Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 3 / 6B / Solution

Let $p(x)={\prod }_{i=0}^n(x-x_i)$ be the monic node polynomial. Suppose first that the quadrature is exact through degree $2n+1$. For any polynomial $r$ of degree at most $n$, $pr$ has degree at most $2n+1$, while every sampled value is zero. Exactness givesThus $p$ is an orthogonal polynomial of degree $n+1$, proving necessity in the orthogonal-node criterion for Gaussian quadrature.

Conversely, suppose the nodes are the zeros of such a polynomial. Rescale it to be monic without changing its zeros or orthogonality. For any $f$ of degree at most $2n+1$, polynomial division givesOrthogonality makes $\int pq=0$; the quadrature of $pq$ is zero because $p(x_i)=0$. The assumed degree-$n$ exactness applies to $r$, soHence degree-$2n+1$ exactness is equivalent to orthogonal nodes. The given distinct-zero property ensures the $n+1$ node values are independent interpolation data; the degree-$n$ exactness fixes the corresponding interpolatory weights. This is the construction of Gaussian quadrature.