Solve:$z^7+az^3+2b$

Question

In my script a complex number is defined like that:
$z^7+az^3+2b$
if ($z=-1+i$) is a root, find ($a+b=?$)

Answer 1

Observe that $z^4=-4$ is real, so that $$z^8+az^4+2bz=0$$

immediately implies $$b=0.$$

Then

$$a=-z^4=4.$$

Answer 2

Hint: $z^2 =-2i$ so $$z^7 = 8i(-1+i)=-8-8i$$

and $$z^3 = -2i(-1+i)=2+2i$$ Plug this in to your equation and voila...