Question

Sum of common roots of the equations $z^3+2z^2+2z+1=0$ and $z^{1985}+z^{100}+1=0$ is

• A. $-1$
• B. $1$
• C. $0$
• D. $2$

Answer 1

The correct option is A $-1$

We have, $z^3+2z^2+2z+1=0$

$\Rightarrow (z^3+1)+2z(z+1)=0$

$\Rightarrow (z+1)(z^2+z+1)=0$

$\Rightarrow z=-1,\omega ,{\omega }^2$

Since $z=-1$ does not satisfy $z^{1985}+z^{100}+1=0$ while $z=\omega ,{\omega }^2$satisfy it; hence, sum is $\omega +{\omega }^2=-1$.