Solve the equation in complex plane

Question

I am having a problem with a complex equation:

$$z^7 - 2iz^4 - iz^3 - 2 = 0.$$

I do not know where to begin. I tried to multiply out the complex part by the $z$ substituting it with $x+iy$. I also tried converting into polar form and solving. I know that in the end I have equate the complex part with the real part, however I do not know how to get there.

Any help will be welcome.

Answer 1

$$ z^7-2iz^4-iz^3-2 = (z^7-iz^3)-2i(z^4-i) = z^3(z^4-i)-2i(z^4-i)=(z^4-i)(z^3-2i) $$ so the roots are $e^{\pi i/8}i^k$ for $k\in[1,4]$ and $\sqrt[3]{2}e^{\pi i/6}\omega^j$ for $j\in[1,3]$.

Answer 2

Hint: The equation can be written in the form $$(z^3 - 2 i) (z^4 - i) = 0.$$

(I found this by noting by inspection that all fourth roots of $i$ solve the equation.)