Prove $(3 + 3 w + 5 w^{2})^{6} - (2 + 6 w + 2 w^{2})^{3} = 0$

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Solution

let $w, w^{2}, 1$ be the cube roots of unity
$w + w^{2} + 1 = 0$
$= ((3 w + 3 w^{2} + 3) + 2 w^{2})^{6} - ((2 + 2 w + 2 w^{2}) + 4 w)^{3}$
$= 2^{6} (w^{6})^{2} - 2^{6} w^{3}$
$= 2^{6} (w^{6})^{2} - 2^{6} w^{3}$ , $(w^{3} = 1)$
$= 2^{6} - 2^{6} = 0$
Hence Proved

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