sup>.x<\/strong><br>P &ndash; Q = M<br>M = ?<\/p><ol class=\"wp-block-list\">\n<li>Inicialmente, descrevem-se as opera&ccedil;&otilde;es e inverte-se o sinal dos coeficientes no subtraendo (polin&ocirc;mio Q):<br>M = -5x + 8x<sup>2<\/sup>y + 15x<sup>2<\/sup> &ndash; ( + 7y<sup>2<\/sup>x &ndash; 12yx<sup>2 <\/sup>+ x &ndash; 5x<sup>2 <\/sup>&ndash; 9.y<sup>2<\/sup>.x)<br>M = -5x + 8x<sup>2<\/sup>y + 15x<sup>2<\/sup> &ndash; 7y<sup>2<\/sup>x + 12yx<sup>2 <\/sup>&ndash; x + 5x<sup>2 <\/sup>+ 9.y<sup>2<\/sup>.x<\/li>\n\n\n\n<li>Agora, podemos agrupar os termos semelhantes, igual o racioc&iacute;nio desenvolvido no exemplo anterior:<br>M = -5x &ndash; x + 15x<sup>2 <\/sup>+ 5x<sup>2 <\/sup>+ 8x<sup>2<\/sup>y + 12yx<sup>2<\/sup> &ndash; 7y<sup>2<\/sup>x <sup>&nbsp;&nbsp;<\/sup>+ 9.y<sup>2<\/sup>.x<\/li>\n\n\n\n<li>Feita a organiza&ccedil;&atilde;o l&oacute;gica e matem&aacute;tica da express&atilde;o, pode-se dar continuidade e encontrar M:<\/li>\n<\/ol><p class=\"wp-block-paragraph\"><strong>M = -6x + 20x<\/strong><strong><sup>2 <\/sup><\/strong><strong>+ 20x<\/strong><strong><sup>2<\/sup><\/strong><strong>y + 2.y<\/strong><strong><sup>2<\/sup><\/strong><strong>.x<\/strong><\/p><h3 class=\"wp-block-heading\" id=\"h-divisao-de-polinomios\">Divis&atilde;o de polin&ocirc;mios<\/h3><p class=\"wp-block-paragraph\">A <span id=\"urn:enhancement-3200d97f-d248-4839-8b15-bd508a017c32\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/divisao\">divis&atilde;o<\/span> de <span id=\"urn:enhancement-9463f022-972d-445a-b4c1-dfdd4c4782b6\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mios<\/span> pode ser feita pelo m&eacute;todo da chave, como quando fracionamos n&uacute;meros inteiros. O <span id=\"urn:enhancement-3dd1f62d-b516-4ea8-8b8a-bbda9efb4f94\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> P(x) ser&aacute; divido pela <span id=\"urn:enhancement-3e8ebf26-d28a-457f-9e40-2e4b31cb9a0c\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o<\/span> D(x), resultando em um quociente Q(x).<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/prynujWLQewnPPrHx-gK0kmw6erNdv5jYTBKCMcruXazoXKxgfYhWu1zFI6mGyLtQ1K0zoseed5quuErpKSlVWJTMmO4yIJtx8yerBR4lrvdyrnXNeUYrNHibdMWvigFrQPbU9d7\" alt=\"Divis&atilde;o de polin&ocirc;mios\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Por exemplo, se tivermos um <span id=\"urn:enhancement-88a74a66-cdee-45fd-bf4e-72d1f03088fd\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> P(x) = 2x<sup>3<\/sup> &ndash; 7x<sup>2<\/sup> + 4x + 1 e um divisor D(x) = x &ndash; 1, ser&aacute; necess&aacute;rio encontrar o Q(x):<\/p><p class=\"wp-block-paragraph\">Como descreve a imagem abaixo, o primeiro termo a ser eliminado ser&aacute; 2x<sup>3<\/sup>, que possui o maior grau na <span id=\"urn:enhancement-b5a850fa-91fd-4034-9926-552c6d80f12f\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o<\/span>. Para isso, pensa-se em um valor que, quando multiplicado por D(x) resulte em 2x<sup>3<\/sup> &mdash; no caso, ser&aacute; 2x<sup>2<\/sup>, o primeiro termo de Q(x).<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/1FtiIPP3lVzClW99yQvyK0jO9H9h38HnCmCzzub_KFeUlBzbXah5NL-ylpz2BNgpg4V6q0SWatZUZXp3j978V1HbHMujtCsPEapz4O8I8cX1qdNa90eHBA_awokLFDK60T9C3SmB\" alt=\"Divis&atilde;o de polin&ocirc;mios - chave\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Prossegue-se a <span id=\"urn:enhancement-ddf806a3-cc4e-4cbb-833b-38c78fa0e438\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/divisao\">divis&atilde;o<\/span>, subtraindo o valor encontrado na <span id=\"urn:enhancement-f70973d0-7542-4270-bd07-631e935ef491\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/multiplicacao\">multiplica&ccedil;&atilde;o<\/span> entre 2x<sup>2<\/sup> e D(x) do <span id=\"urn:enhancement-53f9100f-643a-4fa1-8afb-c0c9dc4ff8d7\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/polinomios\">polin&ocirc;mio<\/span> P (x).&nbsp;<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/F69etnGh4aybcpvW7aBGrj1evDHXw4g6BT_mLZSFELXRuPnFYV4iUWjXingVysnl9hXwPruiAC2PFqdyMPrDv1eRilePMxtuRQb1eqbz0sGiz0pp4VRrf-SAZikx5W8LBaSTdZJ3\" alt=\"continua&ccedil;&atilde;o Divis&atilde;o de polin&ocirc;mios\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Voc&ecirc; pode acompanhar nas imagens que o fracionamento prossegue com o mesmo racioc&iacute;nio.<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/odZUj_bfrJ2Wmmuxs15n2ClKdbFOZhXwtL2WClwwhKXSn_9xJ4eZpIWh95tlDiHgMdqXv5lhyvddHnghEVhuoVwT0_tywDYyh_T-6iMvStO7Btk20pv67AwbAnxYfA_smfM4IAHM\" alt=\"polin&ocirc;mios\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Por fim, na figura abaixo est&aacute; representado o final da <span id=\"urn:enhancement-fa1dbcf7-6a3e-4b4e-9d0e-f4c30c32d09a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/divisao\">divis&atilde;o<\/span>, que apresenta resto R(x)=0 e:<\/p><p class=\"wp-block-paragraph\"><strong>Q(x) = 2x<\/strong><strong><sup>2<\/sup><\/strong><strong> &ndash; 5x- 1<\/strong><\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/JxHTpzY2sRam2wMPWLVG_NGoCnhS8RDs2qB2RGqtNz2xbAxo-iHw1hkx2s2W3sBrb644lCyMP_jOoTC7-Xek27AwLWF_2P2pRk1FfgwoNjDthQhv1G3mQ6VjyCAD1y9PMv_We8zW\" alt=\"Divis&atilde;o polinomial\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Em uma <span id=\"urn:enhancement-0cb676d6-bace-4b34-963b-7217ff74e530\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/divisao\">divis&atilde;o<\/span> polinomial gen&eacute;rica, pode-se dizer que:<\/p><p class=\"wp-block-paragraph\"><strong>P(x) = Q(x).D(x) + R(x)<\/strong><\/p><h3 class=\"wp-block-heading\" id=\"h-multiplicacao-de-polinomios\">Multiplica&ccedil;&atilde;o de polin&ocirc;mios<\/h3><p class=\"wp-block-paragraph\">A <span id=\"urn:enhancement-bfe4b1bf-ddab-4d79-9fbf-53844f826043\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/multiplicacao\">multiplica&ccedil;&atilde;o<\/span> polinomial segue o mesmo princ&iacute;pio que a <span id=\"urn:enhancement-feea31fd-d94f-422e-a096-c0aad4b26a56\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/multiplicacao\">multiplica&ccedil;&atilde;o<\/span> de n&uacute;meros inteiros. Com a distributiva, encontra-se o valor de qualquer P(x).P&rsquo;(x), levando sempre em considera&ccedil;&atilde;o o <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/regra-de-sinais\/\" target=\"_blank\">jogo de sinais<\/a>.<\/p><p class=\"wp-block-paragraph\">Com os mesmos Q(x) e D(x) do exemplo anterior, seguiremos essa representa&ccedil;&atilde;o:<\/p><p class=\"wp-block-paragraph\"><strong>P(x) = Q(x).D(x)&nbsp;<\/strong><br>P(x) = (2x<sup>2<\/sup> &ndash; 5x- 1) . (x-1)<br>P(x) = 2x<sup>3 <\/sup>&ndash; 2x<sup>2<\/sup> &ndash; 5x<sup>2<\/sup>&nbsp; + 5x &ndash; x + 1<br><strong>P(x) = 2x<sup>3 <\/sup>&ndash; 7x<sup>2<\/sup> + 4x + 1<\/strong><\/p><h2 class=\"wp-block-heading\" id=\"h-aprenda-mais-de-polinomios-com-o-estrategia-vestibulares\"><span class=\"ez-toc-section\" id=\"Aprenda-mais-de-polinomios-com-o-Estrategia-Vestibulares\"><\/span>Aprenda mais de polin&ocirc;mios com o Estrat&eacute;gia Vestibulares<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Conhe&ccedil;a os cursos preparat&oacute;rios para o vestibular da Coruja. Nossos pacotes incluem f&oacute;rum de d&uacute;vidas, livros digitais, aulas documentadas, corre&ccedil;&atilde;o ilimitada de reda&ccedil;&otilde;es e mapas mentais sobre matem&aacute;tica e outras mat&eacute;rias! Clique no banner abaixo e confira.<\/p><p class=\"wp-block-paragraph\"><a id=\"cta\" class=\"cta-imagem\" href=\"https:\/\/vestibulares.estrategia.com\/cursos\/\" target=\"blank\">\n                <img decoding=\"async\" width=\"100%\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2021\/12\/EV4.jpg\" alt=\"CTA - Estrat\u00e9gia Vestibulares 4\" title=\"CTA - Estrat\u00e9gia Vestibulares 4\">\n        <\/a>\n\n\n<\/p><\/p>\n","protected":false},"excerpt":{"rendered":"Quando equa&ccedil;&otilde;es possuem inc&oacute;gnitas e n&uacute;meros conhecidos s&atilde;o chamadas de polin&ocirc;mios. As express&otilde;es polinomiais al&eacute;m de fornecerem resultados,&hellip;\n","protected":false},"author":23,"featured_media":62026,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29],"tags":[],"class_list":["post-62025","post","type-post","status-publish","format-standard","has-post-thumbnail","category-matematica"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v25.9 (Yoast SEO v28.2) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Polin\u00f4mios: o que s\u00e3o e como resolver<\/title>\n<meta name=\"description\" content=\"Aprenda sobre os polin\u00f4mios: o que \u00e9, como s\u00e3o constru\u00eddos e como realizar as opera\u00e7\u00f5es de soma, subtra\u00e7\u00e3o, multiplica\u00e7\u00e3o e divis\u00e3o.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" 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