Question

If x = a+b, $y=a\alpha +b\beta$ and $z=a\beta +b\alpha$ where $\alpha$ and $\beta$ are complex cube roots of unity, show that
$xyz=a^3+b^3$

Answer 1

$\alpha =\omega and\beta ={\omega }^2$
Then $xyz=(a+b)(a\omega +b{\omega }^2)(a{\omega }^2+b\omega )=a^3+b^3$