Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 2I / ii / Solution

Use the Lagrange interpolation polynomialIts square has degree $2n-2$, so the exactness already proved givesThe positive weights of Gaussian quadrature follow because ${\ell }_j$ is a nonzero continuous polynomial. Thus these Gaussian quadrature weights are in fact positive, rather than merely nonnegative.