factoring $x^n+x+1$

Question

Is there a way of factoring a polynomial of the general form $$x^n+x+1$$ in the ring $\mathbb C[x]$ or $\mathbb R[x]$ or $\mathbb Z [x]$ for any $n \in \mathbb N$? (Or perhaps with certain conditions on $n$?)

Answer 1

It is easy to show that $x^2+x+1$ is a factor for $n\equiv2\bmod3$ (the other factor has coefficients $1,0,-1,1,0,-1,\dots$).