Intermediate Algebra (11th edition)

Dividing by a Polynomial with a Missing Term

Divide by

Missing term

Remainder

The degree of the remainder, is less than the degree of the divisor,

so the division process is finished. The result is written as follows.

2 r^2 + 3 r+ 2 + Quotient+remainderdivisor

- 2 r+ 11

3 r^2 - 2

- 2 r+11, 3 r^2 - 2,

- 2 r + 11

6 r^2 + 0 r- 4

6 r^2 - 2 r+ 7

9 r^3 + 0 r^2 - 6 r

9 r^3 + 6 r^2 - 8 r

6 r^4 + 0 r^3 - 4 r^2

3 r^2 + 0 r - 2  6 r^4 + 9 r^3 + 2 r^2 - 8 r+ 7

2 r^2 + 3 r+ 2

6 r^4 + 9 r^3 + 2 r^2 - 8 r+ 7 3 r^2 - 2.

EXAMPLE 4

SECTION 5.5 Dividing Polynomials 305

NOW TRY

Stop when the degree of

the remainder is less than

the degree of the divisor.

Finding a Quotient with a Fractional Coefficient

Divide by

Since the remainder is 0, the quotient is p^2 +^32 p- 1. NOW TRY

0

- 2 p- 2

- 2 p- 2

3 p^2 + 3 p

3 p^2 + p

2 p^3 + 2 p^2

2 p+ 2  2 p^3 + 5 p^2 + p- 2

p^2 +

3

2

p- 1

3 p^2

2 p =

3

2 p

2 p^3 + 5 p^2 +p- 2 2 p+2.

EXAMPLE 5

CAUTION When dividing a polynomial by a polynomial of two or more terms:

1. Be sure the terms in both polynomials are written in descending powers.

2. Write any missing terms with 0 placeholders.

OBJECTIVE 3 Divide polynomial functions.We now define the quotient of

two functions.

Dividing Functions

If and define functions, then

Quotient function

The domain of the quotient function is the intersection of the domains of

and g 1 x 2 ,excluding any values of xfor which g 1 x 2 =0.

ƒ 1 x 2

a

ƒ

g

b1x 2 

ƒ 1 x 2

g 1 x 2

.

ƒ 1 x 2 g 1 x 2

NOW TRY
EXERCISE 4
Divide

by. 2 x^2 - 2

2 x^4 + 8 x^3 + 2 x^2 - 5 x- 3

NOW TRY
EXERCISE 5
Divide

by. 3 m- 6

6 m^3 - 8 m^2 - 5 m- 6

NOW TRY ANSWERS

4. 
   

5. 2 m^2 +
   4
   3
   m+ 1

x^2 + 4 x+ 2 +

3 x+ 1

2 x^2 - 2