The Art and Craft of Problem Solving

164 CHAPTER 5 ALGEBRA

5.3.24 Can you generalize the idea used in Exam­

ple 5.3.2 on page 159?

(^1995 1)
VIri· Find
� 1 f(k)·
5.3.25 (AIME 1995) Let f(n) be the integer closest to

5.4 Polynomials

There is much more to polynomials than the mundane operations of adding, subtract­

ing, mUltiplying and dividing. This section contains a few important properties of

polynomials to review and/or learn.

First, some notation and definitions. Let A be a set of numbers that is closed under

addition and multiplication. Define

A[x] = {ao +alx+a 2 .x^2 + ... +an.tt : ai E A,n = 0,1,2,3, ... }

to be the set of polynomials with coefficients in A. The most common coefficient sets

that we use are Z, Q, JR and C. Occasionally we may use Zn, the integers modulo n

(see page 230). We call each expression of the form ajxi a term or monomial.

When writing an arbitrary polynomial , follow the convention of labeling ai as the

coefficient of xi. Consistent notation is clear and helps to avoid errors and confusion

with complicated manipulations. We define the degree of a polynomial to be the

highest exponent with a non-zero exponent. This coefficient is also called the leading

coefficient. If this coefficient is I, the polynomial is called monic. The coefficient ao

is called the constant term.

Polynomial Operations

Much of your early algebra education was devoted to adding, subtracting, multiply­

ing, and dividing polynomials. We won't insult your intelligence by reviewing the first

two operations, but it is worthwhile to think about multiplication and division. Mul­

tiplication is pretty easy, but it is important to use good notation. Make sure that you

understand the following notation, by mUltiplying out a few examples by hand.

If A (x) = }:aixi, B(x) = }:bixi and C(x) = }:Cixi = A (x)B(x) , then

Cj = aobj +a\bj-l + ... + ajbo = }: as bt.

s+t=j;

s,t::::O

Polynomials can be divided just like integers, and the result will be a quotient

and remainder. More formally, polynomials with coefficients in Z, Q, JR, C and Zn all

have a division algorithm that is analogous to the integer version (Problem 3.2. 17 ):

Let f(x) and g(x) be polynomials in K[x] , where K is one of Z,Q,JR, C

or Zn. Then

f(x) = Q(x)g(x) +R(x),

where Q(x),R(x) E K[x] and the degree of R(x) is less than the degree

of g(x). We call Q(x) the quotient and R(x) the remainder.