Show that $\displaystyle (x+1)^n \equiv x^n + 1^n \mod n$ in $\displaystyle \mathbb{Z}[x]$ if and only if n is a prime.

(In other words,all coefficients in the difference polynomial $\displaystyle (x + 1)^n - (x^n + 1^n)$ are divisible by n. This is a relation in the polynomial ring, not for individual integers x.)