HMMT 2005 T11

Let $P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}$ be a polynomial with real coefficients, $a_{n} \neq 0$. Suppose every root of $P$ is a root of unity, but $P(1) \neq 0$. Show that the coefficients of $P$ are symmetric; that is, show that $a_{n}=a_{0}, a_{n-1}=a_{1}, \ldots$