Proving that a given polynomial is a constant multiple of a Legendre Polynomial

Question

If $p(x)$ is a polynomial of degree $n\ge1$ such that

$$\int_{-1}^1 (x^k)p(x)\,dx = 0$$

for $k = 0,1,2,3, \ldots, (n-1)$.

Show that $p(x) = cP_n(x)$ for some constant $c$.