{"id":7215,"date":"2011-11-27T22:00:45","date_gmt":"2011-11-27T22:00:45","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7215"},"modified":"2021-12-28T12:51:28","modified_gmt":"2021-12-28T12:51:28","slug":"escreva-sob-a-forma-de-polinomio","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7215","title":{"rendered":"Escreva sob a forma de polin\u00f3mio"},"content":{"rendered":"<p><ul id='GTTabs_ul_7215' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7215' class='GTTabs_curr'><a  id=\"7215_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7215' ><a  id=\"7215_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_7215'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Escreva sob a forma de polin\u00f3mio as express\u00f5es:<\/p>\n<ol>\n<li>${{(2+x)}^{4}}$<\/li>\n<li>${{(1-2x)}^{5}}$<\/li>\n<li>${{(\\sqrt{2}+x)}^{6}}$<\/li>\n<li>${{(\\sqrt{2}x-\\sqrt{3})}^{4}}$<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7215' onClick='GTTabs_show(1,7215)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7215'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Ora,<br \/>\n$$\\begin{array}{*{35}{l}}<br \/>\n{{(2+x)}^{4}} &amp; = &amp; \\sum\\limits_{k=0}^{4}{{}^{4}{{C}_{k}}\\times {{2}^{4-k}}\\times {{x}^{k}}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; {}^{4}{{C}_{0}}\\times {{2}^{4}}\\times {{x}^{0}}+{}^{4}{{C}_{1}}\\times {{2}^{3}}\\times {{x}^{1}}+{}^{4}{{C}_{2}}\\times {{2}^{2}}\\times {{x}^{2}}+{}^{4}{{C}_{3}}\\times {{2}^{1}}\\times {{x}^{3}}+{}^{4}{{C}_{4}}\\times {{2}^{0}}\\times {{x}^{4}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 4\\times 16\\times 1+4\\times 8\\times x+6\\times 4\\times {{x}^{2}}+4\\times 2\\times {{x}^{3}}+1\\times 1\\times {{x}^{4}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 64+32x+24{{x}^{2}}+8{{x}^{3}}+{{x}^{4}}\u00a0 \\\\<br \/>\n\\end{array}$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{35}{l}}<br \/>\n{{(1-2x)}^{5}} &amp; = &amp; \\sum\\limits_{k=0}^{5}{{}^{5}{{C}_{k}}\\times {{1}^{5-k}}\\times {{(-2x)}^{k}}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; {}^{5}{{C}_{0}}\\times {{1}^{5}}\\times {{(-2x)}^{0}}+{}^{5}{{C}_{1}}\\times {{1}^{4}}\\times {{(-2x)}^{1}}+{}^{5}{{C}_{2}}\\times {{1}^{3}}\\times {{(-2x)}^{2}}+{}^{5}{{C}_{3}}\\times {{1}^{2}}\\times {{(-2x)}^{3}}+{}^{5}{{C}_{4}}\\times {{1}^{1}}\\times {{(-2x)}^{4}}+{}^{5}{{C}_{5}}\\times {{1}^{0}}\\times {{(-2x)}^{5}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 1\\times 1\\times 1+5\\times 1\\times (-2x)+10\\times 1\\times 4{{x}^{2}}+10\\times 1\\times (-8{{x}^{3}})+5\\times 1\\times 16{{x}^{4}}+1\\times 1\\times (-32{{x}^{5}})\u00a0 \\\\<br \/>\n{} &amp; = &amp; 1-10x+40{{x}^{2}}-80{{x}^{3}}+80{{x}^{4}}-32{{x}^{5}}\u00a0 \\\\<br \/>\n\\end{array}$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{35}{l}}<br \/>\n{{(\\sqrt{2}+x)}^{6}} &amp; = &amp; \\sum\\limits_{k=0}^{6}{{}^{6}{{C}_{k}}\\times {{(\\sqrt{2})}^{6-k}}\\times {{x}^{k}}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; {{(\\sqrt{2})}^{6}}+6\\times {{(\\sqrt{2})}^{5}}x+15\\times {{(\\sqrt{2})}^{4}}{{x}^{2}}+20\\times {{(\\sqrt{2})}^{3}}{{x}^{3}}+15\\times {{(\\sqrt{2})}^{2}}{{x}^{4}}+6\\times {{(\\sqrt{2})}^{1}}{{x}^{5}}+{{x}^{6}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 8+24\\sqrt{2}x+60{{x}^{2}}+40\\sqrt{2}{{x}^{3}}+30{{x}^{4}}+6\\sqrt{2}{{x}^{5}}+{{x}^{6}}\u00a0 \\\\<br \/>\n\\end{array}$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{35}{l}}<br \/>\n{{(\\sqrt{2}x-\\sqrt{3})}^{4}} &amp; = &amp; \\sum\\limits_{k=0}^{4}{{}^{4}{{C}_{k}}\\times {{(\\sqrt{2}x)}^{4-k}}\\times {{(-\\sqrt{3})}^{k}}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; {{(\\sqrt{2}x)}^{4}}+4\\times {{(\\sqrt{2}x)}^{3}}\\times (-\\sqrt{3})+6\\times {{(\\sqrt{2}x)}^{2}}\\times {{(-\\sqrt{3})}^{2}}+4\\times {{(\\sqrt{2}x)}^{1}}\\times {{(-\\sqrt{3})}^{3}}+{{(-\\sqrt{3})}^{4}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 4{{x}^{4}}-8\\sqrt{6}{{x}^{3}}+36{{x}^{2}}-12\\sqrt{6}x+9\u00a0 \\\\<br \/>\n\\end{array}$$<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7215' onClick='GTTabs_show(0,7215)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Escreva sob a forma de polin\u00f3mio as express\u00f5es: ${{(2+x)}^{4}}$ ${{(1-2x)}^{5}}$ ${{(\\sqrt{2}+x)}^{6}}$ ${{(\\sqrt{2}x-\\sqrt{3})}^{4}}$ Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19524,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,257],"series":[],"class_list":["post-7215","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-probabilidades-e-combinatoria","tag-12-o-ano","tag-binomio-de-newton"],"views":1547,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Binomio_de_Newton.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7215","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=7215"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7215\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19524"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7215"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7215"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7215"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7215"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}