If q(x) is an arbitrary polynomial in it follows from Exercise 60(b) that (1) for some
Chapter 6, Problem 6.2.61(choose chapter or problem)
If q(x) is an arbitrary polynomial in it follows from Exercise 60(b) that (1) for some scalars (a) Show that for and deduce that 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),(i) (1, 6), (2, 1), and (3, 2)(ii) (1, 10), (0, 5), and (3, 2)6 . . . ,(an, cn) with distinct first components, the functionq(x) defined by equation (1) is the uniquepolynomial of degree at most n that passesthrough all of the points. This formula is knownas the Lagrange interpolation formula. (Comparethis formula with in Exploration:Geometric Applications of Determinantsin Chapter 4.)(c) Use the Lagrange interpolation formula to find thepolynomial of degree at most 2 that passesthrough the points
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