Beginning Algebra, 11th Edition

SECTION 5.7 Dividing Polynomials^341

The parts of a division problem are named here.

Dividend

Quotient

Divisor

Dividing a Polynomial by a Monomial

Divide by

Use the preceding rule,

with replaced by.

Quotient rule

CHECK Multiply:

Original polynomial

Divisor Quotient (Dividend)

Because division by 0 is undefined, the quotient is undefined if

or m=0.From now on, we assume that no denominators are 0.

5 m 5 m (^2) = 0,
(^5) - 10 m 3
5 m^2
5 m^2 # 1 m^3 - 2 m 2 = 5 m^5 - 10 m^3.

=m^3 - 2 m

= + -

5 m^5

5 m^2

-

10 m^3

5 m^2

5 m^5 - 10 m^3

5 m^2

5 m^5 - 10 m^35 m^2.

EXAMPLE 1

12 x^2 + 6 x

6 x

= 2 x+ 1

This becomes

(^2) a ,not 2 a.

The quotient is nota polynomial because of the presence of the

expression which has a variable in the denominator. While the sum, difference, and

product of two polynomials are always polynomials, the quotient of two polynomials

may not be a polynomial.

CHECK

Divisor Quotient

should equal Dividend.

Distributive property

= 16 a^5 - 12 a^4 + 8 a^2 ✓ Dividend NOW TRY

= 4 a^314 a^22 + 4 a^31 - 3 a 2 + 4 a^3 a

2

a

b

*

4 a^3 a 4 a^2 - 3 a+

2

a

b

2

a^ ,

4 a^2 - 3 a+

2

a

NOW TRY
EXERCISE 1
Divide by 4 16 a^6 - 12 a^4 a^2.

NOW TRY ANSWERS

1. 4 a^4 - 3 a^2

NOW TRY
EXERCISE 2
Divide.

36 x^5 + 24 x^4 - 12 x^3

6 x^4

2. 6 x+ 4 -^2 x

CAUTION The most frequent error in a problem like that in Example 2is with

the last term of the quotient.

8 a^2

4 a^3

=

8

4

a^2 -^3 = 2 a-^1 = 2 a

1

a

b =

2

a

NOW TRY

Dividing a Polynomial by a Monomial

Divide.

Divide each term by

= 4 a^2 - 3 a+ Quotient rule

2

a

= 4 a^3.

16 a^5

4 a^3

-

12 a^4

4 a^3

+

8 a^2

4 a^3

16 a^5 - 12 a^4 + 8 a^2

4 a^3

EXAMPLE 2