If $x=a+b, y=a \alpha+b \beta, z=a \beta+b \alpha$ where $\alpha, \beta$ are complex cube roots of u

Question

If $x=a+b, y=a \alpha+b \beta, z=a \beta+b \alpha$ where $\alpha, \beta$ are complex cube roots of unity then $x y z=$

Professor PC Thomas Classes · Answer

Since $\alpha, \beta$ are complex cube roots of unity\ $	herefore \quad 1+\alpha+\beta=0, \quad \alpha \beta=1$\ $x y z=(a+b)(a \alpha+b \beta)(a \beta+b \alpha)$\ $=(a+b)\left[a^{2} \alpha \beta+a b\left(\alpha^{2}+\beta^{2}ight)+b^{2} \alpha \betaight]$\ $=(a+b)\left[a^{2}-a b+b^{2}ight]=a^{3}+b^{3}$.\