 Nodal-polynomial criterion for quadrature exactness

Let an $(n+1)$-node quadrature rule already be exact on all polynomials of polynomial degree at most $n$, for integration against a specified measure $\mu$. Its nodal polynomial isFor an integer $k\ge 0$, the rule is exact through polynomial degree $n+1+k$ if and only iffor every polynomial $r$ of polynomial degree at most $k$, assuming the needed moments exist. Indeed, polynomial division expresses every candidate $p$ as $p=Q_{n+1}r+s$, with $\mathrm{deg}s\le n$. The quadrature rule sees only $s$, since $Q_{n+1}$ vanishes at every quadrature node; exactness on $s$ leaves precisely the displayed orthogonality condition. Conversely, exactness applied to each product $Q_{n+1}r$ forces that condition. This is the mechanism behind the higher precision of Gaussian quadrature.