Solution
= Solution
For odd $u$, define $q(x)=-p_n(-x)$. It is another <polynomial> in $\Pi_n$. Changing variables $x\mapsto-x$, and using evenness of the weight $x^2$, gives $\|u-q\|=\|u-p_n\|$. Thus $q$ is also a best <polynomial>. Uniqueness from part (b) gives $p_n(x)=-p_n(-x)$, proving
$$
\boxed{p_n\text{ is odd}.}
$$
Equivalently, the even component of a candidate is <orthogonal> to the odd residual and can only increase the squared error.