If $x + y + z = 0$ , show that $x^{3} + y^{3} + z^{3} = 3 x y z$ .

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Solution

To show
$x^{3} + y^{3} + z^{3} = x y z$
Proof
As we know,
$x^{3} + y^{3} + z^{3} - 3 x y z = (x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - z x)$
Putting $x + y + z = 0,$ we get
$x^{3} + y^{3} + z^{3} - 3 x y z = 0 (x^{2} + y^{2} + z^{2} - x y - y z - z x)$
$x^{3} + y^{3} + z^{3} - 3 x y z = 0$
$\Rightarrow x^{3} + y^{3} + z^{3} = 3 x y z$
Hence Proved

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