31. Letbe defined as in Problem 30 above and show:
(a) if f(x) is a polynomial of degree less than n.
(b) if f(x) = a₀xⁿ + a₁x^{n -1} + ... + a_n.

32. Let f(x) = a + bx + cx². Let r = f(0), s = f(1) - f(0), and t = f(2) - 2f(1) + f(0).

(a) Show that f(x) = r + sx + tx(x - 1)/2.

(b) Generalize this problem.

33. Let f(x) = 5x⁴ - 6x³ - 3x² + 8x + 2. Use repeated synthetic division to find numbers a, b, c, d, and e such that
 

34. Use the method of the alternate solution for Example 3 in Section 8.1 to do Problem 33.

35. Let f(x) = x³ + ax² + bx + c, and let a, b, c, and r be complex numbers. Show that
 

and express s in terms of a and r.

36. Let f(x) = x³ + ax² + bx + c, and let r be a root of f(x) = 0. Show that f(x) is divisible by
(x - r)² if and only if 3r² + 2ar + b = 0.  [If m is an integer greater than 1 and a polynomial f(x) is divisible by (x - r)^m but not by (x - r)^m+1, we say that r is a multiple root of f(x) = 0 with multiplicity m.]

37. Let f(x) = x³ + ax² + bx + c, g(x) = 3x² + 2ax + b, and h(x) = 6x + 2a. Show that
f(x) = (x - r)³ if and only if f(r) = g(r) = h(r) = 0.

38. Let f(x) = x⁴ + ax³ + bx² + cx + d. Find s, t, and u in terms of a, b, c, and r such that
 

39. Do the methods of this chapter enable you to solve x³ - 3x + 1 = 0?