If q(x) is an arbitrary polynomial in Pn' it follows from Exercise 60(b) that
q (x) = c0p0(x) + . . . + cnpn(x).....................(1)
for some scalars c0, . . . , cn.
(a) Show that ci = q (ai) for i = 0, . . . , n, and deduce that q(x) = q(a0)p0(x) + · · · + q(an)Pn(x) is the unique representation of q(x) with respect to the basis B.
(b) Show that for any n + 1 points (a0, c0), (a1, c1), . . . , (an, en) with distinct first components, the function q(x) defined by Equation (1) is the unique polynomial of degree at most n that passes through all of the points. This formula is known as the Lagrange interpolation formula. (Compare this formula with Problem 19 in Exploration: Geometric Applications of Determinants in Chapter 4.)
(c) Use the Lagrange interpolation formula to find the polynomial of degree at most 2 that passes through the points
(i) (1, 6), (2, - 1), and (3, - 2)
(ii) (- 1, 10), (0, 5), and (3, 2)