roots of polynomial equation

Question

How to find the roots of $x^5-2^5$ by hand. I see that we get a root of $x=2$ and 4 complex roots (should come in pairs). Not sure how to work out the complex roots. Do we need to convert to polar? Would that make it easier to see the other roots?

Answer 1

Write as $x^5 - 2^5e^{2\pi ik} = 0$

So your solutions are $x = 2{e^{2\pi ik/5}}$ for $k=-2,-1,0,1,2$

Answer 2

Yes, polar form would make it easier to see the other roots. Write $x=re^{i\theta},$ so that you're solving $$r^5e^{5i\theta}=2^5.$$ It's easy enough to find $r$ simply by taking the modulus, and if you know a few basic properties of the complex exponential function, you can find values of $\theta$ that give you all five solutions.