Let a0, a1, . . . , an be n + 1 distinct real numbers. Define polynomials p0(x), p1 (x), . . . , Pn(x) by

These are called the Lagrange polynomials associated with a0, a1, . . . , an [Joseph-Louis Lagrange (1736- 1813) was born in Italy but spent most of his life in Germany and France. He made important contributions to such fields as number theory, algebra, astronomy, mechanics, and the calculus of variations. In 1773, Lagrange was the first to give the volume interpretation of a determinant (see Chapter 4).]
(a) Compute the Lagrange polynomials associated with a0 = 1, a1 = 2, a2 = 3.
(b) Show, in general, that

(x -a)xa,xa)..(x- a,) pi(x) = p,(q) = 0 I if i = j