$Multiply x^{2} + 2 y by x^{3} - 2 x y + y^{3} and find the value of the product for x = 1 and y = - 1.$

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Solution

$(x^{2} + 2 y) \times (x^{3} - 2 x y + y^{3})$

$= x^{2} (x^{3} - 2 x y + y^{3}) + 2 y (x 3^{-} 2 x y + y^{3})$

$= x^{5} - 2 x^{3} y + x^{2} y^{3} + 2 x^{3} y - 4 x y^{2} + 2 y^{3}$

$= x^{5} + x^{2} y^{3} - 4 x y^{2} + 2 y^{4}$

Put x = 1 and y = -1

$= (1)^{5} + (1)^{2} (- 1)^{3} - 4 (1) (- 1)^{2} + 2 (- 1)^{4}$

$= 1 + (1) (- 1) - 4 (1) (1) + 2 (1)$

$= 1 - 1 - 4 + 2$

$= - 2$

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