Beginning Algebra, 11th Edition

SECTION 7.1 The Fundamental Property of Rational Expressions 423

NOTE The numerator orthe denominator could have been factored in the first step

in Example 5.Factor - 1 from the numerator, and confirm that the result is the same.

(b)

Factor. (Step 1)

Factor the numerator completely.

=y- 2 Fundamental property (Step 2)

=

21 y+ 221 y- 22

21 y+ 22

=

21 y^2 - 42

21 y+ 22

2 y^2 - 8

2 y+ 4

since the

denominator is 0

for this value.

yZ-2,

(c)

Factor. (Step 1)

= Fundamental property (Step 2)

m+ 4

2 m+ 3

=

1 m+ 421 m- 22

12 m + 321 m- 22

m^2 + 2 m- 8

2 m^2 - m- 6

NOW TRY

We write statements of equality of rational expressions with the understanding

that they apply only to real numbers that make neither denominator equal to 0.

CAUTION Rational expressions cannot be written in lowest terms until after

the numerator and denominator have been factored.

Numerator cannot be factored.

6 +x

4 x

6 x+ 9

4 x+ 6

=

312 x+ 32

212 x+ 32

=

3

2

Divide out the

common factor.

⎧⎨⎩

Already in lowest terms

Writing in Lowest Terms (Factors Are Opposites)

Write in lowest terms.

To get a common factor, the denominator can be factored as follows.

Factor out.

=- 11 x-y 2 Commutative property

= - 11 - y+x 2 - 1

y-x

y-x

x-y

y-x

EXAMPLE 5

With this result in mind, we simplify.

from above.

= Fundamental property NOW TRY

1

- 1

, or - 1

= y-x=- 11 x-y 2

11 x-y 2

• 11 x-y 2

x-y

y-x

NOW TRY
EXERCISE 4
Write each rational expres-
sion in lowest terms.

(a)

(b)

k^2 - 36

k^2 + 8 k+ 12

3 x+ 15

5 x+ 25

NOW TRY ANSWERS

4. (a)^35 (b)kk-+^62

NOW TRY
EXERCISE 5
Write in lowest terms.

10 - a^2

a^2 - 10

5. 
   
   
   
   • 1
   
   
   
   

mZ-^32 , mZ 2

We are factoring out ,

NOTmultiplying by it.

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