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.