{"id":1673,"date":"2022-01-30T18:07:31","date_gmt":"2022-01-30T21:07:31","guid":{"rendered":"http:\/\/localhost\/?page_id=1673"},"modified":"2023-01-06T14:52:48","modified_gmt":"2023-01-06T17:52:48","slug":"equacoes-polinomiais-do-segundo-e-terceiro-grau","status":"publish","type":"page","link":"http:\/\/localhost\/index.php\/ensino\/fundamentos-de-algebra\/equacoes-polinomiais-do-segundo-e-terceiro-grau\/","title":{"rendered":"Equa\u00e7\u00f5es polinomiais do segundo grau"},"content":{"rendered":"
\nConsidere um polin\u00f4mio
\n\\[
\nf(x)=x^2+bx+c\\in\\F[x]
\n\\]
\nonde $\\F$ \u00e9 corpo arbitr\u00e1rio no qual $1+1\\neq 0$. Queremos determinar as ra\u00edzes de $f(x)$. Note que
\n\\[
\nx^2+bx+c=(x+b\/2)^2-b^2\/4+c,
\n\\]
\ne assim a equa\u00e7\u00e3o $f(x)=0$ \u00e9 equivalente \u00e0 equa\u00e7\u00e3o
\n\\[
\n(x+b\/2)^2=b^2\/4-c=\\frac{b^2-4c}4;
\n\\]
\nou seja
\n\\[
\nx+b\/2=\\pm\\frac{\\sqrt{b^2-4c}}2.
\n\\]
\nAssim as ra\u00edzes do polin\u00f4mio s\u00e3o $x_1$ e $x_2$ onde
\n\\[
\nx_1=\\frac{-b+\\sqrt{b^2-4c}}2\\quad\\mbox{e}\\quad x_2=\\frac{-b-\\sqrt{b^2-4c}}2.
\n\\]
\nAs duas ra\u00edzes s\u00e3o frequentemente escritas na forma
\n\\[
\nx_{1,2}=\\frac{-b\\pm\\sqrt{b^2-4c}}2.
\n\\]<\/p>\n

Quando o polin\u00f4mio $f(x)\\in\\F[x]$ est\u00e1 na forma mais geral $f(x)=ax^2+bx+c$ com $a\\neq 0$, as ra\u00edzes de $f(x)$ s\u00e3o as mesmas que as ra\u00edzes de
\n\\[
\nx^2+(b\/a)x+c\/a
\n\\]
\nque s\u00e3o
\n\\[
\n\\frac{-b\/a\\pm\\sqrt{(b\/a)^2-4c\/a}}2=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}.
\n\\]<\/p>\n

\nVamos achar todos os n\u00fameros inteiros $x\\in\\Z$ para os quais $2x^2+3x-7\\equiv 0\\pmod{11}$. A congru\u00eancia \u00e9 equivalente \u00e0 equa\u00e7\u00e3o quadr\u00e1tica
\n\\[
\n\\overline 2x^2+\\overline 3x+\\overline 4=\\overline 0
\n\\]
\nsobre $\\Z_{11}$. As solu\u00e7\u00f5es nesta equa\u00e7\u00e3o s\u00e3o
\n\\begin{align*}
\nx_{1,2}&=\\frac{-\\overline 3\\pm\\sqrt{\\overline 3^2-4\\cdot \\overline 2\\cdot \\overline 4}}{\\overline 4}\\\\&=\\frac{-\\overline 3\\pm\\sqrt{-\\overline {23}}}{\\overline 4}\\\\&=\\frac{-\\overline 3\\pm\\sqrt{-\\overline 1}}{\\overline 4}.
\n\\end{align*}
\nComo $11\\equiv 3\\pmod 4$, $-\\overline 1$ n\u00e3o \u00e9 quadrado em $\\Z_{11}$; ou seja, $\\sqrt{-\\overline 1}$ n\u00e3o existe em $\\Z_{11}$ e a equa\u00e7\u00e3o n\u00e3o possui solu\u00e7\u00f5es. Portanto n\u00e3o existe $x\\in\\Z$ tal que $2x^2+3x-7\\equiv 0\\pmod{11}$.<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Considere um polin\u00f4mio \\[ f(x)=x^2+bx+c\\in\\F[x] \\] onde $\\F$ \u00e9 corpo arbitr\u00e1rio no qual $1+1\\neq 0$. Queremos determinar as ra\u00edzes de $f(x)$. Note que \\[ x^2+bx+c=(x+b\/2)^2-b^2\/4+c, \\] e assim a equa\u00e7\u00e3o $f(x)=0$ \u00e9 equivalente \u00e0 equa\u00e7\u00e3o \\[ (x+b\/2)^2=b^2\/4-c=\\frac{b^2-4c}4; \\] ou seja \\[ x+b\/2=\\pm\\frac{\\sqrt{b^2-4c}}2. \\] Assim as ra\u00edzes do polin\u00f4mio s\u00e3o $x_1$ e $x_2$ onde \\[ x_1=\\frac{-b+\\sqrt{b^2-4c}}2\\quad\\mbox{e}\\quad … Continue reading Equa\u00e7\u00f5es polinomiais do segundo grau<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":1193,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"_links":{"self":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1673"}],"collection":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/comments?post=1673"}],"version-history":[{"count":4,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1673\/revisions"}],"predecessor-version":[{"id":2013,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1673\/revisions\/2013"}],"up":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1193"}],"wp:attachment":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/media?parent=1673"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}