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

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Solution

Consider the following question.

$x^{3} + y^{3} = (x + y) (x^{2} + y^{2} - x y)$
We know that,
${(x + y)}^{3} = x^{3} + y^{3} + 3 x y (x + y)$
$x^{3} + y^{3} = {(x + y)}^{3} - 3 x y (x + y)$
$= (x + y) (x^{2} + y^{2} + 2 x y - 3 x y)$
$= (x + y) (x^{2} + y^{2} - x y)$

Hence, this is the required answer.

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