2010 AMC 10B Problem 25

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

P(1) = P(3) = P(5) = P(7) = a, and
P(2) = P(4) = P(6) = P(8) = -a.

What is the smallest possible value of a?

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

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.