TITLE: Factoring $x^5 + x + 1$
QUESTION [6 upvotes]: Factor. $\displaystyle x^5 + x + 1$.

I'm trying my hand on these types of expressions. In the book, it mentions that for these types you should rewrite the $x$ term such as $2x-x$. But I don't know how to make any use of this technique.
What is this type of factoring called by the way?

REPLY [4 votes]: You can verify that both primitive $3$rd roots of unity are roots of this equation, since $5$ is $2$ mod $3$. This means the corresponding cyclotomic polynomial, $x^2+x+1$, divides $x^5+x+1$, and you can do polynomial division to find the other factor. 
The answer is $(x^2+x+1)(x^3-x^2+1)$.