Question

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

Answer 1

= (x + y)⁴ - 2xy . [ (x + y)² - xy]

= (x + y)².(x + y)² - 2xy. [ x² + 2xy + y² - xy]

= (x² + 2xy + y²).(x² + 2xy + y²) - 2xy.[x² + xy + y²]

= x².x² + x².2xy + x².y² + 2xy.x² + 2xy.2xy + 2xy.y² + y².x² + y².2xy + y².y² - 2xy.x² - 2xy.xy - 2xy.y²

= x⁴ + 2x³y + x²y² + 2x³y + 4x²y² + 2xy³ + x²y² + 2xy³ + y⁴ - 2x³y - 2x²y² - 2xy³

= x⁴ +  2x³y + 4x²y² + 2xy³ + y⁴

Answer 2

$\Big(x+y\Big)^\Big4-\Big2xy\Big[\Big(x\Big+y\Big)^\Big2 - xy \Big]$

→ Eu irei desenvolver os fatores e agrupá-los:

$\Big(x^2+2xy+y^2\Big).\Big(x+y\Big)^\Big2 - 2xy.\Big(x^2+y^2+xy\Big)$
$\Big(x+y\Big)^\Big2.\Big(x^2+y^2\Big) + 2xy\Big(x+y\Big)^\Big2 -2xy\Big(x^2+y^2\Big) - 2xy.\Big(xy\Big)$
$\Big(x^2+y^2\Big).\Big[\Big(x+y\Big)^\Big2 - 2xy.\Big] + 2xy\Big(x^2+y^2+2xy\Big) - 2xy.\Big(xy\Big)$
$\Big(x^2+y^2).\Big(x^2+y^2+2xy-2xy\Big) + 2xy\Big(x^2+y^2\Big) + 2xy\Big(2xy - xy \Big)$
$\Big(x^4+2x^2y^2+y^4\Big) + 2x^3y+2xy^3 + 2x^2y^2$
$x^4 +y^4 + 2x^3y+2xy^3+4x^2y^2$
$x^4+2x^3y+4x^2y^2+2xy^3+y^4$
 
 → Ou ainda :

$x^4+y^4 + 2xy\Big(x^2+2xy+y^2\Big)$
$x^4 + y^4 + 2xy\Big(x+y\Big)^\Big2$