Consider the inner product space $P_2$ with $\langle p, q \rangle = \int_{-1}^1 p(x)q(x) dx$. If $p_0(x) = 1$ and $p_1(x) = x$, find the third orthogonal polynomial $p_2(x)$ in the sequence of Legendre polynomials (normalized such that $p_2(1) = 1$).