Here is the question:
Suppose $P_0, P_1, P_2, \dots$ are polynomials orthonormal with respect to the inner product $$(f,g)=\int_a^b f(x)g(x)W(x)dx,$$ where $W(x) > 0$ is a weight function and $P_n$ is of degree $n$. Is it true that $P_n$ has $n$ distinct roots in $(a,b)$?
Clearly $P_0$ has no roots and since $(P_0,P_1)=0$, I know $P_1$ must cross the $x$-axis at least once otherwise the integral would not equal $0$, so $P_1$ has one root in $(a,b)$. However, I'm not sure how to prove this for an arbitrary $n$-value (if it is true). I would appreciate any advice.