Consider the polynomial $L \in P_5$ defined by $$L(x)=\frac{d}{dx^3}(1-x^2)^4$$ My question is to use a technique similar to the one used in the analysis of the Gaussian quadrature to show that Q has degree of precision 7
What The degree of precision of the quadrature formula is $\leq 2n-1$. We let $P$ be a polynomial of degree $\leq 2n-1$. We divide $P$ by $L$ be a polynomial. $P(x) = Q(x) L(x) + R(x)$ where Q and R are degree n−1 . We want to show that P has degree 7 but we know that L is degree 5. Does this mean that $Q$ and $R$ must be degree 4?