(a) Prove that a polynomial p(x) = a0 + a1x + a2x2 + ∙∙∙∙∙ + anxn of degree < n vanishes at n-fl distinct points, so p(x1) = p(x2) =∙∙∙∙∙∙∙∙∙∙∙= p(xn+1) = 0, if and only if p(x) = 0 is the zero polynomial.
(b) Prove that the monomials I,x,x2,... ,xn are linearly independent.
(c) Explain why a polynomial p(x) = 0 if and only if all its coefficients α0 = a1 =•••=«" = 0. Use Lemma 4.12 and Exercise 2.3.37.