{"id":41087,"date":"2015-06-26T00:08:53","date_gmt":"2015-06-26T03:08:53","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=41087"},"modified":"2023-03-22T23:23:44","modified_gmt":"2023-03-23T02:23:44","slug":"problema-o-que-e-um-inteiro-algebrico","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-o-que-e-um-inteiro-algebrico\/","title":{"rendered":".Problema: O que \u00e9 um inteiro alg\u00e9brico?"},"content":{"rendered":"<p style=\"margin:10px\">\n<div class=\"g1n\">\n<p class=\"tit\" style=\"line-height:25px;margin:10px\">Problema<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\nUm n\u00famero \u00e9 dito inteiro alg\u00e9brico quando ele \u00e9 raiz de alguma equa\u00e7\u00e3o polinomial de coeficientes inteiros.<br \/>\nPor exemplo, o 2 \u00e9 inteiro alg\u00e9brico, pois \u00e9 raiz da equa\u00e7\u00e3o [tex]x &#8211; 2 = 0[\/tex]; do mesmo modo, [tex]\\sqrt{2}[\/tex] tamb\u00e9m \u00e9 inteiro alg\u00e9brico, pois \u00e9 raiz de [tex]x^2 &#8211; 2 = 0[\/tex].<br \/>\n<strong>Prove que [tex]\\sqrt{2 + \\sqrt{3\\textcolor{white}{(}}}[\/tex] \u00e9 um inteiro alg\u00e9brico.<br \/>\n<\/strong><\/div>\n<p style=\"margin:40px\">\n<div class=\"g1n\">\n<p class=\"tit\" style=\"line-height:30px;margin:10px\">Solu\u00e7\u00e3o<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\nConsidere [tex]x =\\sqrt{2 + \\sqrt{3\\textcolor{white}{(}}}[\/tex] e observe que:<\/p>\n<ul>\n<li>[tex] x^2 = 2 + \\sqrt{3}[\/tex];<\/li>\n<li>[tex] x^2 &#8211; 2 = \\sqrt{3}[\/tex];<\/li>\n<li>[tex] (x^2 &#8211; 2)^2 = (\\sqrt{3})^2[\/tex];<\/li>\n<li>[tex] x^4 &#8211; 4x^2 + 4 = 3[\/tex];<\/li>\n<li>[tex] x^4 &#8211; 4x^2 + 1 = 0[\/tex].<\/li>\n<\/ul>\n<p>Portanto, [tex]\\sqrt{2 + \\sqrt{3\\textcolor{white}{(}}}[\/tex] \u00e9 um inteiro alg\u00e9brico.<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 15px;border: 0 none\"\/>\n<p class=\"g2\" style=\"line-height:20px;\">Solu\u00e7\u00e3o elaborada pelos <strong>Moderadores do Blog<\/strong> .<\/p>\n<\/div>\n<p style=\"margin:10px\">\n","protected":false},"excerpt":{"rendered":"<p>Problema Um n\u00famero \u00e9 dito inteiro alg\u00e9brico quando ele \u00e9 raiz de alguma equa\u00e7\u00e3o polinomial de coeficientes inteiros. Por exemplo, o 2 \u00e9 inteiro alg\u00e9brico, pois \u00e9 raiz da equa\u00e7\u00e3o [tex]x &#8211; 2 = 0[\/tex]; do mesmo modo, [tex]\\sqrt{2}[\/tex] tamb\u00e9m \u00e9 inteiro alg\u00e9brico, pois \u00e9 raiz de [tex]x^2 &#8211; 2 = 0[\/tex]. Prove que [tex]\\sqrt{2 &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problema-o-que-e-um-inteiro-algebrico\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":179,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-41087","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/41087","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/users\/179"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=41087"}],"version-history":[{"count":30,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/41087\/revisions"}],"predecessor-version":[{"id":185127,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/41087\/revisions\/185127"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=41087"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}