Find a factorization for $P(z)=z^5+z+1$ with $z \in \mathbb{C}$.

Question

Find a factorization for $P(z)=z^5+z+1$ with $z \in \mathbb{C}$.

I am a bit confused actually. Is anyone is able to give me a hint to solve the problem involving complex numbers?

I think I can use a finite geometric series.

Answer 1

I like actual trial factorization. If it factors over the rationals (with no rational root) then Gauss content lemma says that it factors over the integers. There are just two possibilities, one of them works. $$ (x^3 + A x^2 + B x + 1)(x^2 + C x + 1), $$ $$ (x^3 + A x^2 + B x - 1)(x^2 + C x - 1). $$ See if you can solve for $A,B,C$ integers in either one