356 CHAPTER 6
6.3
Rational Expressions
Simplifying Complex Fractions
OBJECTIVES
1 Simplify Complex Fractions
by Simplifying the Numerator
and Denominator and Then
Dividing.
2 Simplify Complex Fractions
by Multiplying by a Common
Denominator.
3 Simplify Expressions with
A rational expression whose numerator, denominator, or both contain one or more
rational expressions is called a complex rational expression or a complex fraction.
Complex Fractions
x
1
1
x +
2y 2
y
a
b
6x - 2
y + 1
2
9y
The parts of a complex fraction are
x
f d Numerator of complex fraction
y + 2
Negative Exponents.
1
7 + f
y
d Main fraction bar
d Denominator of complex fraction
Our goal in this section is to simplify complex fractions. A complex fraction is simpliP
fied when it is in the form , where P and Q are polynomials that have no common
Q
factors. Two methods of simplifying complex fractions are introduced. The first method
evolves from the definition of a fraction as a quotient.
OBJECTIVE
1
Simplifying Complex Fractions: Method 1
Simplifying a Complex Fraction: Method I
Simplify the numerator and the denominator of the complex fraction so
that each is a single fraction.
Step 2. Perform the indicated division by multiplying the numerator of the complex fraction by the reciprocal of the denominator of the complex fraction.
Step 3. Simplify if possible.
Step 1.
EXAMPLE 1
a.
Simplify each complex fraction.
2x
27y 2
5x
x + 2
b.
10
x - 2
6x2
9
1
x
+
y
y2
c.
y
1
+
2
x
x
Solution
a. The numerator of the complex fraction is already a single fraction, and so is the
2x
denominator. Perform the indicated division by multiplying the numerator,
, by
27y 2
6x2
. Then simplify.
the reciprocal of the denominator,
9
2x
27y 2
2x
6x2
=
,
9
27y 2
6x2
9
2x # 9
27y 2 6x2
2x # 9
=
27y 2 # 6x2
1
=
9xy 2
=
Multiply by the reciprocal of
6x 2
.
9
Section 6.3
Helpful Hint
Both the numerator and denominator are single fractions, so we
perform the indicated division.
5x
x + 2
5x
10
5x # x - 2
=
b.
,
=
x + 2
x - 2
x + 2
10
10
e
x - 2
5x1x - 22
= #
2 51x + 22
x1x - 22
=
21x + 22
Simplifying Complex Fractions 357
e
Multiply by the reciprocal of
10
.
x - 2
Simplify.
c. First simplify the numerator and the denominator of the complex fraction separately
so that each is a single fraction. Then perform the indicated division.
1#y
x
1
x
+
+ #
2
2
y
y y
y
y
=
y
y
1
1#x
+
+
x
x#x
x2
x2
x + y
=
=
=
=
y2
y + x
x2
x + y
y
2
Simplify the numerator. The LCD is y 2 .
Simplify the denominator. The LCD is x 2 .
Add.
#
x2
y + x
Multiply by the reciprocal of
y + x
x2
.
x2 1x + y2
y 2 1y + x2
x2
y2
Simplify.
PRACTICE
1
Simplify each complex fraction.
5k
36m
a.
15k
9
8x
x - 4
b.
3
x + 4
5
b
+ 2
a
a
c.
5a
1
+
2
b
b
CONCEPT CHECK
5
y
Which of the following are equivalent to ?
2
z
5
2
5#z
5
z
,
,
a.
c.
b.
y
y 2
y
z
2
OBJECTIVE
Answer to Concept Check:
a and b
2
Simplifying Complex Fractions: Method 2
Next we look at another method of simplifying complex fractions. With this method,
we multiply the numerator and the denominator of the complex fraction by the LCD
of all fractions in the complex fraction.
358 CHAPTER 6
Rational Expressions
Simplifying a Complex Fraction: Method 2
Multiply the numerator and the denominator of the complex fraction by
the LCD of the fractions in both the numerator and the denominator.
Step 2. Simplify.
Step 1.
EXAMPLE 2
Simplify each complex fraction.
x
1
+
2
y
y
b.
y
1
+
2
x
x
5x
x + 2
a.
10
x - 2
Solution
a. Notice we are reworking Example 1b using method 2. The least common denomina5x
10
5x
tor of
and
is 1x + 221x - 22 . Multiply both the numerator,
,
x + 2
x - 2
x + 2
10
and the denominator,
, by the LCD.
x - 2
5x
5x
a
b # 1x + 2 2 1x - 22
x + 2
x + 2
=
10
10
a
b # 1x + 22 1x - 22
x - 2
x - 2
Multiply numerator and
denominator by the LCD.
=
5 x # 1x - 22
2 # 5 # 1x + 22
Simplify.
=
x1x - 22
21x + 22
Simplify.
b. Here, we are reworking Example 1c using method 2. The least common denominax 1 y
1
tor of 2 , , 2 , and is x2y 2 .
x
y y x
x
1
x
1
+
a 2 + b # x 2y 2
2
y
y
y
y
=
y
y
1
1
+
a 2 + b # x 2y 2
x
x
x2
x
x # 2 2
1 # 2 2
x y +
x y
y
y2
=
y
# x2 y2 + 1 # x2 y2
2
x
x
3
2
x + xy
= 3
y + xy 2
x 21 x + y 2
= 2
y1y + x2
x2
= 2
y
PRACTICE
2
8x
x - 4
a.
3
x + 4
Use method 2 to simplify:
b
1
+
a
a2
b.
a
1
+
b
b2
Multiply the numerator and
denominator by the LCD.
Use the distributive property.
Simplify.
Factor.
Simplify.
Section 6.3
Simplifying Complex Fractions 359
OBJECTIVE
3
Simplifying Expressions with Negative Exponents
If an expression contains negative exponents, write the expression as an equivalent
expression with positive exponents.
EXAMPLE 3
Simplify.
x -1 + 2xy -1
x -2 - x -2y -1
Solution This fraction does not appear to be a complex fraction. If we write it by using
only positive exponents, however, we see that it is a complex fraction.
x -1 + 2xy -1
x -2 - x -2y -1
1
2x
+
y
x
=
1
1
- 2
x2
xy
1 2x 1
1
The LCD of , , 2 , and 2 is x2y. Multiply both the numerator and denominator
x y x
x
y
2
by x y.
a
=
2x # 2
1
+
b xy
y
x
1
1
- 2 b # x 2y
x2
xy
1# 2
2x # 2
xy +
xy
y
x
=
1 # 2
1
x y - 2 # x 2y
2
x
xy
x1y + 2x2 2
xy + 2x3
=
or
y - 1
y - 1
a
PRACTICE
3
Simplify:
EXAMPLE 4
Solution
Helpful Hint
Don’t forget that 12x2
but 2x -1 = 2 #
1
2
= .
x
x
-1
1
,
=
2x
Simplify.
3x -1 + x -2y -1
y -2 + xy -1
Simplify:
12x2 -1 + 1
2x -1 - 1
1
+ 1
2x
=
2
- 1
- 1
x
(2x)-1 + 1
2x -1
Apply the
distributive
property.
Write using positive exponents.
1
+ 1b # 2x
2x
=
2
a - 1b # 2x
x
1 #
2x + 1 # 2x
2x
=
2#
2x - 1 # 2x
x
a
=
1 + 2x
4 - 2x
or
1 + 2x
212 - x2
The LDC of
1
2
and is 2x.
2x
x
Use distributive property.
Simplify.
360 CHAPTER 6
Rational Expressions
PRACTICE
4
Simplify:
13x2 -1 - 2
5x -1 + 2
Vocabulary, Readiness & Video Check
Complete the steps by writing the simplified complex fraction.
1.
7
x
z
1
+
x
x
=
7
xa b
x
z
1
xa b + xa b
x
x
=
2.
x
4
x2
1
+
2
4
=
x
4a b
4
x2
1
4a b + 4a b
2
4
=
Write with positive exponents.
3. x - 2 =
4. y -3 =
5. 2x -1 =
6. 12x2 -1 =
7. 19y2 -1 =
8. 9y -2 =
Martin-Gay Interactive Videos
Watch the section lecture video and answer the following questions.
OBJECTIVE
1
9. From Example 2, before you can rewrite the complex fraction as
division, describe how it must appear.
OBJECTIVE
2
10. How does finding an LCD in method 2, as in Example 3, differ from
finding an LCD in method 1? In your answer, mention the purpose of
the LCD in each method.
OBJECTIVE
3
See Video 6.3
6.3
11. Based on Example 4, what connection is there between negative
exponents and complex fractions?
Exercise Set
Simplify each complex fraction. See Examples 1 and 2.
10
3x
1.
5
6x
2
5
3.
3
2 +
5
4
x - 1
5.
x
x - 1
2
1 x
7.
4
x +
9x
2.
1 +
15
2x
5
6x
1
7
4
3 7
x
x + 2
2
x + 2
3
5 x
2
x +
3x
2 +
4.
6.
8.
4x 2 - y 2
xy
9.
2
1
y
x
x + 1
3
11.
2x - 1
6
2
3
+ 2
x
x
13.
4
9
x
x2
2
1
+ 2
x
x
15.
8
x + 2
x
10.
12.
14.
16.
x 2 - 9y 2
xy
1
3
y
x
x + 3
12
4x - 5
15
2
1
+
2
x
x
4
1
x
x2
1
3
+ 2
y
y
27
y + 2
y
Section 6.3
4
+
5 - x
17.
2
+
x
x
x + 2
x
19.
x + 1
+
x
2
+ 3
x
21.
4
- 9
x2
x
1 y
23. 2
x
- 1
y2
- 2x
x - y
25.
y
5
x 3
- 5
2
x x +
x -
5
1
1
1
18.
20.
22.
24.
26.
2
x
1
2
+ 2
x
x
27.
y
28.
x2
x
1
9
x
29.
3
1 +
x
x - 1
x2 - 4
31.
1
1 +
x - 2
2
4
+
x + 5
x + 3
33.
3x + 13
2
x + 8x + 15
30.
32.
34.
2
3
x - 4
4 - x
2
2
x - 4
x
1
5
a+2
a-2
3
6
+
2+a
2-a
1
2 +
x
1
4x x
2
1 x
4
x x
7y
x 2 + xy
y2
x2
5
2
2
x
x
1
+ 2
x
x
4
4
x
4
1 x
x + 3
x2 - 9
1
1 +
x - 3
2
6
+
x + 2
x + 7
4x + 13
2
x + 9x + 14
Simplifying Complex Fractions 361
REVIEW AND PREVIEW
Simplify. See Sections 5.1 and 5.2.
51.
53.
3x 3y 2
12x
144x 5y 5
- 16x 2y
-36xb3
9xb2
48x 3y 2
54.
- 4xy
52.
Solve the following. See Section 2.6.
55. 0 x - 5 0 = 9
56. 0 2y + 1 0 = 1
CONCEPT EXTENSIONS
Solve. See the Concept Check in this section.
x + 1
9
?
57. Which of the following are equivalent to
y - 2
5
y - 2
x + 1
x + 1#y - 2
x + 1# 5
a.
,
b.
c.
9
5
9
5
9
y - 2
a
7
58. Which of the following are equivalent to
?
b
13
a# b
a
b
a
13
a 13
a.
b.
,
c.
,
d. #
7 13
7
13
7
b
7 b
59. When the source of a sound is traveling toward a listener,
the pitch that the listener hears due to the Doppler effect is
a
given by the complex rational compression
, where
s
1 770
a is the actual pitch of the sound and s is the speed of the
sound source. Simplify this expression.
Simplify. See Examples 3 and 4.
35.
37.
39.
41.
43.
45.
47.
49.
x -1
x + y -2
2a -1 + 3b -2
-2
a -1 - b -1
1
x - x -1
a -1 + 1
a -1 - 1
3x -1 + 12y2 -1
2a
-1
a
5x
-1
-1
x -2
+ 12a2 -1
+ 2a
+ 2y
-2
36.
38.
40.
42.
44.
46.
-1
x -2y -2
5x -1 - 2y -1
25x -2 - 4y -2
48.
50.
a -3 + b -1
a -2
-1
x + y -1
3x -2 + 5y -2
x -2
x + 3x -1
a -1 - 4
4 + a -1
5x -2 - 3y -1
x -1 + y -1
a -1 + 2a -2
2a -1 + 12a2 -1
x -2y -2
5x -1 + 2y -1
3x -1 + 3y -1
4x -2 - 9y -2
60. In baseball, the earned run average (ERA) statistic gives
the average number of earned runs scored on a pitcher
E
per game. It is computed with the following expression: ,
I
9
where E is the number of earned runs scored on a pitcher
and I is the total number of innings pitched by the pitcher.
Simplify this expression.
362 CHAPTER 6
Rational Expressions
1
x
61. Which of the following are equivalent to ?
3
y
y
1#y
3
1
1
a.
,
c.
,
b.
y
x
x 3
x
3
5
62. Which of the following are equivalent to ?
2
a
5
2
1
2
5#2
a.
,
b.
,
c.
1
a
5
a
1 a
63. In your own words, explain one method for simplifying a
complex fraction.
64. Explain your favorite method for simplifying a complex
fraction and why.
Simplify.
1
65.
1 + 11 + x2
x
67.
1
1 1
1 +
x
5
3
2
- 2
xy
y2
x
69.
7
3
2
+
+ 2
2
xy
y
x
-1
6.4
66.
71.
72.
31a + 12 -1 + 4a -2
1a 3 + a 2 2 -1
9x
-1
- 51x - y2 -1
41x - y2 -1
In the study of calculus, the difference quotient
f 1a + h2 - f 1a2
h
is often found and simplified. Find and simplify this quotient for
each function f(x) by following steps a through d.
a. Find 1a + h2 .
b. Find f(a).
c. Use steps a and b to find
f 1a + h2 - f 1a2
h
d. Simplify the result of step c.
73. f 1x2 =
1
x
74. f 1x2 =
5
x
1x + 22 -1 + 1x - 22 -1
1x 2 - 42 -1
x
68.
1
1 1
1 x
2
1
1
- 2
xy
x2
y
70.
3
2
1
+ 2
2
xy
x
y
75.
3
x + 1
76.
2
x2
Dividing Polynomials: Long Division and Synthetic Division
OBJECTIVE
OBJECTIVES
1 Divide a Polynomial by a
Monomial.
2 Divide by a Polynomial.
3 Use Synthetic Division to
Divide a Polynomial by a
Binomial.
4 Use the Remainder Theorem to
Evaluate Polynomials.
1
Dividing a Polynomial by a Monomial
Recall that a rational expression is a quotient of polynomials. An equivalent form of a
rational expression can be obtained by performing the indicated division. For example,
the rational expression
10x3 - 5x2 + 20x
5x
can be thought of as the polynomial 10x3 - 5x2 + 20x divided by the monomial 5x. To
perform this division of a polynomial by a monomial (which we do on the next page),
recall the following addition fact for fractions with a common denominator.
v
a
b
a + b
+
=
c
c
c
If a, b, and c are monomials, we might read this equation from right to left and gain
insight into dividing a polynomial by a monomial.
Dividing a Polynomial by a Monomial
Divide each term in the polynomial by the monomial.
a
b
a + b
= + , where c ⬆ 0
c
c
c