Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 6C (Numerical Analysis)

Let ${\ell }_i$ be the degree-$n-1$ cardinal polynomial with ${\ell }_i(c_j)={\delta }_{ij}$. Exactness applied to the nonnegative polynomial ${\ell }_i^2$ givesbecause $w>0$ and ${\ell }_i$ is nonzero. Thus every Gaussian quadrature weight is positive.

Symmetry gives nodes $\pm c$ with equal weights $b$. Exactness for $1$ and $x^2$ requiresThereforeOdd moments vanish automatically, so the rule is exact through degree three.