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,
共x ⫹ 1兲 ⫹ 共x ⫹ 2兲 and 2x ⫹ 3 are equivalent because
共x ⫹ 1兲 ⫹ 共x ⫹ 2兲 ⫽ x ⫹ 1 ⫹ x ⫹ 2
⫽x⫹x⫹1⫹2
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.
⫽ 2x ⫹ 3.
Example 1
Finding the Domain of an Algebraic Expression
a. The domain of the polynomial
2x 3 ⫹ 3x ⫹ 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
冪x ⫺ 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
x⫹2
x⫺3
is the set of all real numbers except x ⫽ 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 ⫺ 1
,
x⫹1
or
x2 ⫺ 1
x2 ⫹ 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
⭈ c ⫽ a,
⭈c 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.
x⫹6 x⫹6
⫽
⫽x⫹2
3
3
Remember that to simplify
fractions, divide out common
factors, not terms.
Write
Simplifying a Rational Expression
x 2 ⫹ 4x ⫺ 12
in simplest form.
3x ⫺ 6
Solution
x2 ⫹ 4x ⫺ 12 共x ⫹ 6兲共x ⫺ 2兲
⫽
3x ⫺ 6
3共x ⫺ 2兲
⫽
x⫹6
,
3
x⫽2
Factor completely.
Divide out common factors.
Note that the original expression is undefined when x ⫽ 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 ⫽ 2.
Now try Exercise 33.
Sometimes it may be necessary to change the sign of a factor by factoring out 共⫺1兲
to simplify a rational expression, as shown in Example 3.
Example 3
Write
Simplifying Rational Expressions
12 ⫹ x ⫺ x2
in simplest form.
2x2 ⫺ 9x ⫹ 4
Solution
12 ⫹ x ⫺ x2
共4 ⫺ x兲共3 ⫹ x兲
⫽
2x2 ⫺ 9x ⫹ 4 共2x ⫺ 1兲共x ⫺ 4兲
⫽
⫺ 共x ⫺ 4兲共3 ⫹ x兲
共2x ⫺ 1兲共x ⫺ 4兲
⫽⫺
3⫹x
, x⫽4
2x ⫺ 1
Factor completely.
共4 ⫺ x兲 ⫽ ⫺ 共x ⫺ 4兲
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 Example 3, for instance, the restriction x ⫽ 4 is listed with the simplified expression
1
to make the two domains agree. Note that the value x ⫽ 2 is excluded from both
domains, so it is not necessary to list this value.
Appendix A.4
Rational Expressions
A41
Operations with Rational Expressions
To multiply or divide rational expressions, use the properties of fractions
discussed in Appendix A.1. Recall that to divide fractions, you invert the divisor and
multiply.
Example 4
Multiplying Rational Expressions
2x2 ⫹ x ⫺ 6
x2 ⫹ 4x ⫺ 5
⭈
x3 ⫺ 3x2 ⫹ 2x 共2x ⫺ 3兲共x ⫹ 2兲
⫽
4x2 ⫺ 6x
共x ⫹ 5兲共x ⫺ 1兲
⫽
⭈
x共x ⫺ 2兲共x ⫺ 1兲
2x共2x ⫺ 3兲
共x ⫹ 2兲共x ⫺ 2兲
, x ⫽ 0, x ⫽ 1, x ⫽ 32
2共x ⫹ 5兲
Now try Exercise 53.
In Example 4, the restrictions x ⫽ 0, x ⫽ 1, and x ⫽ 32 are listed with the simplified
expression in order to make the two domains agree. Note that the value x ⫽ ⫺5 is
excluded from both domains, so it is not necessary to list this value.
Example 5
Dividing Rational Expressions
x 3 ⫺ 8 x 2 ⫹ 2x ⫹ 4 x 3 ⫺ 8
⫼
⫽ 2
x2 ⫺ 4
x3 ⫹ 8
x ⫺4
⫽
x3 ⫹ 8
⭈ x 2 ⫹ 2x ⫹ 4
Invert and multiply.
共x ⫺ 2兲共x2 ⫹ 2x ⫹ 4兲 共x ⫹ 2兲共x2 ⫺ 2x ⫹ 4兲
⭈ 共x2 ⫹ 2x ⫹ 4兲
共x ⫹ 2兲共x ⫺ 2兲
⫽ x 2 ⫺ 2x ⫹ 4, x ⫽ ± 2
Divide out
common factors.
Now try Exercise 55.
To add or subtract rational expressions, you can use the LCD (least common
denominator) method or the basic definition
a c
ad ± bc
± ⫽
,
b d
bd
b ⫽ 0, d ⫽ 0.
Basic definition
This definition provides an efficient way of adding or subtracting two fractions that
have no common factors in their denominators.
Example 6
WARNING / CAUTION
When subtracting rational
expressions, remember to
distribute the negative sign to
all the terms in the quantity that
is being subtracted.
Subtracting Rational Expressions
x
2
x共3x ⫹ 4兲 ⫺ 2共x ⫺ 3兲
⫺
⫽
x ⫺ 3 3x ⫹ 4
共x ⫺ 3兲共3x ⫹ 4兲
Basic definition
⫽
3x 2 ⫹ 4x ⫺ 2x ⫹ 6
共x ⫺ 3兲共3x ⫹ 4兲
Distributive Property
⫽
3x 2 ⫹ 2x ⫹ 6
共x ⫺ 3兲共3x ⫹ 4兲
Combine like terms.
Now try Exercise 65.
A42
Appendix A
Review of Fundamental Concepts of Algebra
For three or more fractions, or for fractions with a repeated factor in the denominators, the LCD method works well. Recall that the least common denominator of
several fractions consists of the product of all prime factors in the denominators, with
each factor given the highest power of its occurrence in any denominator. Here is a
numerical example.
1 3 2 1⭈2 3⭈3 2⭈4
⫹ ⫺ ⫽
⫹
⫺
6 4 3 6⭈2 4⭈3 3⭈4
⫽
2
9
8
⫹
⫺
12 12 12
⫽
3
12
⫽
1
4
The LCD is 12.
Sometimes the numerator of the answer has a factor in common with the
denominator. In such cases the answer should be simplified. For instance, in the
3
example above, 12
was simplified to 14.
Example 7
Combining Rational Expressions: The LCD Method
Perform the operations and simplify.
2
x⫹3
3
⫺ ⫹ 2
x⫺1
x
x ⫺1
Solution
Using the factored denominators 共x ⫺ 1兲, x, and 共x ⫹ 1兲共x ⫺ 1兲, you can see that the
LCD is x共x ⫹ 1兲共x ⫺ 1兲.
3
2
x⫹3
⫺ ⫹
x⫺1
x
共x ⫹ 1兲共x ⫺ 1兲
⫽
共x ⫹ 3兲共x兲
3共x兲共x ⫹ 1兲
2共x ⫹ 1兲共x ⫺ 1兲
⫺
⫹
x共x ⫹ 1兲共x ⫺ 1兲
x共x ⫹ 1兲共x ⫺ 1兲
x共x ⫹ 1兲共x ⫺ 1兲
⫽
3共x兲共x ⫹ 1兲 ⫺ 2共x ⫹ 1兲共x ⫺ 1兲 ⫹ 共x ⫹ 3兲共x兲
x共x ⫹ 1兲共x ⫺ 1兲
⫽
3x 2 ⫹ 3x ⫺ 2x 2 ⫹ 2 ⫹ x 2 ⫹ 3x
x共x ⫹ 1兲共x ⫺ 1兲
Distributive Property
⫽
3x 2 ⫺ 2x 2 ⫹ x 2 ⫹ 3x ⫹ 3x ⫹ 2
x共x ⫹ 1兲共x ⫺ 1兲
Group like terms.
⫽
2x2 ⫹ 6x ⫹ 2
x共x ⫹ 1兲共x ⫺ 1兲
Combine like terms.
⫽
2共x 2 ⫹ 3x ⫹ 1兲
x共x ⫹ 1兲共x ⫺ 1兲
Factor.
Now try Exercise 67.
Appendix A.4
Rational Expressions
A43
Complex Fractions and the Difference Quotient
Fractional expressions with separate fractions in the numerator, denominator, or both
are called complex fractions. Here are two examples.
冢x冣
冢x冣
1
x2 ⫹ 1
1
and
冢x
2
1
⫹1
冣
To simplify a complex fraction, combine the fractions in the numerator into a single
fraction and then combine the fractions in the denominator into a single fraction.
Then invert the denominator and multiply.
Example 8
Simplifying a Complex Fraction
2 ⫺ 3共x兲
x
⫽
1
1共x ⫺ 1兲 ⫺ 1
1⫺
x⫺1
x⫺1
冢 x ⫺ 3冣
冤
2
冢
冥
冣 冤
冥
Combine fractions.
2 ⫺ 3x
冢 x 冣
⫽
x⫺2
冢x ⫺ 1冣
Simplify.
x⫺1
⫽
2 ⫺ 3x
x
⫽
共2 ⫺ 3x兲共x ⫺ 1兲
, x⫽1
x共x ⫺ 2兲
⭈x⫺2
Invert and multiply.
Now try Exercise 73.
Another way to simplify a complex fraction is to multiply its numerator and
denominator by the LCD of all fractions in its numerator and denominator. This method
is applied to the fraction in Example 8 as follows.
冢 x ⫺ 3冣
冢 x ⫺ 3冣
2
冢
1
1⫺
x⫺1
2
⫽
冣 冢
1
1⫺
x⫺1
x共x ⫺ 1兲
冣
⭈ x共x ⫺ 1兲
冢2 ⫺x 3x冣 ⭈ x共x ⫺ 1兲
⫽
冢xx ⫺⫺ 21冣 ⭈ x共x ⫺ 1兲
⫽
共2 ⫺ 3x兲共x ⫺ 1兲
, x⫽1
x共x ⫺ 2兲
LCD is x共x ⫺ 1兲.
A44
Appendix A
Review of Fundamental Concepts of Algebra
The next three examples illustrate some methods for simplifying rational
expressions involving negative exponents and radicals. These types of expressions
occur frequently in calculus.
To simplify an expression with negative exponents, one method is to begin by
factoring out the common factor with the smaller exponent. Remember that when
factoring, you subtract exponents. For instance, in 3x⫺5兾2 ⫹ 2x⫺3兾2 the smaller
exponent is ⫺ 52 and the common factor is x⫺5兾2.
3x⫺5兾2 ⫹ 2x⫺3兾2 ⫽ x⫺5兾2关3共1兲 ⫹ 2x⫺3兾2⫺ 共⫺5兾2兲兴
⫽ x⫺5兾2共3 ⫹ 2x1兲
⫽
Example 9
3 ⫹ 2x
x 5兾2
Simplifying an Expression
Simplify the following expression containing negative exponents.
x共1 ⫺ 2x兲⫺3兾2 ⫹ 共1 ⫺ 2x兲⫺1兾2
Solution
Begin by factoring out the common factor with the smaller exponent.
x共1 ⫺ 2x兲⫺3兾2 ⫹ 共1 ⫺ 2x兲⫺1兾2 ⫽ 共1 ⫺ 2x兲⫺3兾2关 x ⫹ 共1 ⫺ 2x兲(⫺1兾2)⫺(⫺3兾2)兴
⫽ 共1 ⫺ 2x兲⫺3兾2关x ⫹ 共1 ⫺ 2x兲1兴
⫽
1⫺x
共1 ⫺ 2x兲 3兾2
Now try Exercise 81.
A second method for simplifying an expression with negative exponents is shown
in the next example.
Example 10
Simplifying an Expression with Negative Exponents
共4 ⫺ x 2兲1兾2 ⫹ x 2共4 ⫺ x 2兲⫺1兾2
4 ⫺ x2
⫽
共4 ⫺ x 2兲1兾2 ⫹ x 2共4 ⫺ x 2兲⫺1兾2 共4 ⫺ x 2兲1兾2
⭈ 共4 ⫺ x 2兲1兾2
4 ⫺ x2
⫽
共4 ⫺ x 2兲1 ⫹ x 2共4 ⫺ x 2兲 0
共4 ⫺ x 2兲 3兾2
⫽
4 ⫺ x2 ⫹ x2
共4 ⫺ x 2兲 3兾2
⫽
4
共4 ⫺ x 2兲 3兾2
Now try Exercise 83.
Appendix A.4
Example 11
Rational Expressions
A45
Rewriting a Difference Quotient
The following expression from calculus is an example of a difference quotient.
冪x ⫹ h ⫺ 冪x
h
Rewrite this expression by rationalizing its numerator.
Solution
冪x ⫹ h ⫺ 冪x
h
You can review the techniques
for rationalizing a numerator in
Appendix A.2.
⫽
冪x ⫹ h ⫺ 冪x
h
冪x ⫹ h ⫹ 冪x
⭈ 冪x ⫹ h ⫹ 冪x
⫽
共冪x ⫹ h 兲2 ⫺ 共冪x 兲2
h共冪x ⫹ h ⫹ 冪x 兲
⫽
h
h共冪x ⫹ h ⫹ 冪x 兲
⫽
1
,
冪x ⫹ h ⫹ 冪x
h⫽0
Notice that the original expression is undefined when h ⫽ 0. So, you must exclude
h ⫽ 0 from the domain of the simplified expression so that the expressions are
equivalent.
Now try Exercise 89.
Difference quotients, such as that in Example 11, occur frequently in calculus. Often,
they need to be rewritten in an equivalent form that can be evaluated when h ⫽ 0. Note
that the equivalent form is not simpler than the original form, but it has the advantage
that it is defined when h ⫽ 0.
A.4
EXERCISES
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
VOCABULARY: Fill in the blanks.
1. The set of real numbers for which an algebraic expression is defined is the ________ of the expression.
2. The quotient of two algebraic expressions is a fractional expression and the quotient of
two polynomials is a ________ ________.
3. Fractional expressions with separate fractions in the numerator, denominator, or both are called ________
fractions.
4. To simplify an expression with negative exponents, it is possible to begin by factoring out the common factor
with the ________ exponent.
5. Two algebraic expressions that have the same domain and yield the same values for all numbers
in their domains are called ________.
6. An important rational expression, such as
a ________ ________.
共x ⫹ h兲2 ⫺ x2
, that occurs in calculus is called
h
A46
Appendix A
Review of Fundamental Concepts of Algebra
SKILLS AND APPLICATIONS
44. Error Analysis
In Exercises 7–22, find the domain of the expression.
7. 3x 2 ⫺ 4x ⫹ 7
9. 4x 3 ⫹ 3, x ⱖ 0
11.
1
3⫺x
13.
⫺1
x2 ⫺ 2x ⫹ 1
8. 2x 2 ⫹ 5x ⫺ 2
10. 6x 2 ⫺ 9, x > 0
12.
x2
14.
x⫹6
3x ⫹ 2
x2
⫺ 5x ⫹ 6
x2 ⫺ 4
x2 ⫺ 2x ⫺ 3
15. 2
x ⫺ 6x ⫹ 9
x2 ⫺ x ⫺ 12
16. 2
x ⫺ 8x ⫹ 16
17. 冪x ⫹ 7
19. 冪2x ⫺ 5
18. 冪4 ⫺ x
20. 冪4x ⫹ 5
21.
1
22.
冪x ⫺ 3
x2
1
冪x ⫹ 2
25.
15x 2
10x
26.
18y 2
60y 5
45.
3xy
xy ⫹ x
28.
2x
xy ⫺ y
29.
4y ⫺ 8y 2
10y ⫺ 5
30.
9x 2 ⫹ 9x
2x ⫹ 2
x⫺5
10 ⫺ 2x
32.
x 3 ⫹ 5x 2 ⫹ 6x
35.
x2 ⫺ 4
y 2 ⫺ 7y ⫹ 12
37. 2
y ⫹ 3y ⫺ 18
x 2 ⫺ 7x ⫹ 6
38. 2
x ⫹ 11x ⫹ 10
2 ⫺ x ⫹ 2x 2 ⫺ x 3
x2 ⫺ 4
z3 ⫺ 8
41. 2
z ⫹ 2z ⫹ 4
39.
43. Error Analysis
5x3
2x3 ⫹ 4
⫽
x2 ⫺ 9
⫹ x 2 ⫺ 9x ⫺ 9
y 3 ⫺ 2y 2 ⫺ 3y
42.
y3 ⫹ 1
40.
1
2
3
4
x3
Describe the error.
5x3
5
5
⫽
⫽
3
2x ⫹ 4 2 ⫹ 4 6
5
6
x2 ⫺ 2x ⫺ 3
x⫺3
x⫹1
46.
0
x
1
2
3
4
5
6
x⫺3
x2 ⫺ x ⫺ 6
1
x⫹2
GEOMETRY In Exercises 47 and 48, find the ratio of the
area of the shaded portion of the figure to the total area of
the figure.
47.
48.
x+5
2
r
2x + 3
x+5
2
12 ⫺ 4x
x⫺3
x 2 ⫺ 25
34.
5⫺x
x 2 ⫹ 8x ⫺ 20
36. 2
x ⫹ 11x ⫹ 10
y 2 ⫺ 16
33.
y⫹4
0
x
2y
27.
31.
x共x ⫹ 5兲共x ⫺ 5兲 x共x ⫹ 5兲
⫽
共x ⫺ 5兲共x ⫹ 3兲
x⫹3
In Exercises 45 and 46, complete the table. What can you conclude?
3共 䊏 兲
3
24. ⫽
4 4共x ⫹ 1兲
In Exercises 25–42, write the rational expression in simplest
form.
x共x 2 ⫹ 25兲
x 3 ⫹ 25x
⫽
⫺ 2x ⫺ 15 共x ⫺ 5兲共x ⫹ 3兲
⫽
In Exercises 23 and 24, find the missing factor in the numerator
such that the two fractions are equivalent.
5
5共䊏兲
23.
⫽
2x
6x 2
Describe the error.
x+5
In Exercises 49–56, perform the multiplication or division
and simplify.
5
x⫺1
⭈
x ⫺ 1 25共x ⫺ 2兲
r
r2
51.
⫼ 2
r⫺1 r ⫺1
49.
x共x ⫺ 3兲
5
4⫺y
4y ⫺ 16
52.
⫼
5y ⫹ 15 2y ⫹ 6
50.
x ⫹ 13
x 3共3 ⫺ x兲
t2 ⫺ t ⫺ 6
t⫹3
⭈
t 2 ⫹ 6t ⫹ 9 t 2 ⫺ 4
x
x 2 ⫹ xy ⫺ 2y 2
54.
⭈
3
2
2
x ⫹x y
x ⫹ 3xy ⫹ 2y 2
53.
55.
x 2 ⫺ 36 x 3 ⫺ 6x 2
⫼ 2
x
x ⫹x
56.
x 2 ⫺ 14x ⫹ 49 3x ⫺ 21
⫼
x 2 ⫺ 49
x⫹7
⭈
Appendix A.4
In Exercises 57–68, perform the addition or subtraction and
simplify.
57. 6 ⫺
5
x⫹3
5
x
⫹
x⫺1 x⫺1
3
5
61.
⫹
x⫺2 2⫺x
59.
63.
4
x
⫺
2x ⫹ 1 x ⫹ 2
58.
2x ⫺ 1 1 ⫺ x
⫹
x⫹3
x⫹3
5
2x
62.
⫺
x⫺5 5⫺x
60.
64.
3x1兾3 ⫺ x⫺2兾3
3x⫺2兾3
⫺x 3共1 ⫺ x 2兲⫺1兾2 ⫺ 2x共1 ⫺ x 2兲1兾2
84.
x4
1
2
1
⫹ 2
⫹ 3
x
x ⫹1 x ⫹x
2
1
2
68.
⫹
⫹
x ⫹ 1 x ⫺ 1 x2 ⫺ 1
67. ⫺
x ⫹ 4 3x ⫺ 8 x ⫹ 4 ⫺ 3x ⫺ 8
⫺
⫽
x⫹2
x⫹2
x⫹2
⫺2x ⫺ 4 ⫺2共x ⫹ 2兲
⫽
⫽
⫽ ⫺2
x⫹2
x⫹2
6⫺x
x⫹2
8
70.
⫹
⫹ 2
2
x共x ⫹ 2兲
x
x 共x ⫹ 2兲
x共6 ⫺ x兲 ⫹ 共x ⫹ 2兲 2 ⫹ 8
⫽
x 2共x ⫹ 2兲
6x ⫺ x 2 ⫹ x 2 ⫹ 4 ⫹ 8
⫽
x 2共x ⫹ 2兲
6共x ⫹ 2兲
6
⫽ 2
⫽
x 共x ⫹ 2兲 x 2
69.
In Exercises 71–76, simplify the complex fraction.
x
73.
共x ⫺ 2兲
x2
共x ⫹ 1兲2
冤
冥
x
冤 共x ⫹ 1兲 冥
3
冢冪x ⫺ 2冪x冣
1
75.
冪x
In Exercises 85– 88, simplify the difference quotient.
1
1
1
1
⫺ 2
⫺
2
共
x
⫹
h
兲
x
x⫹h
x
85.
86.
h
h
x
⫹
h
x
1
1
⫺
⫺
x⫹h⫹1 x⫹1
x⫹h⫺4 x⫺4
87.
88.
h
h
冢
ERROR ANALYSIS In Exercises 69 and 70, describe the
error.
共x ⫺ 4兲
x
4
⫺
4
x
x2 ⫺ 1
x
74.
共x ⫺ 1兲2
x
t2
⫺ 冪t 2 ⫹ 1
冪t 2 ⫹ 1
76.
t2
72.
2x共x ⫺ 5兲⫺3 ⫺ 4x 2共x ⫺ 5兲⫺4
2x 2共x ⫺ 1兲1兾2 ⫺ 5共x ⫺ 1兲⫺1兾2
4x 3共2x ⫺ 1兲3兾2 ⫺ 2x共2x ⫺ 1兲⫺1兾2
83.
1
x
⫺
x 2 ⫺ x ⫺ 2 x 2 ⫺ 5x ⫹ 6
10
2
66. 2
⫹ 2
x ⫺ x ⫺ 2 x ⫹ 2x ⫺ 8
71.
78. x5 ⫺ 5x⫺3
x 5 ⫺ 2x⫺2
x 2共x 2 ⫹ 1兲⫺5 ⫺ 共x 2 ⫹ 1兲⫺4
In Exercises 83 and 84, simplify the expression.
5x
2
⫹
x ⫺ 3 3x ⫹ 4
65.
冢 2 ⫺ 1冣
In Exercises 77–82, factor the expression by removing the
common factor with the smaller exponent.
77.
79.
80.
81.
82.
3
⫺5
x⫺1
冢
冢
冤
冢
A47
Rational Expressions
冣
冣
冥
冣
冤
冣
冢
冥
冢
冣
冣
In Exercises 89–94, simplify the difference quotient by
rationalizing the numerator.
89.
91.
93.
94.
冪x ⫹ 2 ⫺ 冪x
90.
2
冪t ⫹ 3 ⫺ 冪3
92.
t
冪z ⫺ 3 ⫺ 冪z
3
冪x ⫹ 5 ⫺ 冪5
x
冪x ⫹ h ⫹ 1 ⫺ 冪x ⫹ 1
h
冪x ⫹ h ⫺ 2 ⫺ 冪x ⫺ 2
h
PROBABILITY In Exercises 95 and 96, consider an experiment in which a marble is tossed into a box whose base is
shown in the figure. The probability that the marble will
come to rest in the shaded portion of the box is equal to the
ratio of the shaded area to the total area of the figure. Find
the probability.
95.
96.
x
2
x
2x + 1
x+4
x
x
x+2
4
x
(x + 2)
97. RATE A digital copier copies in color at a rate of
50 pages per minute.
(a) Find the time required to copy one page.
A48
Appendix A
Review of Fundamental Concepts of Algebra
(b) Find the time required to copy x pages.
(c) Find the time required to copy 120 pages.
98. RATE After working together for t hours on a
common task, two workers have done fractional parts
of the job equal to t兾3 and t兾5, respectively. What
fractional part of the task has been completed?
102. INTERACTIVE MONEY MANAGEMENT The table
shows the projected numbers of U.S. households (in
millions) banking online and paying bills online from
2002 through 2007. (Source: eMarketer; Forrester
Research)
FINANCE In Exercises 99 and 100, the formula that approximates the annual interest rate r of a monthly installment
loan is given by
24共NM ⴚ P兲
[
]
N
rⴝ
冸P ⴙ NM
12 冹
where N is the total number of payments, M is the monthly
payment, and P is the amount financed.
4t 2 ⫹ 16t ⫹ 75
2 ⫹ 4t ⫹ 10
冢t
冣
0
2
4
6
8
10
14
16
18
2002
2003
2004
2005
2006
2007
21.9
26.8
31.5
35.0
40.0
45.0
13.7
17.4
20.9
23.9
26.7
29.1
Number banking online ⫽
⫺0.728t2 ⫹ 23.81t ⫺ 0.3
⫺0.049t2 ⫹ 0.61t ⫹ 1.0
and
Number paying bills online ⫽
4.39t ⫹ 5.5
0.002t2 ⫹ 0.01t ⫹ 1.0
where t represents the year, with t ⫽ 2 corresponding
to 2002.
(a) Using the models, create a table to estimate the
projected numbers of households banking online
and the projected numbers of households paying
bills online for the given years.
(b) Compare the values given by the models with the
actual data.
(c) Determine a model for the ratio of the projected
number of households paying bills online to the
projected number of households banking online.
(d) Use the model from part (c) to find the ratios for
the given years. Interpret your results.
20
TRUE OR FALSE? In Exercises 103 and 104, determine
whether the statement is true or false. Justify your answer.
103.
x 2n ⫺ 12n
⫽ x n ⫹ 1n
x n ⫺ 1n
104.
x 2 ⫺ 3x ⫹ 2
⫽ x ⫺ 2, for all values of x
x⫺1
12
T
t
Paying Bills
EXPLORATION
where T is the temperature (in degrees Fahrenheit) and
t is the time (in hours).
(a) Complete the table.
t
Banking
Mathematical models for these data are
99. (a) Approximate the annual interest rate for a
four-year car loan of $20,000 that has monthly
payments of $475.
(b) Simplify the expression for the annual interest rate
r, and then rework part (a).
100. (a) Approximate the annual interest rate for a fiveyear car loan of $28,000 that has monthly
payments of $525.
(b) Simplify the expression for the annual interest rate
r, and then rework part (a).
101. REFRIGERATION When food (at room temperature)
is placed in a refrigerator, the time required for the
food to cool depends on the amount of food, the air
circulation in the refrigerator, the original temperature
of the food, and the temperature of the refrigerator.
The model that gives the temperature of food that has
an original temperature of 75⬚F and is placed in a
40⬚F refrigerator is
T ⫽ 10
Year
22
T
(b) What value of T does the mathematical model
appear to be approaching?
105. THINK ABOUT IT How do you determine whether a
rational expression is in simplest form?
106. CAPSTONE In your own words, explain how to
divide rational expressions.