SECTION 7.1 Rational Expressions and Functions; Multiplying and Dividing 365
Look again at the rational expression from Example 2(b).
or
In this expression, acan take any value except or since these values make the
denominator 0. In the simplified expression acannot equal Thus,
for all values of aexcept or
From now on, such statements of equality will be made with the understanding that
they apply only to those real numbers which make neither denominator equal 0. We
will no longer state such restrictions.
Writing Rational Expressions in Lowest TermsWrite each rational expression in lowest terms.
(a) Here, the numerator and denominator are opposites.
To write this expression in lowest terms, write the denominator as.
The numerator could have been rewritten instead to get the same result. (Try this.)
(b)
Factor the difference of squares
in the numerator.Write as.Fundamental property= - 1 r+ 42 , or -r- 4 Lowest terms NOW TRY
=
r+ 4
- 1
= 4 - r - 11 r- 42
1 r + 421 r - 42
- 11 r- 42
=
1 r + 421 r - 42
4 - r
r^2 - 16
4 - r
m- 3
3 - m
=
m- 3
- 11 m- 32
=
1
- 1
=- 1
- 11 m- 32
m- 3
3 - m
EXAMPLE 3
- 3 - 2.
a^2 - a- 6
a^2 + 5 a+ 6
=
a- 3
a+ 3
,
- 3.
a- 3a+ 3 ,
- 3 - 2,
1 a- 321 a+ 22
1 a+ 321 a+ 22
a^2 - a- 6
a^2 + 5 a+ 6
,
NOW TRY
EXERCISE 2
Write each rational
expression in lowest terms.
(a)
(b)
(c)
am-bm+an-bn
am+bm+an+bnt^3 + 8
t+ 23 a^2 - 7 a+ 2
a^2 + 2 a- 8CAUTION Be careful! When using the fundamental property of rational
numbers, only common factors may be divided.For example,
and
because the 2 in is not a factorof the numerator. Remember to factor before
writing a fraction in lowest terms.
y- 2
y- 2
2
Zy- 1
y- 2
2
Zy
NOW TRY
EXERCISE 3
Write each rational expression
in lowest terms.
(a) (b)
81 - y^2
y- 9a- 10
10 - aNOW TRY ANSWERS
- (a)
(b)
(c) - (a)
(b)- 1 y+ 92 , or -y- 9- 1
a-b
a+bt^2 - 2 t+ 43 a- 1
a+ 4(f )
This expression cannot be simplified further and is in lowest terms. NOW TRY
8 +k
16
Be careful. The numerator
cannot be factored.