Chapter 3.2, Problem LC

LINKING concepts... For Individual or Group Exploration

Horner’s Method

A fourth-degree polynomial in x such as 3x⁴+ 5x³+ 4x²+ 3x + 1contains all of the powers of x from the first through the fourth. However, any polynomial can he written without powers of x. Evaluating a polynomial without powers of x (Horner’s method) is somewhat easier than evaluating a polynomial with powers.

a) Show that {[(3x + 5)x + 4]x + 3}x + 1 = 3x⁴ + 5x³ + 4x² + 3x + 1 is an identity.

b) Rewrite the polynomial P(x) = 6x⁵ –3x⁴ + 9x³ + 6x² –8x 12 without powers of x as in part (a).

c) Find P(2) without a calculator using both forms of the polynomial.

d) For which form did you perform fewer arithmetic operations?

e) Explain in detail how to rewrite any polynomial without powers of x.

f) Explain how this new form relates to synthetic division and the remainder theorem.