{"id":8963,"date":"2012-05-06T23:14:12","date_gmt":"2012-05-06T22:14:12","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8963"},"modified":"2021-12-29T01:04:58","modified_gmt":"2021-12-29T01:04:58","slug":"considere-o-polinomio","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8963","title":{"rendered":"Considere o polin\u00f3mio"},"content":{"rendered":"<p><ul id='GTTabs_ul_8963' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8963' class='GTTabs_curr'><a  id=\"8963_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8963' ><a  id=\"8963_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_8963'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>$$P(z) = 2{z^4} &#8211; 3{z^3} + 6{z^2} &#8211; 12z &#8211; 8\\,\\,,z \\in \\mathbb{C}$$<\/p>\n<ol>\n<li>Determine os n\u00fameros reais $a$, $b$ e $c$ tais que, para todo o n\u00famero complexo $z$, $$P(z) = \\left( {{z^2} + 4} \\right)\\left( {a{z^2} + bz + c} \\right)$$<\/li>\n<li>Resolva, em $\\mathbb{C}$, a equa\u00e7\u00e3o $P(z) = 0$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8963' onClick='GTTabs_show(1,8963)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8963'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Ora, $$\\begin{array}{*{20}{l}}<br \/>\n{P(z)}&amp; = &amp;{\\left( {{z^2} + 4} \\right)\\left( {a{z^2} + bz + c} \\right)} \\\\<br \/>\n{}&amp; = &amp;{a{z^4} + b{z^3} + c{z^2} + 4a{z^2} + 4bz + 4c} \\\\<br \/>\n{}&amp; = &amp;{a{z^4} + b{z^3} + \\left( {4a + c} \\right){z^2} + 4bz + 4c}<br \/>\n\\end{array}$$<br \/>\nPara que os polin\u00f3mios sejam id\u00eanticos, os coeficientes dos termos de igual grau t\u00eam de ser iguais: $$\\begin{array}{*{20}{l}}<br \/>\n{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{a = 2} \\\\<br \/>\n{b =\u00a0 &#8211; 3} \\\\<br \/>\n{4a + c = 6} \\\\<br \/>\n{4b =\u00a0 &#8211; 12} \\\\<br \/>\n{4c =\u00a0 &#8211; 8}<br \/>\n\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{a = 2} \\\\<br \/>\n{b =\u00a0 &#8211; 3} \\\\<br \/>\n{4a + c = 6} \\\\<br \/>\n{b =\u00a0 &#8211; 3} \\\\<br \/>\n{c =\u00a0 &#8211; 2}<br \/>\n\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{a = 2} \\\\<br \/>\n{b =\u00a0 &#8211; 3} \\\\<br \/>\n{c =\u00a0 &#8211; 2}<br \/>\n\\end{array}} \\right.}<br \/>\n\\end{array}$$<br \/>\n\u00ad<\/li>\n<li>Substituindo $a$, $b$ e $c$ pelos valores encontrados, vem: $$\\begin{array}{*{20}{l}}<br \/>\n{P(z) = 0}&amp; \\Leftrightarrow &amp;{\\left( {{z^2} + 4} \\right)\\left( {2{z^2} &#8211; 3z &#8211; 2} \\right) = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{{z^2} + 4 = 0}&amp; \\vee &amp;{2{z^2} &#8211; 3z &#8211; 2 = 0}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{{z^2} =\u00a0 &#8211; 4}&amp; \\vee &amp;{z = \\frac{{3 \\pm \\sqrt {9 + 16} }}{4}}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{z =\u00a0 &#8211; 2i}&amp; \\vee &amp;{z = 2i}&amp; \\vee &amp;{z =\u00a0 &#8211; \\frac{1}{2}}&amp; \\vee &amp;{z = 2}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8963' onClick='GTTabs_show(0,8963)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado $$P(z) = 2{z^4} &#8211; 3{z^3} + 6{z^2} &#8211; 12z &#8211; 8\\,\\,,z \\in \\mathbb{C}$$ Determine os n\u00fameros reais $a$, $b$ e $c$ tais que, para todo o n\u00famero complexo $z$, $$P(z) =&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19170,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,310],"tags":[427,18],"series":[],"class_list":["post-8963","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-numeros-complexos-12--ano","tag-12-o-ano","tag-numeros-complexos"],"views":1325,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat61.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8963","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=8963"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8963\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19170"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8963"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8963"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8963"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8963"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}