I am studying numerical analysis and I came across this question:
Let $P_{n+1}(x)\in\Pi_{n+1}$ be orthogonal to $\Pi_n$ relative to a weight function $w(x)\geq 0$ on $[a,b].$ Denote by $x_0,x_1,\ldots,x_n$ the $(n+1)$ real roots of $P_{n+1}$ all in (a,b). Let $Q(f) = \sum_{i=0}^n A_i f(x_i)$ be a quadrature formula to approximate $\int_a^b w(x)f(x)dx$. Prove that if $Q(f)$ has precision $\geq n$, then the precision of $Q(f)$ is actually at least $2n+1$.
I am not sure how to go about solving this. I do know that if $Q(f)$ has precision $\geq n$ then it is exact for any polynomial with degree $\leq n$. Then I'm given information about $P_{n+1}$ and that it is orthogonal to any such polynomial relative to $w(x)$ but I am not sure how to proceed from here.
I should note that this is not for homework. This is self study.
Thank you for any help or comments in advance.