Solving a fourth degree polynomial over C.

Question

I have the following polynomial: $$2z^4 - 4z^3 + 21z^2 - 36z + 27$$

One factor is $z - 3i$. I need to find all the other roots. How would I go about doing this?

Thanks in advance.

Answer 1

the other factor must be $$z+3i$$ since the coefficients are all real your equation can be written as $$(9+z^2)(3-4z+2z^2)=0$$

Answer 2

As the coefficients are real, $z+3i$ is also a factor. So the polynomial is divisible by $z^2+9$.