$x + y + z = 0$ find $(x^{3} + y^{3} + z^{3} - 3 x y z)$

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Solution

Let $x + y + z = 0$
or, $x + y = - z$
Cubing both sides we get,
$x^{3} + y^{3} + 3 x y (x + y) = - z^{3}$
or, $x^{3} + y^{3} - 3 x y z = - z^{3}$ [ Since $x + y = - z$ ]
or, $x^{3} + y^{3} + z^{3} - 3 x y z = 0$ .

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