Writing Rational Expressions in Lowest Terms

Write each rational expression in lowest terms.

(a)

Commutative property

Fundamental property

Lowest terms

(b)

Factor the numerator.

Factor the denominator.

Fundamental property

Lowest terms

(c)

Factor the difference of squares in the

numerator. Factor the denominator.

Lowest terms

(d)

Factor the difference of cubes

in the numerator.

Lowest terms

(e)

Group the terms.

Factor within the groups.

Factor by grouping.

= Lowest terms

r +s

r -s

=

1 p+ q 21 r+s 2

1 p+ q 21 r-s 2

=

r 1 p+ q 2 + s 1 p+q 2

r 1 p+ q 2 - s 1 p+q 2

=

1 pr+ qr 2 + 1 ps+qs 2

1 pr+ qr 2 - 1 ps+qs 2

pr+qr+ ps+ qs

pr+qr- ps- qs

= x^2 + 3 x+ 9

=

1 x- 321 x^2 + 3 x+ 92

x- 3

x^3 - 27

x- 3

=

y- 2

2

=

1 y+ 221 y- 22

21 y+ 22

y^2 - 4

2 y+ 4

=

a- 3

a+ 3

=

a- 3

a+ 3

1

=

1 a- 321 a+ 22

1 a+ 321 a+ 22

a^2 - a- 6

a^2 + 5 a+ 6

=

x+ 5

x- 3

=

x+ 5

x- 3

1

=

1 x+ 521 x+ 22

1 x- 321 x+ 22

1 x+ 521 x+ 22

1 x+ 221 x- 32

EXAMPLE 2

364 CHAPTER 7 Rational Expressions and Functions