One task performed by scientists and engineers is finding the roots of a polynomial f(x), i.e. given
f(x) = cnxn + cn1 xn1 + cn2 xn2 + . . . + c2x2 + c1x1 + c0x0 = 0
Find x0 such that f(x0) = 0. Recall that a polynomial f(x) of degree n has a maximum n distinct roots. Some roots may be real numbers and some may be complex numbers. A quadratic function, for example, may have a graph similar to one of the following:
Factoring is one way of determining the roots of a polynomial f(x), if the polynomial is factorable.
f(x) = 2x4 7x3 + 4x2 + 7x 6 = 0
f(x) = 2x4 + 4x2 6 7x3 + 7x = 0
f(x) = 2 ( x4 + 2x2 3 ) 7x ( x2 1 ) = 0
f(x) = 2 ( x2 + 3 ) ( x2 1 ) 7x ( x2 1 ) = 0
f(x) = [ 2 ( x2 + 3 ) 7x ] ( x2 1 ) = 0
f(x) = ( 2x2 + 6 7x ) ( x2 1 ) = 0
f(x) = ( 2x2 7x + 6 ) ( x2 1 ) = 0
f(x) = ( 2x 3 ) ( x 2 ) ( x + 1 ) ( x 1 ) = 0
x1 = 3/2 x2 = 2 x3 = 1 x4 = 1
However, not all polynomials are factorable.
EXAMPLE: f(x) = x5 2x3 5x2 + 2 = 0 is not factorable.
If a nonfactorable polynomial is a quadratic function, then the quadratic formula is used. For polynomials of higher degrees, other more general techniques are required. We shall look at several techniques for estimating roots of polynomials, namely the Bisection Method, RegulaFalsi Method, and Newton's Method. To simplify the problem, find the first positive root of a given polynomial f(x). For the initial interval [a, b], find the unit interval containing the first positive root.
However, let us first take a look at an efficient way of evaluating a polynomial using Horner's Method.