Past exam of the numerical analysis course of the University of Oxford / 2014 / 2 / d / Solution

By parity preservation by polynomial least-squares projection, $p_2(x)=\beta x$: the constant and quadratic coefficients vanish. The remaining normal equation isThereforeThe residual is automatically orthogonal to $1$ and $x^2$ by parity and to $x$ by construction, so it is orthogonal to every element of ${\Pi }_2$. This checks the full linear least squares condition, rather than only a necessary coefficient equation.