Solution
= 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 coefficient
$$
\boxed{P_l(x)=\frac{(2l)!}{2^l(l!)^2}x^l+\text{lower-degree terms}.}
$$
It therefore has degree exactly $l$, including the constant case $P_0=1$.