Context:
Let $\left\{\hat{p}_n\left(x\right)\right\}_{n=0}^{\infty}$ be the sequence of orthogonal monic polynomials with respect to the inner product $$\left(f,g\right):=\int_{a}^{b}f\left(x\right)g\left(x\right)dx$$
Question:
Prove that the degree $n$ polynomial $\hat{p}_n\left(x\right)$ does not have a repeated root. (i.e. that it has distinct roots $x_{i}$).
Attempt
Here I attempted to prove this by contraction:
Lets assume that $\hat{p}_n\left(x\right)$ does have a repeated root. Ifso then we can represent it as follows:
$$\hat{p}_n\left(x\right)=\left(x-x^{*}\right)^{2}q\left(x\right)$$
where $q(x)$ has degree $n-2$ which implies that $q(x)$ is orthogonal to $\hat{p}_n\left(x\right)$.
Thus: $$\left(\hat{p}_n,q\right)=0$$ $$\left(\left(x-x^{*}\right)^{2}q\left(x\right),q\left(x\right)\right)=0$$
If I could show that $\left(\left(x-x^{*}\right)^{2}q\left(x\right),q\left(x\right)\right)>0$ then I could achieve the contradiction.
I know from the properties of this inner product that $(q(x),q(x))>0$. I am unable to untangle the $\left(x-x^{*}\right)^{2}$ from $(q(x),q(x))$ though!
Question
How can I achieve this contradiction?
How can I show that $\left(\left(x-x^{*}\right)^{2}q\left(x\right),q\left(x\right)\right)>0$ ?
hints appreciated!
