Question

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

• A. $a+b$
• B. $a^2+b^2$
• C. $a^3+b^3$
• D. $0$

Answer 1

The correct option is C $a^3+b^3$

$x=a+b$

$y=\alpha a+\beta b$

$z=\beta a+\alpha b$

Now, $\alpha$ and $\beta$ are complex cube roots of unity.

So,

$\alpha =\omega$

$\beta ={\omega }^2$

$xyz=(a+b)(\omega a+{\omega }^{2}b)({\omega }^{2}a+\omega b)$

$=(a+b)\{{{\omega }^{3}a}^2+{\omega }^{2}ab+{\omega }^{4}ab+{\omega }^3{b}^2\}$

$=(a+b)\{{{\omega }^{3}a}^2+{\omega }^{2}ab+{\omega }^{1}ab+{\omega }^3{b}^2\}$

$=(a+b)\{a^2-ab+b^2\}$

$=a^3+b^3$