The question is:
Establish that alpha=2+3i is a root of the quintic equation
3z^5 + 2z^4 - 64z^3 +384z^2 - 667z +182 =0
and find the remaining roots.
Now, I know that as it's a quintic there will be five roots. As I'm told that 2+3i is a root, I can know that 2-3i is also a root. And once I get so far I will be able to divide, take the one real root and the other two complex roots.
But I'm falling at the first hurdle, how do I show that 2+3i is a root?
This is the part I don't get, or at least I'm being thick.
I've got an example written down:
p(z) = z^4 - 4z^3 +55z^2 -8z +106
2+7i and 2-7i are roots.
So p(z)=(z^2-4z+53)(z^2+2)
I understand how the division and eventual solutions are obtained, I jsut don't quite understand the step to what the quadratic from the solutions is?
It's probably something incredibly simple isn't it?