Prove that there is only one polynomial P3(x) among all polynomials of degree < 3 that satisfy the interpolating conditions 11. (а) P3(x;) = yi, i = 0, 1, 2, 3 where the x;'s are distinct. Hint: Generalize the proof given in and following (4.10) for the uniqueness of P2(x). (b) Give a proof of uniqueness for P(x) in Theorem 4.1.5.

Given the following set S of polynomials, find a polynomial p(x) in P2 so that SU {p(x)} spans P₂ and a non-zero polynomial q(x) in P2 so that SU{q(x)} does not span P2. Use the '^' character to indicate an exponent, e.g. 5x^2–2x+1. 5-|-5x²+4x-9, −8x²-6x+8] p(x) = 0 q(x) = 0