Unknown

1

Fundamentals

1.1 The Anatomy of a Polynomial of a Single

Variable

3t3 - 7t2 + 4t + 1 and 8t6 - t5 + fit” + &it - 1 are polynomials. So are
(t2 - l)/(t - 1) for t # 1 an d cos2(arccost) for -1 5 t 5 1. But t1i3 and
sint are not polynomials. What do we mean by a polynomial?

A function of a single variable t is a polynomial on its domain

if we can put it in the form

UJ” + un-p +. 1. + a1t + do

where ura, an-l,... , al, uo are constants.

This definition says that every polynomial can be expressed as a finite
sum of monomial terms of the form akt” in which the variable is raised
to a nonnegative integer power. We use the convention that to = 1, so
that uot’ = uc. To begin with, we will look at polynomials for which the
constants ui are real or complex numbers.
With this definition in hand, we can immediately agree that the first two
functions are polynomials. For the next two, we have to remove a disguise:

(t2 - l)/(t - 1) = t + 1

cos2(arccost) = cos20 = 2cos20 - 1 = 2t2 - 1

where t = cos6, 0 5 6 5 ?r. The last two, t’13 and sint, do not look
like polynomials, but how can we decide for sure? One way is to look for
properties which distinguish these functions from polynomials. One of the
tasks of this book will be to provide a number of such characteristics to
assist in this sort of classification question.
In the title of this section, we promised you some anatomy. Here it is. For
the polynomial, u,t” + u,-#-’ +... + ult + uc, with a, # 0, the numbers
si (0 < i < n) are called coefficients. a, is the leading coefficient, and a#
the leading term. uo is the constant term or the constant coeficient. al is
the linear coefficient and art the linear term. When the leading coefficient
a, is 1, the polynomial is said to be manic.
The nonnegative integer n is the degree of the polynomial; we write
degp = n. A constant polynomial has but a single term, uc. A nonzero