When we can freely "chose" points $\displaystyle x_k$ anywhere then we can optimally place them...thus we get the Gaussian Quadrature formulas.

My question is what is the Degree of Precision for Gaussian Quadrature scheme?

that uses $\displaystyle (n+1)$ points (n sub intervals)?

$\displaystyle x^*_k$ and $\displaystyle a^*_k $ denote the Gaussian quad points...