Prove that $\frac{(x+2)^p-2^p}{x}$ is irreducible in $\mathbb{Z[x]}$

Question

Prove that $\frac{(x+2)^p-2^p}{x}$ is irreducible in $\mathbb{Z[x]}$

Can I get some help with this one? I mean my gut is obviously telling me to start with a binomial expansion, but I just don't know where to go from there. Any insight is appreciated!

Answer 1

$$\frac{(x+2)^p-2^p}{x}=x^{p-1}+\sum_{n=1}^{p-1}\binom{p}{n}2^{p-n}x^{n-1}$$ where $p\mid\binom pn$ for $0<n<p$ and $p^2\nmid \binom{p}{1}2^{p-1}$ if $p\neq 2$, hence Eisestein's criterion applies.

On the other hand, for $p=2$ then $\frac{(x+2)^p-2^p}{x}$ has degree $1$ hence it's irreducible.

Answer 2

Hint: Eisenstein's irreducibility criterion.