Factorize:
$x^{3} + y^{3} + z^{3} = 3 x y z .$

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Solution

$x^{3} + y^{3} + z^{3} - 3 x y z = 0$
$\Rightarrow x^{3} + y^{3} + 3 x^{2} y + 3 x y^{2} + z^{3} - 3 x y z - 3 x^{2} y - 3 x y^{2} = 0$
$\Rightarrow (x + y)^{3} + z^{3} - 3 x y (x + y + z) = 0$
$\Rightarrow (x + y + z) ((x + y)^{2} + z^{2} - (x + y) z) - 3 x y (x + y + z) = 0$
$\Rightarrow (x + y + z) (x^{2} + 2 x y + y^{2} + z^{2} - x y - x z - 3 x y) = 0$
$\Rightarrow (x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - z x) = 0$ .

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