Simplifique ao máximo a expressão a seguir:.... Pergunta de ideia deLukyo

January 2020 1 234 Report

Simplifique ao máximo a expressão a seguir:
$~$
$\large\begin{array}{l}\mathsf{(x+y)^4-2xy\left[2(x+y)^2-xy\right]} \end{array}$
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Verified answer

$\Big(x+y\Big)^\Big4-2xy.\Big[2.\Big(x+y\Big)^\Big2-xy\Big]$

→ Vou desenvolver os termos e agrupá-los :

$=\Big(x^2+2xy+y^2\Big).\Big(x+y\Big)^\Big2 - 2xy.\Big[2.\Big(x^2+2xy+y^2\Big) - xy\Big]$
$=\Big(x+y\Big)^\Big2.\Big(x^2+y^2\Big) + 2xy\Big(x+y\Big)^\Big2 - 2xy.\Big(2x^2+2y^2+3xy\Big)$
$=\Big(x+y\Big)^\Big2.\Big(x^2+y^2\Big) + 2xy.\Big(x+y\Big)^\Big2 - 4xy.\Big(x^2+y^2\Big) - 6x^2y^2$
$=\Big(x^2+y^2\Big).\Big[\Big(x+y\Big)^\Big2 - 4xy\Big] + 2xy.\Big(x+y\Big)^\Big2 - 6x^2y^2$
$=\Big(x^2+y^2\Big).\Big[x^2+y^2+2xy-4xy\Big] + 2xy.\Big(x^2+y^2\Big) + 2xy.\Big(2xy\Big) - 6x^2y^2$
$=\Big(x^2+y^2\Big).\Big[x^2-2xy + y^2\Big] + 2xy.\Big(x^2+y^2\Big)-2x^2.y^2$
$=\Big(x^2+y^2\Big).\Big[x^2-2xy+y^2+2xy\Big] - 2x^2y^2$
$=\Big(x^2+y^2\Big).\Big[x^2+y^2\Big] - 2x^2y^2$
$=x^4+2x^2y^2+y^4-2x^2y^2$
$=x^4 + y^4$

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