Question

Sum of the common roots of $z^{2006}+z^{100}+1=0$
and $z^3+2z^2+2z+1=0$ is

• A. 0
• B. -1
• C. 1
• D. 2

Answer 1

The correct option is B

-1

$(z^3+1)+(2z^2+2z)=0$
$\Rightarrow (z+1)(z^2-z+1)+2z(z+1)=0\Rightarrow (z+1)(z^2+z+1)=0\Rightarrow z=-1,w,w^2$
Out of these $w$ and $w^2$ satisfy
$z^{2006}+z^{100}+1=0$
$\therefore \text{Sum of common roots}=w+w^2=-1$