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356 CHAPTER 6 Rational Expressions
OBJECTIVE
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 1 a b 2
x 2 y^2 6 x - 2 9 y
x +
y y + 1
The parts of a complex fraction are x y + 2 f
y f
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
Negative Exponents.
6.3 Simplifying Complex Fractions
d (^) Numerator of complex fraction d (^) Main fraction bar d (^) Denominator of complex fraction
Our goal in this section is to simplify complex fractions. A complex fraction is simpli- fied when it is in the form
P
Q
, where P and Q are polynomials that have no common factors. Two methods of simplifying complex fractions are introduced. The first method evolves from the definition of a fraction as a quotient.
1 Simplifying Complex Fractions: Method 1
Simplifying a Complex Fraction: Method I Step 1. 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 com- plex fraction by the reciprocal of the denominator of the complex fraction. Step 3. Simplify if possible.
E X A M P L E 1 Simplify each complex fraction.
a.
2 x 27 y^2 6 x^2 9
b.
5 x x + 2 10 x - 2
c.
x y^2
y y x^2
x
Solution
a. The numerator of the complex fraction is already a single fraction, and so is the denominator. Perform the indicated division by multiplying the numerator, 2 x 27 y^2
, by the reciprocal of the denominator, 6 x^2 9
. Then simplify. 2 x 27 y^2 6 x^2 9
2 x 27 y^2
6 x^2 9
2 x 27 y^2
6 x^2
2 x #^9
27 y^2 #^6 x^2
9 xy^2
Multiply by the reciprocal of 6 x^2
Section 6.3 Simplifying Complex Fractions 357
b.
e 5 x x + 2
e
x - 2
5 x x + 2
x - 2
5 x x + 2
x^ -^2
10
5 x 1 x - 22 2 #^51 x + 22
=
x 1 x - 22 21 x + 22
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.
x y^2
y y x^2
x
x y^2
1 #^ y y #^ y y x^2
1 #^ x x #^ x
x + y y^2 y + x x^2
=
x + y y^2
x
2 y + x
=
x^2 1 x + y 2 y^2 1 y + x 2
= x^2 y^2
Simplify the numerator. The LCD is y^2.
Simplify the denominator. The LCD is x^2.
Add.
Multiply by the reciprocal of y + x x^2 .
Simplify.
PRACTICE 1 Simplify each complex fraction.
a.
5 k 36 m 15 k 9
b.
8 x x - 4 3 x + 4
c.
a +^
b a^2 5 a b^2
b
Helpful Hint Both the numerator and denomi- nator are single fractions, so we perform the indicated division.
CONCEPT CHECK
Which of the following are equivalent to
y 2 z
a.
y ,^
z b.^
y
z
2 c.
y ,^
z 2
OBJECTIVE
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.
Answer to Concept Check: a and b
Section 6.3 Simplifying Complex Fractions 359
PRACTICE
OBJECTIVE
3 Simplifying Expressions with Negative Exponents
If an expression contains negative exponents, write the expression as an equivalent expression with positive exponents.
E X A M P L E 3 Simplify.
x -^1 + 2 xy -^1 x -^2 - x -^2 y -^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 + 2 xy -^1 x -^2 - x -^2 y -^1
x +^
2 x y 1 x^2
x^2 y
The LCD of
x ,
2 x y ,
x^2 , and
x^2 y is x^2 y. Multiply both the numerator and denominator by x^2 y.
a
x +^
2 x y b^
(^) x^2 y
a
x^2
x^2 y
b #^ x^2 y
x
(^) x^2 y + 2 _x
y_
(^) x^2 y
1 x^2
(^) x^2 y - 1
x^2 y
(^) x^2 y
xy + 2 x^3 y - 1 or
x 1 y + 2 x^22 y - 1
Apply the distributive property.
Simplify.
3 Simplify:
3 x -^1 + x -^2 y -^1 y -^2 + xy -^1
E X A M P L E 4 Simplify:
12 x 2 -^1 + 1 2 x -^1 - 1
Solution
(2 x )-^1 + 1 2 x -^1 - 1
2 x
x -^1
a
2 x
a
x -^1 b^
(^2) x
2 x
(^2) x + 1 # (^2) x
2 x
(^2) x - 1 # (^2) x
1 + 2 x 4 - 2 x or 1 + 2 x 212 - x 2
Write using positive exponents.
The LDC of^1 2 x and^2 x is 2 x.
Use distributive property.
Simplify.
Helpful Hint
Don’t forget that 12 x 2 -^1 = (^21) x ,
but 2 x -^1 = 2 #^1 x = 2 x .
360 CHAPTER 6 Rational Expressions
PRACTICE 4 Simplify:
13 x 2 -^1 - 2 5 x -^1 + 2
Vocabulary, Readiness & Video Check
Complete the steps by writing the simplified complex fraction.
x 1 x
z x
x a
x b
x a
x b + x a z x b
x 4 x^2 2
4 a x 4 b
4 a x^2 2 b + 4 a
b
Write with positive exponents.
3. x -^2 = 4. y -^3 = 5. 2 x -^1 = 6. 12 x 2 -^1 = 7. 19 y 2 -^1 = 8. 9 y -^2 =
Martin-Gay Interactive Videos
See Video 6.
OBJECTIVE 1 OBJECTIVE 2
OBJECTIVE 3
Watch the section lecture video and answer the following questions.
9. From Example 2, before you can rewrite the complex fraction as division, describe how it must appear. 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. 11. Based on Example 4, what connection is there between negative exponents and complex fractions?
Simplify each complex fraction. See Examples 1 and 2.
1.
10 3 x 5 6 x
2.
15 2 x 5 6 x
3.
1 + (^25)
2 + 3 5
4.
2 + (^17)
3 - 4 7
5.
4 x - 1 x x - 1
6.
x x + 2 2 x + 2
7.
1 - 2 x x + (^94) x
8.
5 - 3 x x + (^32) x
9.
4 x^2 - y^2 xy 2 y -^
1 x
10.
x^2 - 9 y^2 xy 1 y -^
3 x
11.
x + 1 3 2 x - 1 6
12.
x + 3 12 4 x - 5 15
13.
2 x +^
3 x^2 4 x^2
14.
2 x^2
15.
1 x +^
2 x^2 x + 8 x^2
16.
1 y +^
3 y^2 y + 27 y^2
6.3 Exercise Set
362 CHAPTER 6 Rational Expressions
71. 31 a + 12 -^1 + 4 a -^2 1 a^3 + a^22 -^1 72. 9 x -^1 - 51 x - y 2 -^1 41 x - y 2 -^1 In the study of calculus, the difference quotient
f 1 a + h 2 - f 1 a 2 h is often found and simplified. Find and simplify this quotient for each function f(x) by following steps a through d****. a. Find 1 a + h 2. b. Find f ( a ). c. Use steps a and b to find f 1 a + h 2 - f 1 a 2 h d. Simplify the result of step c.
73. f 1 x 2 = 1 x 74. f 1 x 2 = 5 x 75. (^) x^3 + 1 61. Which of the following are equivalent to
1 x 3 y
?
a.^1 x , 3 y b.^1 x
# y
3 c.^1 x , y 3
62. Which of the following are equivalent to^52 a
?
a.^51 , 2 a b.^15 , 2 a c.^51 #^2 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. 65.^1 1 + 11 + x 2 -^1 66.
1 x + 22 -^1 + 1 x - 22 -^1 1 x^2 - 42 -^1
67. x 1 - 1 1 + 1 x 68. x 1 - 1 1 - 1 x
69.
2 y^2
70.
2 x^2
OBJECTIVE
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 10 x^3 - 5 x^2 + 20 x 5 x can be thought of as the polynomial 10 x^3 - 5 x^2 + 20 x divided by the monomial 5 x. 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. a c
b c
a + b 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.
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.
6.4 Dividing Polynomials: Long Division and Synthetic Division
Dividing a Polynomial by a Monomial Divide each term in the polynomial by the monomial. a + b c
a c
b c , where c 0
v
76. 2 x^2
