Unknown

56 2. Evaluation, Division, and Expansion

For the polynomial 4r3 + 2r2 - r - 1, verify that the Horner’s table is

4 2 -1 -1

0 0 0

4 2 -1 -1

4 6

4 6 5

8

4 14

and use this to determine a factorial power expansion of the given cubic.

(For convenience, we can simply delete the first two lines of the table.) [See

also Exercise 7.1.16-17, Explorations E.54 and E.57.l

2.2 Division of Polynomials

In Exercise 1.8, we observed that the bottom line of Horner’s table gave us
the coefficients of the quotient q(t) w h en we divided the polynomial p(t)
by (t - c) to obtain an identity of the form

p(t) = q(t)@ - c) + k.
By analogy with numbers, we can look upon this equation as representing
a division. The polynomial p(t) is divided by (t-c), yielding a quotient q(t)
and a remainder k. However, there is no reason to restrict our attention to
divisors of degree 1.
The exercises in this section will sketch in the details of a theory of
division for polynomials which is similar to that for integers. The extent
to which we can discuss division of one polynomial by another depends on
the domain from which the coefficients are taken, so let us establish some
terminology:
Let D be an integral domain, and D[t] be the set of all polynomials in
the variable t with coefficients in D. For short, if f(t) belongs to D[t], we
say that f(t) is a polynomial over D. For any pair f(t), g(t) of polynomials
in D[t], we say that g(t) divides f(t) (in symbols: g(t) ] f(t)) if there is a
polynomial h(t) in D[t] for which f(t) = g(t)h(t). In this situation, g(t) is
a divisor or factor of f(t) and f(t) a multiple of g(t).

Exercises

1. Let p(t) be any polynomial over an integral domain D and c be any
   element of D. Consider the equation

p(t) = (t - c)q(t) + k