Intermediate Algebra (11th edition)

SECTION 5.2 Adding and Subtracting Polynomials 279

NOW TRY
EXERCISE 1
Write the polynomial in
descending powers of the
variable. Then give the
leading term and the leading
coefficient.

• 2 x^3 - 2 x^5 + 4 x^2 + 7 - x

Term Numerical Coefficient Degree

12 9

or 3 1

or 0

or 4

(^12)
3
x^2
3
=^1 x
2
3
=^1
3
x^2

• x^4 , - 1 x^4 - 1


• 6, - 6 x^0 - 6

3 x, 3 x^1

12 x^9

axn

Any nonzero constant

has degree 0.

NOTE The number 0 has no degree, since 0 times a variable to any power is 0.

A polynomial containing only the variable xis called a polynomial in x.(Other

variables may be used.) A polynomial in one variable is written in descending powers

of the variable if the exponents on the variable decrease from left to right.

Descending powers of x

x^5 - 6 x^2 + 12 x- (^5) Think of 12xas
and as .- 5 - 5 x^0
12 x^1

When written in descending powers of the variable, the greatest-degree term is

written first and is called the leading termof the polynomial. Its coefficient is the

leading coefficient.

Writing Polynomials in Descending Powers

Write each polynomial in descending powers of the variable. Then give the leading

term and the leading coefficient.

(a) is written as

(b) is written as

Each leading term is shown in color. In part (a), the leading coefficient is 8, and in

part (b) it is-1. NOW TRY

- 2 + m+ 6 m^2 - m^3 - m^3 + 6 m^2 +m-2.

y- 6 y^3 + 8 y^5 - 9 y^4 + 12 8 y^5 - 9 y^4 - 6 y^3 +y+12.

EXAMPLE 1

Some polynomials with a specific number of terms are given special names.

• A polynomial with exactly three terms is a trinomial.


• A two-term polynomial is a binomial.


• A single-term polynomial is a monomial.

Although many polynomials contain only one variable, polynomials may have more

than one variable. The degree of a term with more than one variable is defined to be

the sum of the exponents on the variables. The degree of a polynomialis the greatest

degree of all of its terms. The table gives examples.

Type of Polynomial Example Degree

Monomial

7

Binomial

3

2

Trinomial^5

12 (The terms have degrees 12,

5, and 2, and 12 is the greatest.)

x^3 y^9 + 12 xy^4 + 7 xy

• 3 + 2 k^5 + 9 z^4

t^2 + 11 t+ 4

11 y+ 8 1 1 y=y^12

6 + 2 x^3

5 x^3 y^710  13 + 7 = 102

0 17 = 7 x^02

NOW TRY ANSWER

1. ;
   
   
   • 2 x^5 ; - 2
   
   
   • 2 x^5 - 2 x^3 + 4 x^2 - x+ 7