 Orthogonal-node criterion for Gaussian quadrature

An $(n+1)$-node quadrature exact on degree $n$ is exact through degree $2n+1$ precisely when its node polynomial $p={\prod }_i(x-x_i)$ is orthogonal to every degree-$n$ polynomial for the integration weight. Necessity applies the rule to $pr$. Sufficiency divides an arbitrary degree-$2n+1$ polynomial as $pq+r$, with both $q,r$ of degree at most $n$; the integral and quadrature of $pq$ vanish while both agree on $r$.