Question

The common roots of the equations $z^3 + 2z^2 + 2z + 1 = 0$ and $z^{2000} + z^{202} + 1 = 0$ are

• A. $-1, \omega, \omega^2$
• B. $1, \omega, \omega^2$
• C. $\omega, \omega^2$
• D. $-\omega, \omega^2$

Answer 1

Answer: (C)

$ z^3 + 2z^2 + 2z + 1 = 0$
$\Rightarrow (z + 1) (z^2 + z + 1) = 0$
$\Rightarrow z = - 1, \omega, \omega^2$
Now $z = - 1$ does not satisfies $z^{2000} + z^{202} + 1 = 0$
but $\omega, \omega^2$ satisfies $z^{2000} + z^{202} + 1 = 0 = f(z)$
$\therefore f(\omega) = \omega^{2000} + \omega^{202} + 1$
$= (\omega^3)^{666} \omega^2 + (\omega^3)^{67} \omega^1 + 1$
$= \omega^2 + \omega + 1 = 0$
Similarly $\omega^2$ also satisfies the second equation, so $\omega, \omega^2$ are common roots.