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Appendix A.4

Rational Expressions

A39

A.4 RATIONAL EXPRESSIONS
What you should learn
• Find domains of algebraic
expressions.
• Simplify rational expressions.
• Add, subtract, multiply, and divide
rational expressions.
• Simplify complex fractions and
rewrite difference quotients.

Domain of an Algebraic Expression
The set of real numbers for which an algebraic expression is defined is the domain of the expression. Two algebraic expressions are equivalent if they have the same domain and yield the same values for all numbers in their domain. For instance, ͑x ϩ 1͒ ϩ ͑x ϩ 2͒ and 2x ϩ 3 are equivalent because

͑x ϩ 1͒ ϩ ͑x ϩ 2͒ ϭ x ϩ 1 ϩ x ϩ 2
ϭxϩxϩ1ϩ2

Why you should learn it
Rational expressions can be used to
solve real-life problems. For instance,
in Exercise 102 on page A48, a rational
expression is used to model the
projected numbers of U.S. households
banking and paying bills online from
2002 through 2007.

ϭ 2x ϩ 3.

Example 1

Finding the Domain of an Algebraic Expression

a. The domain of the polynomial
2x 3 ϩ 3x ϩ 4
is the set of all real numbers. In fact, the domain of any polynomial is the set of all real numbers, unless the domain is specifically restricted.
b. The domain of the radical expression
Ίx Ϫ 2

is the set of real numbers greater than or equal to 2, because the square root of a negative number is not a real number.
c. The domain of the expression
xϩ2
xϪ3
is the set of all real numbers except x ϭ 3, which would result in division by zero, which is undefined.
Now try Exercise 7.
The quotient of two algebraic expressions is a fractional expression. Moreover, the quotient of two polynomials such as
1
,
x

2x Ϫ 1
,
xϩ1

or

x2 Ϫ 1
x2 ϩ 1

is a rational expression.

Simplifying Rational Expressions
Recall that a fraction is in simplest form if its numerator and denominator have no factors in common aside from ± 1. To write a fraction in simplest form, divide out common factors.
a
b

и c ϭ a,
иc b

c

0

A40

Appendix A

Review of Fundamental Concepts of Algebra

The key to success in simplifying rational expressions lies in your ability to factor polynomials. When simplifying rational expressions, be sure to factor each polynomial completely before concluding that the numerator and denominator have no factors in common.

Example 2

WARNING / CAUTION
In Example 2, do not make the
mistake of trying to simplify
further by dividing out terms.
xϩ6 xϩ6
ϭ
ϭxϩ2
3
3
Remember that to simplify
fractions, divide out common
factors, not terms.

Write

Simplifying a Rational Expression

x 2 ϩ 4x Ϫ 12
in simplest form.
3x Ϫ 6

Solution
x2 ϩ 4x Ϫ 12 ͑x ϩ 6͒͑x Ϫ 2͒
ϭ
3x Ϫ 6
3͑x Ϫ 2͒
ϭ

xϩ6
,
3

x

Factor completely.

2

Divide out common factors.

Note that the original expression is undefined when x ϭ 2 (because division by zero is undefined). To make sure that the simplified expression is equivalent to the original expression, you must restrict the domain of the simplified expression by excluding the value x ϭ 2.

Now try Exercise 33.
Sometimes it may be necessary to change the sign of a factor by factoring out ͑Ϫ1͒ to simplify a rational expression, as shown in Example 3.

Example 3
Write

Simplifying Rational Expressions

12 ϩ x Ϫ x2
in simplest form.
2x2 Ϫ 9x ϩ 4

Solution
12 ϩ x Ϫ x2
͑4 Ϫ x͒͑3 ϩ x͒
ϭ
2x2 Ϫ 9x ϩ 4 ͑2x Ϫ 1͒͑x Ϫ 4͒
ϭ

Factor completely.

Ϫ ͑x Ϫ 4͒͑3 ϩ x͒
͑2x Ϫ 1͒͑x Ϫ 4͒

ϭϪ

3ϩx
, x
2x Ϫ 1

4

͑4 Ϫ x͒ ϭ Ϫ ͑x Ϫ 4͒

Divide out common factors.

Now try Exercise 39.
In this text, when a rational expression is written, the domain is usually not listed with the expression. It is implied that the real numbers that make the denominator zero are excluded from the expression. Also, when performing operations with rational expressions, this text follows the convention of listing by the simplified expression all values of x that must be specifically excluded from the domain in order to make the domains of the simplified and original expressions agree. In Ex...

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