Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 6C / a / Solution

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.