Question

If $x=a+b,y=a\beta +b\gamma ,z=a\gamma +b\beta$, where $\gamma$ and $\beta$ are cube roots of unity, then $xyz=a^3+b^3.$ If this is true enter 1, else enter 0.

Answer 1

Let $\gamma =w$ and $\beta =w^2$ where $w^3=1$
And $1+w+w^2=0$ ...(i)
Now $yz$
$(a\gamma +b\beta )(a\beta +b\gamma )$
$=(aw+b{w}^2)(a{w}^2+bw)$
$=w^2(a+bw)(aw+b)$
$=w^2(a^{2}w+ab+ab{w}^2+b^{2}w)$
$=w^2\left(\right(a^2+b^2)w+ab(1+w^2\left)\right)$
$=w^2\left(\right(a^2+b^2)w-ab(w\left)\right)$
$=w^3(a^2-ab+b^2$
$=(a^2-ab+b^2)$
$=\frac{a^3+b^3}{a+b}$
Hence
$xyz=a^3+b^3$.