Past exam of the options II course of the University of Oxford / 2011 / P / P3 / d

By part (c), the best polynomial of degree at most $2$ is odd, hence has the form $p_2(x)=cx$. The orthogonal projection condition with $q=x$ givesThe residual is odd, so its weighted inner products with the even polynomials $1$ and $x^2$ vanish automatically. Thus all three normal equations hold and