Question

Recall that (by the Fundamental Theorem of Algebra) the only polynomial P(t) of degree n−1 that vanishes at n distinct points t₁,...,t_n ∈ R is P(t) ≡ 0. Using this, show that given any values b₁,...,b_n ∈ R, there is a polynomial Q(t) = ξ₁ + ξ₂t + ... + ξ_nt^n-1 such that Q(t_i) = b_i (and thus Q(t) takes precisely the given values at the given points). Hint: Show that the coefficient matrix of the corresponding system is nonsingular.

Answer 1