$x^n+x+1$ is irreducible over $\mathbb{Z}[x]$ if and only if $n=2$ or $n \not\equiv 2 \mod 3.$

Question

I have disposed of the only if part quite easily using the fact that $x^2+x+1$ is a factor of $x^{3n+2}+x+1.$

Eisenstein's criterion fails to be useful for the if part. Cohen's irreducibility criterion is ridiculous as well given that the Bunyakovsky Conjecture is still open. Any more ideas?

The proof of this statement is necessary to solve a problem intended for high schoolers, so surely there must be an elegant, simple solution. If not, please tell me, and I will send my grievances to the director.