Writing a Rational Expression in Lowest Terms

Step 1 Factor the numerator and denominator.

Step 2 Use the fundamental property to divide out
common factors.

Write in lowest terms.

=

x+ 1

x- 1

=

1 x- 121 x+ 12

1 x- 121 x- 12

x^2 - 1

1 x- 122

CONCEPTS EXAMPLES

7.2 Multiplying and Dividing Rational

Expressions

Multiplying or Dividing Rational Expressions

Step 1 Note the operation. If the operation is division,
use the definition of division to rewrite as
multiplication.

Step 2 Multiply numerators and multiply denominators.

Step 3 Factor numerators and denominators completely.

Step 4 Write in lowest terms, using the fundamental
property.

Note: Steps 2 and 3 may be interchanged based on
personal preference.

Multiply.

Factor.

= Lowest terms

31 x+ 22

x- 3

=

31 x+ 321 x- 521 x+ 22

1 x- 521 x+ 321 x- 32

=

13 x+ 921 x^2 - 3 x- 102

1 x- 521 x^2 - 92

3 x+ 9

x- 5

#x

(^2) - 3 x- 10
x^2 - 9
Divide.
Factor.
= Lowest terms
x- 5
3 x- 2

12 x+ 121 x+ 521 x- 52
1 x+ 5212 x+ 1213 x- 22

12 x+ 121 x^2 - 252
1 x+ 5216 x^2 - x- 22

2 x+ 1
x+ 5

x

(^2) - 25
6 x^2 - x- 2
2 x+ 1
x+ 5
,
6 x^2 - x- 2
x^2 - 25
Multiply numerators
and denominators.
Multiply by the
reciprocal of the
divisor.
Multiply numerators
and denominators.
7.3 Least Common Denominators
Finding the LCD
Step 1 Factor each denominator into prime factors.
Step 2 List each different factor the greatest number
of times it appears.
Step 3 Multiply the factors from Step 2 to get the LCD.
Writing a Rational Expression with a Specified
Denominator
Step 1 Factor both denominators.
Step 2 Decide what factor(s) the denominator must be
multiplied by in order to equal the specified
denominator.
Step 3 Multiply the rational expression by that factor
divided by itself. (That is, multiply by 1.)
Find the LCD for and
Find the numerator.
must be multiplied by 2 zin order to obtain
5
2 z 1 z- 32
#^2 z
2 z

10 z
4 z^21 z- 32

10 z
4 z^3 - 12 z^2
2 z 1 z- 32 4 z^21 z- 32.
5
2 z 1 z- 32

?
4 z^21 z- 32
5
2 z^2 - 6 z

?
4 z^3 - 12 z^2
= 4 k 1 k- 422
LCD= 1 k- 422 # 4 #k
4 k^2 - 16 k= 4 k 1 k- 42
k^2 - 8 k+ 16 = 1 k- 422
1
4 k^2 - 16 k
.
3
k^2 - 8 k+ 16
Factor each
denominator.
(continued)
484 CHAPTER 7 Rational Expressions and Applications
⎧
⎨
⎩
http://www.ebook777.com
http://www.ebook777.com