Irska

Each term in $p(x)$ is of the form $a_i{x}^i$, where $a_i$ is rational. Expanding the expression $a_i(x+n{)}^i-a_i{x}^i$, we see that $n$ is a factor in all terms. Thus it suffices to pick $n$ to equal the least common multiple of the denominators of the coefficients $a_i$.