Past exam of the mathematics course of the University of Cambridge / 2009 / ib / Paper 4 / 8C / Solution

For distinct Gaussian quadrature nodes, let ${\ell }_i$ be the Lagrange interpolation polynomial with ${\ell }_i(x_j)={\delta }_{ij}$. Since ${\ell }_i^2$ has degree $2n-2$, exactness givesThe last inequality follows from $w>0$ on the interval and the nonzero polynomial ${\ell }_i$ being nonzero on a subinterval. It does not require prior knowledge of the signs of the weights.

For $n=2$, the nodes are the roots of the supplied orthogonal polynomial $p_2$:Indeed exactness for $1$ gives $b_1+b_2={\int }_{-1}^1(1+x^2)dx=8/3$, and exactness for $x$ gives equality of the weights.