Nodal-polynomial criterion for quadrature exactness
= Nodal-polynomial criterion for quadrature exactness
An $(n+1)$-node <quadrature rule> already exact through degree $n$ is exact through degree $n+1+k$ exactly when its nodal polynomial
$$
Q_{n+1}(x)=\prod_{i=0}^n(x-x_i)
$$
is orthogonal to every polynomial of degree at most $k$.