7.3 POLYNOMIAL EQUATIONS

We now apply the discussion of the previous section to the special case where f(x) is a polynomial of degree n. We want to find the roots (real or complex) of the polynomial equation

(7.3 1a,b)

where a _n ?0, n>0.

We note the following facts regarding Equations (7.3 1a,b) with real or complex coefficients.

1. There are n (not necessarily distinct) real or complex roots.

2. If n is odd and all coefficients are real, there is at least one real root.

3. If all the coefficients are real and complex roots exist, they occur as conjugate pairs.

4. If x ₀ is a root of Equation (7.3 1a,b), then necessarily

   (7.3 2)

   where g(x) is a polynomial of degree (n ?1).

Newton s Method

To use Newton s method for computing the root of an equation [Equation (7.2 11)], we need to evaluate f(x _k) and f'(x _k). We describe a method, known as Horner s method, for calculating f(x _k) and f ?(x _k) where f is a polynomial. Recall that for any polynomial as given in Equations (7.3 1a,b) we can write

(7.3 3)

where

(7.3 4)

and R is a constant. Equations (7.3 1a,b, 3, 4) are compatible provided

(7.3 5a)

(7.3 5b)

(7.3 5c)

(7.3 5d)

Thus if (x ?x ₀) is not a factor of f(x), we have from Equation (7.3 3)

(7.3 6a,b)

In Equations (7.3 5a to d), the b _i are given in terms of a _n and x ₀. Therefore, starting from Equation (7.3 5a), we...