Download P.4 Rational Expressions

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Section P.4
Rational Expressions
P.4 Rational Expressions
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, the expressions 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 ⫽ 2x ⫹ 3.
Example 1 Finding the Domain of an Algebraic Expression
a. The domain of the polynomial
2x 3 ⫹ 3x ⫹ 4
What you should learn
䊏
䊏
䊏
䊏
Find domains of algebraic expressions.
Simplify rational expressions.
Add, subtract, multiply, and divide rational
expressions.
Simplify complex fractions.
Why you should learn it
Rational expressions are useful in estimating
the temperature of food as it cools. For
instance, a rational expression is used in
Exercise 96 on page 46 to model the temperature of food as it cools in a refrigerator set
at 40⬚F.
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
Dwayne Newton/PhotoEdit
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 5.
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⭈c a
⫽ ,
b⭈c b
c ⫽ 0.
Alert students to the importance of the
domain of an expression in graphing
functions later in this course and
in calculus.
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Prerequisites
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 Simplifying a Rational Expression
Write
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 27.
It may sometimes 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 Simplifying a Rational Expression
Write
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
,
2x ⫺ 1
x⫽4
Factor completely.
共4 ⫺ x兲 ⫽ ⫺ 共x ⫺ 4兲
Divide out common factors.
Now try Exercise 35.
STUDY TIP
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
beside 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 beside
the simplified expression to
make the two domains agree.
Note that the value x ⫽ 12 is
excluded from both domains,
so it is not necessary to list
this value.
Operations with Rational Expressions
To multiply or divide rational expressions, you can use the properties of fractions
discussed in Section P.1. Recall that to divide fractions you invert the divisor
and multiply.
The factoring technique used in equating
共4 ⫺ x兲 and ⫺ 共x ⫺ 4兲 in Example 3 is not
always obvious to students. Identify the
technique in one or two examples.
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Section P.4
Rational Expressions
39
Example 4 Multiplying Rational Expressions
2x2 ⫹ x ⫺ 6
x2 ⫹ 4x ⫺ 5
⭈
x 3 ⫺ 3x2 ⫹ 2x 共2x ⫺ 3兲共x ⫹ 2兲
⫽
4x2 ⫺ 6x
共x ⫹ 5兲共x ⫺ 1兲
⫽
共x ⫹ 2兲共x ⫺ 2兲
,
2共x ⫹ 5兲
⭈
x共x ⫺ 2兲共x ⫺ 1兲
2x共2x ⫺ 3兲
x ⫽ 0, x ⫽ 1, x ⫽
3
2
Now try Exercise 51.
Example 5 Dividing Rational Expressions
Divide
x2 ⫹ 2x ⫹ 4
x3 ⫺ 8
.
by
x2 ⫺ 4
x3 ⫹ 8
Solution
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
⫽ x2 ⫺ 2x ⫹ 4,
x ⫽ ±2
Divide out common
factors.
Now try Exercise 53.
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 and 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 Subtracting Rational Expressions
Subtract
x
2
.
from
3x ⫹ 4
x⫺3
Solution
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
STUDY TIP
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 57.
When subtracting rational
expressions, remember to
distribute the negative sign to
all the terms in the quantity that
is being subtracted.
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Prerequisites
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
⫹ ⫺ ⫽
6 4 3 6
⭈2⫹3⭈3⫺2⭈4
⭈2 4⭈3 3⭈4
⫽
2
9
8
⫹
⫺
12 12 12
⫽
3
1
⫽
12 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
⫽
共3x2 ⫺ 2x2 ⫹ x2兲 ⫹ 共3x ⫹ 3x兲 ⫹ 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 63.
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Section P.4
Rational Expressions
41
Complex Fractions
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
1
x2 ⫹ 1
and
冢x
2
1
⫹1
冣
A complex fraction can be simplified by combining the fractions in its numerator
into a single fraction and then combining the fractions in its 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
Combine fractions.
冥
2 ⫺ 3x
冢 x 冣
⫽
x⫺2
冢x ⫺ 1冣
Simplify.
⫽
2 ⫺ 3x
x
⫽
共2 ⫺ 3x兲共x ⫺ 1兲
,
x共x ⫺ 2兲
x⫺1
⭈x⫺2
Invert and multiply.
Additional Example
x
x⫹6
3
⫹
4 2
4
⫽
3
2x ⫺ 3
2⫺
x
x
冢
冢
冣 冢
冣 冢
冣
冣
⫽
x⫹6
4
⫽
x共x ⫹ 6兲
,x⫽0
4共2x ⫺ 3兲
x
⭈ 2x ⫺ 3
x⫽1
Now try Exercise 69.
In Example 8, the restriction x ⫽ 1 is added to the final expression to make
its domain agree with the domain of the original expression.
Another way to simplify a complex fraction is to multiply each term in 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2x ⫺ 3冣
冢
1⫺
1
x⫺1
⫽
冢2x ⫺ 3冣
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共x ⫺ 2兲
x⫽1
LCD is x共x ⫺ 1兲.
Combine fractions.
Simplify.
Point out to your students that two different
methods are shown here for simplifying the
same complex fraction. Emphasize that both
yield the same result.
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Prerequisites
The next four 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 ⫽
3 ⫹ 2x
x5兾2
Example 9 Simplifying an Expression with
Negative Exponents
Simplify 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兲共⫺1兾2兲 ⫺ 共⫺3兾2兲兴
⫽ 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 75.
A second method for simplifying this type of expression involves
multiplying the numerator and denominator by a term to eliminate the negative
exponent.
Example 10 Simplifying an Expression with
Negative Exponents
Simplify
共4 ⫺ x 2兲1兾2 ⫹ x 2共4 ⫺ x2兲⫺1兾2
.
4 ⫺ x2
Solution
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
4
⫽
共4 ⫺ x 2兲 3兾2
共4 ⫺ x2兲3兾2
Now try Exercise 79.
Activities
1. What is the domain of 冪3 ⫺ x ?
Answer: The set of all real numbers less
than or equal to 3.
2. The implied domain excludes what
values of x from the domain of
x2 ⫹ 6x ⫹ 9
?
x2 ⫺ 9
Answer: x ⫽ 3, x ⫽ ⫺3
3. Simplify:
冢1x ⫺ x ⫹1 1冣
x⫹1
1
Answer:
x共x ⫹ 1兲2
.
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Section P.4
Example 11 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
⫽
冪x ⫹ h ⫺ 冪x
h
⭈
冪x ⫹ h ⫹ 冪x
冪x ⫹ h ⫹ 冪x
共冪x ⫹ h 兲 ⫺ 共冪x 兲2
h共冪x ⫹ h ⫹ 冪x 兲
2
⫽
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 85.
Difference quotients, like 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.
Example 12 Rewriting a Difference Quotient
Rewrite the expression by rationalizing its numerator.
冪x ⫺ 4 ⫺ 冪x
4
Solution
冪x ⫺ 4 ⫺ 冪x
4
⫽
冪x ⫺ 4 ⫺ 冪x
冪x ⫺ 4 ⫹ 冪x
⭈冪
x ⫺ 4 ⫹ 冪x
4
⫽
共冪x ⫺ 4兲2 ⫺ 共冪x兲2
4共冪x ⫺ 4 ⫹ 冪x 兲
⫽
⫺4
4共冪x ⫺ 4 ⫹ 冪x兲
⫽⫺
1
冪x ⫺ 4 ⫹ 冪x
Now try Exercise 86.
Rational Expressions
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Prerequisites
P.4 Exercises
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
Vocabulary Check
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 _______ .
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 _______ .
In Exercises 1– 16, find the domain of the expression.
1. 3x 2 ⫺ 4x ⫹ 7
2. 2x 2 ⫹ 5x ⫺ 2
3. 4x3 ⫹ 3, x ≥ 0
4. 6x 2 ⫺ 9, x > 0
5.
1
3⫺x
x2 ⫺ 1
⫺ 2x ⫹ 1
x2 ⫺ 2x ⫺ 3
9. 2
x ⫺ 6x ⫹ 9
7.
x2
6.
x⫹6
3x ⫹ 2
x2 ⫺ 5x ⫹ 6
x2 ⫺ 4
2
x ⫺ x ⫺ 12
10. 2
x ⫺ 8x ⫹ 16
27.
4y ⫺ 8y2
10y ⫺ 5
28.
9x 2 ⫹ 9x
2x ⫹ 2
29.
x⫺5
10 ⫺ 2x
30.
12 ⫺ 4x
x⫺3
31.
y2 ⫺ 16
y⫹4
32.
x 2 ⫺ 25
5⫺x
33.
x 3 ⫹ 5x 2 ⫹ 6x
x2 ⫺ 4
34.
x 2 ⫹ 8x ⫺ 20
x 2 ⫹ 11x ⫹ 10
8.
11. 冪x ⫹ 7
12. 冪4 ⫺ x
35.
36.
13. 冪2x ⫺ 5
1
15.
冪x ⫺ 3
14. 冪4x ⫹ 5
1
16.
冪x ⫹ 2
y 2 ⫺ 7y ⫹ 12
y 2 ⫹ 3y ⫺ 18
3⫺x
x 2 ⫹ 11x ⫹ 10
37.
2 ⫺ x ⫹ 2x 2 ⫺ x 3
x⫺2
38.
x2 ⫺ 9
x 3 ⫹ x 2 ⫺ 9x ⫺ 9
40.
y 3 ⫺ 2y 2 ⫺ 3y
y3 ⫹ 1
39.
In Exercises 17–22, find the missing factor in the numerator
such that the two fractions are equivalent.
z2
z3 ⫺ 8
⫹ 2z ⫹ 4
5
5共䊏兲
17.
⫽
2x
6x2
2共
2
兲
18. 2 ⫽ 䊏
3x
3x4
In Exercises 41 and 42, complete the table. What can you
conclude?
3
3共䊏兲
19. ⫽
4 4共x ⫹ 1兲
2
2共
兲
20. ⫽ 䊏
5 5共x ⫺ 3兲
41.
21.
22.
x⫺1
共x ⫺ 1兲共䊏兲
⫽
4共x ⫹ 2兲
4共x ⫹ 2兲2
x
0
1
2
3
4
5
6
0
1
2
3
4
5
6
x2 ⫺ 2x ⫺ 3
x⫺3
共x ⫹ 3兲共䊏兲
x⫹3
⫽
2共x ⫺ 1兲
2共x ⫺ 1兲2
x⫹1
42.
In Exercises 23 – 40, write the rational expression in simplest
form.
15x 2
23.
10x
18y 2
24.
60y 5
3xy
25.
xy ⫹ x
2x2y
26.
xy ⫺ y
x
x⫺3
x2 ⫺ x ⫺ 6
1
x⫹2
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Section P.4
43. Error Analysis
Describe the error.
61.
5x 3
5x3
5
5
⫽ 3
⫽
⫽
3
2x ⫹ 4 2x ⫹ 4 2 ⫹ 4 6
44. Error Analysis
62.
Describe the error.
x2
45
Rational Expressions
1
x
⫺
⫺ x ⫺ 2 x 2 ⫺ 5x ⫹ 6
2
10
⫹
x 2 ⫺ x ⫺ 2 x 2 ⫹ 2x ⫺ 8
1
2
1
⫺
63. ⫺ ⫹ 2
x
x ⫹ 1 x3 ⫹ x
⫹ 25x
x共x2 ⫹ 25兲
⫽
2
x ⫺ 2x ⫺ 15 共x ⫺ 5兲共x ⫹ 3兲
x3
64.
x共x ⫺ 5兲共x ⫹ 5兲 x共x ⫹ 5兲
⫽
⫽
共x ⫺ 5兲共x ⫹ 3兲
x⫹3
2
2
1
⫹
⫹
x ⫹ 1 x ⫺ 1 x2 ⫺ 1
In Exercises 65–72, simplify the complex fraction.
Geometry In Exercises 45 and 46, find the ratio of the area
of the shaded portion of the figure to the total area of the
figure.
冢 2 ⫺ 1冣
x
65.
45.
66.
共x ⫺ 2兲
2
2
3
冤 (x ⫹ h)
1
x+5
2
69.
x+5
2
2
⫺
1
x2
冥
h
冢冪x ⫺ 2冪x冣
x⫹h
70.
1
x+5
71.
2x + 3
In Exercises 47– 54, perform the multiplication or
division and simplify.
x共x ⫺ 3兲
5
冪x
72.
冢x ⫹ h ⫹ 1 ⫺ x ⫹ 1冣
冢冪t
4y ⫺ 16
4⫺y
⫼
5y ⫹ 15 2y ⫹ 6
76. 2x共x ⫺ 5兲⫺3 ⫺ 4x2共x ⫺ 5兲⫺4
51.
t2 ⫺ t ⫺ 6
t 2 ⫹ 6t ⫹ 9
52.
y3 ⫺ 8
2y 3
78. 4x3共2x ⫺ 1兲3兾2 ⫺ 2x共2x ⫺ 1兲⫺1兾2
53.
3共x ⫹ y兲 x ⫹ y
⫼
4
2
54.
x⫹2
x⫺2
⫼
5共x ⫺ 3兲 5共x ⫺ 3兲
t⫹3
⭈ t2 ⫺ 4
4y
⭈ y 2 ⫺ 5y ⫹ 6
In Exercises 55 – 64, perform the addition or subtraction
and simplify.
⫹1
74. x5 ⫺ 5x⫺3
50.
⭈
2
73. x5 ⫺ 2x⫺2
r
r2
⫼ 2
r⫺1 r ⫺1
49.
x⫺1
⭈ 25共x ⫺ 2兲
t
2
⫺ 冪t 2 ⫹ 1
冣
t2
In Exercises 73–78, simplify the expression by removing the
common factor with the smaller exponent.
x ⫹ 13
x 3共3 ⫺ x兲
5
x⫺1
x
h
48.
47.
冣
x2 ⫺ 1
x2
46.
冢
冢 x 冣
68.
共x ⫺ 1兲
冤 x 冥
冤 共x ⫹ 1兲 冥
67.
x
冤 共x ⫹ 1兲 冥
r
共x ⫺ 4兲
x
4
⫺
4
x
75. x2共x2 ⫹ 1兲⫺5 ⫺ 共x2 ⫹ 1兲⫺4
77. 2x2共x ⫺ 1兲1兾2 ⫺ 5共x ⫺ 1兲⫺1兾2
In Exercises 79–84, simplify the expression.
79.
2x3兾2 ⫺ x⫺1兾2
x2
80.
x2共32x ⫺1兾2兲 ⫺ 3x1兾2共x2兲
x4
55.
5
x
⫹
x⫺1 x⫺1
56.
2x ⫺ 1 1 ⫺ x
⫺
x⫹3
x⫹3
81.
⫺x2共x 2 ⫹ 1兲⫺1兾2 ⫹ 2x共x 2 ⫹ 1兲⫺3兾2
x3
57.
6
x
⫺
2x ⫹ 1 x ⫹ 3
58.
3
5x
⫹
x ⫺ 1 3x ⫹ 4
82.
x3共4x1兾2兲 ⫺ 3x2共83x3兾2兲
x6
59.
3
5
⫹
x⫺2 2⫺x
60.
2x
5
⫺
x⫺5 5⫺x
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Chapter P
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Prerequisites
83.
共x2 ⫹ 5兲共12 兲共4x ⫹ 3兲⫺1兾2共4兲 ⫺ 共4x ⫹ 3兲1兾2共2x兲
共x2 ⫹ 5兲2
84.
共2x ⫹ 1兲1兾2共3兲共x ⫺ 5兲2 ⫺ 共x ⫺ 5兲3共12 兲共2x ⫹ 1兲⫺1兾2共2兲
2x ⫹ 1
95. Resistance The formula for the total resistance RT (in
ohms) of a parallel circuit is given by
In Exercises 85–90, rationalize the numerator of the
expression.
85.
87.
89.
冪x ⫹ 2 ⫺ 冪x
86.
2
冪x ⫹ 2 ⫺ 冪2
88.
x
冪x ⫹ 9 ⫺ 3
90.
x
x
2x + 1
冪x ⫹ 5 ⫺ 冪5
x
冪x ⫹ 4 ⫺ 2
x
x+4
x
where R1, R2, and R3 are the resistance values of the first,
second, and third resistors, respectively.
(b) Find the total resistance in the parallel circuit when
R1 ⫽ 6 ohms, R2 ⫽ 4 ohms, and R3 ⫽ 12 ohms.
3
92.
x
2
1
1
1
1
⫹
⫹
R1 R2 R3
(a) Simplify the total resistance formula.
冪z ⫺ 3 ⫺ 冪z
Probability In Exercises 91 and 92, 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.
91.
RT ⫽
x
x + 2 4 (x + 2)
x
93. Rate A photocopier copies at a rate of 16 pages per
minute.
(a) Find the time required to copy 1 page.
(b) Find the time required to copy x pages.
96. 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. Consider the model that
gives the temperature of food that is at 75⬚F and is placed
in a 40⬚F refrigerator as
T ⫽ 10
4t 2 ⫹ 16t ⫹ 75
2 ⫹ 4t ⫹ 10
冢t
冣
where T is the temperature (in degrees Fahrenheit) and t is
the time (in hours).
(a) Complete the table.
t
0
2
4
6
8
10
12
14
16
18
20
22
T
t
T
(c) Find the time required to copy 60 pages.
94. Monthly Payment The formula that approximates the
annual interest rate r of a monthly installment loan is given
by
24共NM ⫺ P兲
N
r⫽
NM
P⫹
12
冤
冢
冥
冣
where N is the total number of payments, M is the monthly
payment, and P is the amount financed.
(a) Approximate the annual interest rate for a five-year car
loan of $20,000 that has monthly payments of $400.
(b) Simplify the expression for the annual interest rate r,
and then rework part (a).
(b) What value of T does the mathematical model appear
to be approaching?
97. Plants The table shows the numbers of endangered and
threatened plant species in the United States for the years
2000 through 2005. (Source: U.S. Fish and Wildlife
Service)
Year
Endangered, E
Threatened, T
2000
2001
2002
2003
2004
2005
565
592
596
599
597
599
139
144
147
147
147
147
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Section P.4
2342.52t ⫹ 565
3.91t2 ⫹ 1
2
and
Threatened plants: T ⫽
(b) Determine a model for the ratio of the number of
marriages to the number of divorces. Use the model to
find this ratio for each of the given years.
243.48t ⫹ 139
1.65t2 ⫹ 1
2
where t represents the year, with t ⫽ 0 corresponding to
2000.
(a) Using the models, create a table to estimate the numbers
of endangered plant species and the numbers of
threatened plant species for the given years. Compare
these estimates with the actual data.
(b) Determine a model for the ratio of the number of
threatened plant species to the number of endangered
plant species. Use the model to find this ratio for each
of the given years.
98. Marriages and Divorces The table shows the rates (per
1000 of the total population) of marriages and divorces in
the United States for the years 1990 through 2004.
(Source: U.S. National Center for Health Statistics)
Year
Marriages, M
Divorces, D
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
9.8
9.4
9.3
9.0
9.1
8.9
8.8
8.9
8.4
8.6
8.3
8.2
7.8
7.5
7.4
4.7
4.7
4.8
4.6
4.6
4.4
4.3
4.3
4.2
4.1
4.2
4.0
4.0
3.8
3.7
Mathematical models for the data are
Marriages: M ⫽
47
(a) Using the models, create a table to estimate the number
of marriages and the number of divorces for each of the
given years. Compare these estimates with the actual
data.
Mathematical models for the data are
Endangered plants: E ⫽
Rational Expressions
8686.635t2 ⫺ 191,897.18t ⫺ 9.8
774.364t2 ⫺ 20,427.65t ⫺ 1
and
Divorces: D ⫽ ⫺0.001t2 ⫺ 0.06t ⫹ 4.8
where t represents the year, with t ⫽ 0 corresponding to
1990.
In Exercises 99–104, simplify the expression.
共x ⫹ h兲2 ⫺ x2
, h⫽0
h
共x ⫹ h兲3 ⫺ x3
100.
, h⫽0
h
1
1
⫺
共x ⫹ h兲2 x2
101.
, h⫽0
h
1
1
⫺
2共x ⫹ h兲 2x
, h⫽0
102.
h
99.
103.
104.
冪2x ⫹ h ⫺ 冪2x
h
冪x ⫹ h ⫺ 冪x
h
h⫽0
,
,
h⫽0
In Exercises 105 and 106, simplify the given expression.
4 n共n ⫹ 1兲共2n ⫹ 1兲
4
⫹ 2n
n
6
n
冢
冣 冢冣
3 n共n ⫹ 1兲共2n ⫹ 1兲
106. 9冢 冣冢
冣 ⫺ n冢3n冣
n
6
105.
Synthesis
True or False? In Exercises 107 and 108, determine
whether the statement is true or false. Justify your answer.
107.
x2n ⫺ 12n
⫽ xn ⫹ 1n
x n ⫺ 1n
108.
x2n ⫺ n2
⫽ xn ⫹ n
xn ⫺ n
109. Think About It How do you determine whether a
rational expression is in simplest form?
110. Think About It Is the following statement true for all
nonzero real numbers a and b? Explain.
ax ⫺ b
⫽ ⫺1
b ⫺ ax
111. Writing Write a paragraph explaining to a classmate
why 冪x ⫹ y ⫽ 冪x ⫹ 冪y.
112. Writing Write a
1
1
why
⫽ ⫹
x⫹y
x
paragraph explaining to a classmate
1
.
y