AMC 12 2010b Problem 21

Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that

What is the smallest possible value of $a$?

\textbf{(A)}\ 105 \textbf{(B)}\ 315 \textbf{(C)}\ 945 \textbf{(D)}\ 7! \textbf{(E)}\ 8!