Solving $z^4-2iz-i$

Question

I am new to the world of complex numbers. I've been going through a few problems in the Complex Numbers: A to Z book. I'm stuck at this problem:

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

I've managed to simplify the equation to

$z^3(z^4-2iz-i) = 2$

$z = 2^{1/3}$

I am not able to solve

$z^4-2iz-i$. Any hints or clues are appreciated!

Answer 1

The equation $z^7-2iz^4-iz^3-2=0$ reduces to

$$z^4(z^3-2i)-i(z^3-2i)=0$$

or

$$(z^3-2i)(z^4-i)=0$$

which is easy to solve.