Question

If $x = a + b, y = a \omega +b \omega ^2$ and $z = a \omega^2 + b \omega$, then which one of the following is true.

• A. $x + y + z \neq 0$
• B. $x^2 + y^2 + z^2 = a^2 + b^2$
• C. $x^3 + y^3 + z^3 = 3(a^3 + b^3)$
• D. $xyz = 2(a^3 + b^3)$

Answer 1

Answer: (C)

$x^3 + y^3 + z^3 = 3(a^3 + b^3)$

Answer 2

We have $x+y+z=a+b+a\omega+b\omega^{2}+a\omega^{2}+b\omega$ $=a\left(1+\omega+\omega^{2}\right)+b\left(1+\omega+\omega^{2}\right)$ $=a\left(0\right)+b\left(0\right)=0$ $\therefore x+y+z=0$ $x^{2}+y^{2}+z^{2}=6\,ab$ (verify this) $xyz=a^3+b^3$ (verify this) $x^3+y^3+z^3=3(a^3+b^3)$ $[\because x^3+y^3+z^3=3xyz$ since $x + y + z = 0]$