Question

If $z=2+i$ and $z^3+3z^2-9z+35=kz$, then the value of $k$ is

Answer 1

$z=2+i\Rightarrow (z-2{)}^2=-1\Rightarrow z^2-4z+5=0\cdots (1)$

Using equation $\left(1\right)$, we get
$z^3+3z^2-9z+35=z(z^2-4z+5)+7z^2-14z+35=0+7(z^2-4z+5)+14z=14z\therefore k=14$