IR-TST 2011 Problem 8

Let $p$ be a prime and $k$ a positive integer such that $k \le p$ . We know that $f(x)$ is a polynomial in $\mathbb Z[x]$ such that for all $x \in \mathbb{Z}$ we have $p^k | f(x)$ . (a) Prove that there exist polynomials $A_0(x),\ldots,A_k(x)$ all in $\mathbb Z[x]$ such that $$f(x)=\sum_{i=0}^{k} (x^p-x)^ip^{k-i}A_i(x),$$(b) Find a counter example for each prime $p$ and each $k > p$ .