Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 2 / 20 / b / Solution

The polynomial $v$ has leading term $x^{2l}$. Its $l$th derivative has leading term $(2l)!x^l/l!$, and all other terms have lower degree. Thus the Rodrigues formula for Legendre polynomials defines a polynomial with nonzero leading coefficientIt therefore has degree exactly $l$, including the constant case $P_0=1$.