Let x1, x2, ..., xn be the roots of the Legendre polynomial Pn. If the Ai's are

Question:

Let x1, x2, ..., xn be the roots of the Legendre polynomial Pn. If the Ai's are defined as in Exercise 15, then the quadrature formula

will be exact for all polynomials of degree less than 2n.
(a) If 1 Pj(x1)A1 + Pj(x2)A2 + ........ + Pj(xn)An = (l, Pj) = 0
(b) Use the results from parts (a) and (b) of Exercise 15 to set up a nonhomogeneous n x n linear system for determining the coefficients A1, A2, ... , An.