Question

If $z_{1}, z_{2}, z_{3}$ are the roots of the equation $z^{3}-z^{2}(1+3 i)$ $+z(3 i-2)+2=0$, then $Im\left(z_{1}\right)+Im\left(z_{2}\right)+Im\left(z_{3}\right)$ is

Answer 1

$\left(z^{3}-z^{2}-2 z+2\right)+3 i\left(z-z^{2}\right)=0$
$(z-1)\left(z^{2}-2\right)+3 i z(1-z)=0$
$(z-1)\left(z^{2}-2-3 i z\right)=0$
$z=1, z=2 i, z=i$
$Im\left(z_{1}\right)+Im\left(z_{2}\right)+Im\left(z_{3}\right)=3$