Let a be any number and let P(x) = a_n xⁿ + a_n−1xⁿ⁻¹ + · · · + a₁x + a₀ be a polynomial of degree n or less.

(a) Show that if P^(j)(a) = 0 for j = 0, 1, . . . , n, then P(x) = 0, that is, a_j = 0 for all j. Use induction, noting that if the statement is true for degree n − 1, then P'(x) = 0.
(b) Prove that T_n is the only polynomial of degree n or less that agrees with ƒ at x = a to order n. If Q is another such polynomial, apply (a) to P(x) = T_n(x) − Q(x).