12.5
Multiply and Divide
Rational Expressions
You multiplied and divided polynomials.
Before
You will multiply and divide rational expressions.
Now
So you can describe football data, as in Ex. 35.
Why?
Multiplying and dividing rational expressions is similar to multiplying and
Key Vocabulary
dividing numerical fractions.
• multiplicative
inverse, p. 103
• polynomial, p. 554
KEY CONCEPT
For Your Notebook
• rational expression,
p. 794
Multiplying and Dividing Rational Expressions
Let a, b, c, and d be polynomials.
a c
ac
Algebra }
p}5}
where b Þ 0 and d Þ 0
b
d
a
b
bd
c
d
a
b
d
ad
bc
}4}5}p}
c 5 } where b Þ 0, c Þ 0, and d Þ 0
3(x 1 2)
12 3
Examples x}
5}
2
x p}
x
x3
EXAMPLE 1
x
x21
4
x
x
x21
x
4
Multiply rational expressions involving monomials
2
2
2x
6x
Find the product }
p}
.
3
3x
2
2
2
12x
2
(2x )(6x )
2x
6x
} p }3 5 }}}}}
3x 12x
(3x)(12x3)
APPLY EXCLUDED
VALUES
4
5 12x
}4
When performing
operations with rational
expressions, remember
that the answer may
have excluded values.
In Example 1, the
excluded value is 0.
✓
Multiply numerators and denominators.
Product of powers property
36x
4
12 p x
5}
4
Factor and divide out common factors.
1
5}
Simplify.
3 p 12 p x
3
GUIDED PRACTICE
for Example 1
Find the product.
2y 3 15y 3
1. }
p}
5
5y
802
x2
4(x 2 1)
}4}5}p}5}
8y
Chapter 12 Rational Equations and Functions
7z2 z3
2. }
p}
3
4z
14z
EXAMPLE 2
Multiply rational expressions involving polynomials
2
2
3x 1 3x
2 4x 1 3
Find the product }}}}}}}
p x}
.
2
2
4x 2 24x 1 36
3x2 1 3x
4x 2 24x 1 36
x 2x
x2 2 4x 1 3
x 2x
p }
}}}}}}}
2
2
(3x2 1 3x)(x2 2 4x 1 3)
5 }}}}}}}}}}
2
2
(4x 2 24x 1 36)(x 2 x)
3x(x 1 1)(x 2 3)(x 2 1)
4x(x 2 3)(x 2 3)(x 2 1)
Factor and divide out common factors.
3(x 1 1)
4(x 2 3)
Simplify.
5 }}}}}}}}}}
5}
CHECK
Multiply numerators and denominators.
Check your simplification using a graphing
calculator.
2
2
3x 1 3x
2 4x 1 3
Graph y1 5 }}
p x}
2
2
4x 2 24x 1 36
x 2x
3(x 1 1)
4(x 2 3)
and y 2 5 } .
The graphs coincide. So, the expressions are
equivalent for all values of x other than the
excluded values (0, 1, and 3).
MULTIPLYING BY A POLYNOMIAL When you multiply a rational expression by
a polynomial, first write the polynomial as a fraction with a denominator of 1.
EXAMPLE 3
Multiply a rational expression by a polynomial
5x
Find the product }
p (x 1 3).
2
x 1 5x 1 6
5x
x 1 5x 1 6
p (x 1 3)
}
2
5x
13
5}
p x}
2
1
x 1 5x 1 6
5x(x 1 3)
5 }}}}}
2
Multiply numerators and denominators.
x 1 5x 1 6
5x(x 1 3)
(x 1 2)(x 1 3)
5 }}
Factor and divide out common factor.
5x
5}
Simplify.
x12
✓
Rewrite polynomial as a fraction.
GUIDED PRACTICE
for Examples 2 and 3
Find the product.
2
2x2 1 2x
1x22
3. x}
p }}}}}}}
2
2
x 1 2x
5x 2 15x 1 10
2
2w
4. }}
p (w 2 4)
2
w 2 7w 1 12
12.5 Multiply and Divide Rational Expressions
803
DIVIDING RATIONAL EXPRESSIONS To divide by a rational expression, multiply
by its multiplicative inverse.
EXAMPLE 4
Divide rational expressions involving polynomials
2
7x 2 7x
x11
Find the quotient }}}}}
4 }}}}}
.
2
2
x 1 2x 2 3
2
7x 2 7x
x 1 2x 2 3
x 2 7x 2 8
x11
x 2 7x 2 8
4 }}}}}
}}}}}
2
2
REVIEW INVERSES
2
7x2 2 7x
2 7x 2 8
5 }}}}}
p x}
2
x11
x 1 2x 2 3
For help with finding
the multiplicative
inverse of a number,
see p. 103.
(7x2 2 7x)(x2 2 7x 2 8)
5 }}}}}}}}}}
2
(x 1 2x 2 3)(x 1 1)
Multiply by multiplicative inverse.
Multiply numerators and denominators.
7x(x 2 1)(x 2 8)(x 1 1)
(x 1 3)(x 2 1)(x 1 1)
Factor and divide out common factors.
7x(x 2 8)
x13
Simplify.
5 }}}}}}}}}}
5}
DIVIDING BY A POLYNOMIAL When you divide a rational expression by a
polynomial, first write the polynomial as a fraction with a denominator of 1.
Then multiply by the multiplicative inverse of the polynomial.
EXAMPLE 5
Divide a rational expression by a polynomial
2
1 16x 1 24
Find the quotient 2x
4 (x 1 6).
}}}}}}}
2
3x
2
2x 1 16x 1 24
4 ( x 1 6)
}}}}}}}
3x2
2x2 1 16x 1 24
16
5 }}}}}}}
4 x}
2
1
3x
2
1 16x 1 24
1
5 2x
p}
}}}}}}}
2
x16
3x
2
1 16x 1 24
5 2x
}}
2
3x (x 1 6)
2(x 1 2)(x 1 6)
5 }}}}}}
2
3x (x 1 6)
2(x 1 2)
5}
2
Multiply by multiplicative inverse.
Multiply numerators and denominators.
Factor and divide out common factor.
Simplify.
3x
"MHFCSB
✓
Rewrite polynomial as fraction.
at classzone.com
GUIDED PRACTICE
for Examples 4 and 5
Find the quotient.
2
6m 2 3m2
4m 1 44
m 24
5. }
4}
2
2m 1 4m
804
Chapter 12 Rational Equations and Functions
2
2 6n 1 9
6. n
} 4 (n 2 3)
12n
EXAMPLE 6
Solve a multi-step problem
ADVERTISING The amount A (in millions of dollars) spent on all advertising
and the amount T (in millions of dollars) spent on television advertising in
the United States during the period 1970–2003 can be modeled by
13,000 1 3700x
1 2 0.015x
A 5 }}}}}}}
and
1 860x
T 5 1800
}}}}}}
1 2 0.016x
where x is the number of years since 1970. Write a model that gives the
percent p (in decimal form) of the amount spent on all advertising that was
spent on television advertising. Then approximate the percent spent on
television advertising in 2003.
Solution
STEP 1 Write a verbal model. Then write an equation.
Percent spent on
Amount spent on
5
4
television advertising
television advertising
5
p
Amount spent on
all advertising
4
T
A
STEP 2 Find the quotient.
p5T4A
Write equation.
13,000 1 3700x
1 2 0.015x
1 860x
5 1800
}} 4 }}}}}}}
Substitute for T and for A.
1 860x
1 2 0.015x
5 1800
}} p }}
Multiply by multiplicative inverse.
1 2 0.016x
1 2 0.016x
13,000 1 3700x
5 }}}}}}}}}}}}
(1800 1 860x)(1 2 0.015x)
(1 2 0.016x)(13,000 1 3700x)
Multiply numerators
and denominators.
20(90 1 43x)(1 2 0.015x)
(1 2 0.016x)(20)(650 1 185x)
Factor and divide out
common factor.
(90 1 43x)(1 2 0.015x)
(1 2 0.016x)(650 1 185x)
Simplify.
5 }}}}}}}}}}}}
5 }}}}}}}}}}}
STEP 3 Approximate the percent spent on television advertising in 2003.
Because 2003 2 1970 5 33, x 5 33. Substitute 33 for x in the model
and use a calculator to evaluate.
(90 1 43 p 33)(1 2 0.015 p 33)
(1 2 0.016 p 33)(650 1 185 p 33)
p 5 }}}}}}}}}}}}} < 0.239
c About 24% of the amount spent on all advertising was spent on television
advertising in 2003.
✓
GUIDED PRACTICE
for Example 6
7. In Example 6, find the values of T and of A separately when x 5 33. Then
divide the value of T by the value of A. Compare your answer with the
answer in Step 3 above.
12.5 Multiply and Divide Rational Expressions
805
12.5
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 5, 15, and 35
★ 5 STANDARDIZED TEST PRACTICE
Exs. 2, 21, 26, 27, 28, 36, and 37
5 MULTIPLE REPRESENTATIONS
Ex. 35
SKILL PRACTICE
1. VOCABULARY Copy and complete: To divide by a rational expression,
multiply by its ? .
2.
EXAMPLES
1, 2, and 3
on pp. 802–803
for Exs. 3–10, 12
★ WRITING Describe how to multiply a rational expression by a polynomial.
MULTIPLYING EXPRESSIONS Find the product.
5
4. }
p}
6
2v 2 2
1 v 2 12
5. v}}}}}
p }}}}}
2
6. }
p}
2
2
7
6p
8q
y22
2
5v 1 10
v 1 5v 1 4
4x 2 20x 2 144
20
5x
7. }}}}}}}
p }}}}}}}
3
2
2x 2 17x 2 9x
23m
9. }}
p (m 2 5)
2
5
r
8. }
p (r 2 1 8)
3
7r 1 56r
2n 2 6
10. }}}}}}
p (3n2 1 14n 1 8)
2
m 2 7m 1 10
on p. 804
for Exs. 11,
13–21
4y2 1 20y
y 24
22y 2 10y
2
EXAMPLES
4 and 5
4q5
3
2
5
3. 9p
} p }4
3n 2 7n 2 6
ERROR ANALYSIS Describe and correct the error in finding the product or
quotient.
15x3
x3
11. }
4}
x
22
12. x}
p}
x15
2
5
x3
5
15x3
2
x22
15x3
2
5
x
22x
} 4 } 5 }3 p }
x
(x 2 2)x
(x 1 5)(2 2 x)
p}5}
}
x15 22x
(x 2 2)x
(x 1 5)(2 2 x)
3
75x
5}
3
5}
75
5}
5}
x15
2x
x
2
DIVIDING EXPRESSIONS Find the quotient.
6
12
16r2
13. }
4}
5r
3
5s
25s12 }
14. }
4
2
w
1 5w
15. 2w
4}
}
2
w 2 81
c 1 c 2 30
3
2
3a 1 18a 2 21
2
2
23x2 1 13x 2 12
15x 2 14x 2 8
2x 2 9x 1 9
18. }}}}}}
4 }}
2
35x 1 14
2
2 9t 2 22
20. t}
4 (5t 2 1 9t 2 2)
1 4k 2 15
19. 4k
}} 4 (2k 1 5)
5t 2 1
2k 2 3
21.
c26
c 2 11c 1 30
c 1c
16. }}}}}
4 }}}}}}
2
2
w19
2
9a 2 18a
1 3a 2 10
17. a}}
4 }}
2
2
a 1 6a 2 7
2
18
2
2
★ MULTIPLE CHOICE What common factor do you divide out when finding
x 2 2 3x 1 2
x 2 2x 2 3
x2 1 4x 1 3
x 2 7x 1 12
the quotient }}}}}
4 }}}}}}
?
2
2
A x21
806
B x23
Chapter 12 Rational Equations and Functions
C x11
D x13
TRANSLATING PHRASES Translate the verbal phrase into a product or
quotient of rational expressions. Then find the product or quotient.
22. The product of x 1 3 and the ratio of x 1 5 to x2 2 9
23. The product of 8x2 and the multiplicative inverse of 2x3
24. The quotient of x2 1 3x 2 18 and the ratio of x 1 6 to 2
25. The quotient of the multiplicative inverse of x2 2 3x 2 4 and twice the
multiplicative inverse of x2 2 1
26.
2
x 21
★ MULTIPLE CHOICE What is the quotient }
4 (x 2 1)?
2(x 1 1)
A 21
B 0
D x2 2 1
C 1
★ OPEN – ENDED
Let a, b, c, and d be different polynomials. Find two
a
c
rational expressions }
and }
that satisfy the given conditions.
b
d
x23
x12
27. The product of the rational expressions is } , and the excluded values
are 22, 21, 4, and 5.
x26
x14
28. The quotient of the rational expressions is } , and the excluded values
are 24, 22, 3, and 6.
GEOMETRY Write an expression for the area of the figure. Find a value
of x less than 5 for which the given dimensions and the area are positive.
29. Rectangle
30. Triangle
2x 2 1 2x 2 24
2x 1 1
x 2 2 6x 1 5
x12
2x 2 2 x 2 1
x23
x2 2 x 2 6
x25
CHALLENGE Let a be a polynomial in the given equation. Find a.
3x2 1 5x 2 2
x24
a
31. }
p }}}}}} 5 6x2 1 7x 2 3
x12
2
2x 1 1
2 2x 2 3
2
32. 8x
}}}}}} 4 }
a 5 12x 2 x 2 6
x25
PROBLEM SOLVING
EXAMPLE 6
33. VEHICLES The total distance M (in billions of miles)
on p. 805
for Exs. 33–35
traveled by all motor vehicles and the distance T
(in billions of miles) traveled by trucks in the
United States during the period 1980–2002 can
be modeled by
M 5 1500 1 63x
and
1 2.2x
T 5 100
}
1 2 0.014x
where x is the number of years since 1980. Write a
model that gives the percent p (in decimal form)
of the total motor vehicle distance that was traveled
by trucks as a function of x. Then approximate the
percent traveled by trucks in 2002.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
12.5 Multiply and Divide Rational Expressions
807
34. CONSUMER SPENDING The average annual amount T (in dollars) spent
on reading and entertainment and the average annual amount E (in
dollars) spent on entertainment by consumers in the United States
during the period 1985–2002 can be modeled by
1300 1 84x
1 1 0.015x
T5 }
and
1100 1 64x
E5}
1 1 0.0062x
where x is the number of years since 1985. Write a model that gives
the percent p (in decimal form) of the amount spent on reading and
entertainment that was spent on entertainment as a function of x. Then
approximate the percent spent on entertainment in 2000.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
35.
MULTIPLE REPRESENTATIONS Football player Emmitt Smith’s career
number Y of rushing yards gained and his career number A of rushing
attempts from 1990 (when he started playing professional football)
through the 2002 football season can be modeled by
860 1 1800x
1 1 0.024x
Y 5 }}
1 380x
A 5 230
}
and
1 1 0.014x
where x is the number of years since 1990.
a. Writing an Equation A football player’s rushing average is the
number of rushing yards gained divided by the number of rushing
attempts. Write a model that gives Smith’s career rushing average R
as a function of x for the period 1990–2002.
b. Making a Table Make a table that shows Smith’s approximate career
rushing average (rounded to the nearest hundredth) for each year
during the period. Describe how the career rushing average changed
over time.
36.
★ SHORT RESPONSE Baseball player Hank Aaron’s
career number B of times at bat and career number H
of hits during the period 1954–1976 can be modeled by
300 1 700x
1 1 0.01x
B5 }
and
62 1 240x
H5}
1 1 0.017x
where x is the number of years since 1954.
a. Model A baseball player’s batting average is the
number of hits divided by the number of times at
bat. Write a model that gives Hank Aaron’s career
batting average A as a function of x.
b. Decide The table shows Aaron’s actual career number of times at bat and
actual career number of hits for three different years. For which year does
the model give the best approximation of A? Explain your choice.
808
Year
1954
1959
1976
Career times
at bat
468
3524
12,364
Career hits
131
1137
3771
5 WORKED-OUT SOLUTIONS
on p. WS1
★ 5 STANDARDIZED
TEST PRACTICE
5 MULTIPLE
REPRESENTATIONS
37.
★ EXTENDED RESPONSE The gross revenue
R (in millions of dollars) from movie tickets sold
and the average movie ticket price P (in dollars)
in the United States during the period 1991–2002
can be modeled by
2 74x
R 5 4700
}
P 5 0.015x2 1 4.1
and
1 2 0.053x
where x is the number of years since 1991.
a. Model Write a model that gives the number T of
movie tickets sold (in millions) as a function of x.
b. Describe Graph the model on a graphing calculator and describe
how the number of tickets sold changed over time. Can you use the
graph to describe how the gross revenue and ticket prices changed
over time? Explain your reasoning.
c. Compare The table shows the actual number of tickets sold for each
year during the period. Make a scatter plot of the data on the same
screen as the graph of the model in part (b). Compare the scatter plot
with the graph of the model.
Year
1991
1992
1993
1994
1995
1996
Tickets (millions)
1141
1173
1244
1292
1263
1339
Year
1997
1998
1999
2000
2001
2002
Tickets (millions)
1388
1481
1465
1421
1487
1639
38. CHALLENGE The total amount F (in billions of dollars) spent on food
other than groceries and the amount E (in billions of dollars) spent at
restaurants in the U.S. during the period 1977–2003 can be modeled by
88 1 9.2x
F 5 }}}}}
and
1 2 0.0097x
54 1 6.5x
E5}
1 2 0.012x
where x is the number of years since 1977. Write a model that gives the
percent p (in decimal form) of the amount spent on food other than
groceries that was spent at restaurants as a function of x. Approximate
the percent that was spent at locations other than restaurants in 2002.
MIXED REVIEW
PREVIEW
Add or subtract. (p. 914)
Prepare for
Lesson 12.6
in Exs. 39–52.
2
2
39. }
1}
5
3
40. }
1}
5
1
41. }
1}
8
7
42. }
1}
7
7
43. }
2}
1
7
44. }
2}
5
9
45. }
2}
2
5
46. }
2}
5
8
3
10
8
15
12
4
4
14
6
21
9
17
21
51
Find the sum, difference, or product.
47. (25x2 2 6x) 1 (4x2 2 5) (p. 554)
48. (7x2 1 5x 1 1) 1 (26x2 1 13x) (p. 554)
49. (2x2 2 x 1 12) 2 (3x 1 8) (p. 554)
50. (7x2 1 16) 2 (8x3 1 3x2 2 7) (p. 554)
51. (5x 2 6)(4x 2 5) (p. 562)
52. (2x 1 9)(3x 2 7) (p. 562)
EXTRA PRACTICE for Lesson 12.5, p. 949
ONLINE QUIZ at classzone.com
809
Extension
Use after Lesson 12.5
Simplify Complex Fractions
GOAL Simplify complex fractions.
Key Vocabulary
• complex fraction
A complex fraction is a fraction that contains a fraction in its numerator,
denominator, or both. To simplify a complex fraction, divide its numerator by
its denominator.
For Your Notebook
KEY CONCEPT
Simplifying a Complex Fraction
Let a, b, c, and d be polynomials where b Þ 0, c Þ 0, and d Þ 0.
READING
The widest fraction
bar separates the
numerator of a complex
fraction from the
denominator.
Algebra
a
b
}
c
}
d
}
a
a d
5}
4 }c 5 }
p}
b
d
b
c
x
}
x
x
x 3
3x
3
2
Example }
x 5}4}5}p}
x5}5}
2
}
3
EXAMPLE 1
3
2
2x
2
Simplify a complex fraction
Simplify the complex fraction.
3x
2
a. }
3
}
26x
3x
5}
4 (26x3)
2
Write fraction as quotient.
1
3x
5}
p}
3
Multiply by multiplicative inverse.
2
26x
3x
212x
5 }3
Multiply numerators and denominators.
1
4x
5 2}2
Simplify.
2
11
21
b. x}
5 (x2 2 1) 4 x}
x21
x11
}
x21
x21
x11
5 (x2 2 1) p }
Multiply by multiplicative inverse.
(x2 2 1)(x 2 1)
x11
Multiply numerators and denominators.
5 }}}}}}}}}
(x 1 1)(x 2 1)(x 2 1)
x11
Factor and divide out common factor.
5 (x 2 1)2
Simplify.
5 }}}}}}
810
Write fraction as quotient.
Chapter 12 Rational Equations and Functions
EXAMPLE 2
Simplify
Simplify a complex fraction
2x2 2 8x
x 1 4x 1 4
.
}}}}}
x3 2 16x
}
x12
}
2
2x2 2 8x
x 1 4x 1 4
}
x3 2 16x
}
x12
}
2
x3 2 16x
2x2 2 8x
5 }}}}}
4}
2
x12
x 1 4x 1 4
Write fraction as quotient.
2x2 2 8x
x12
5 }}}}}
p}
2
3
Multiply by multiplicative inverse.
x 1 4x 1 4
x 2 16x
(x 1 4x 1 4)(x 2 16x)
Multiply numerators and
denominators.
5 }}}
2x(x 2 4)(x 1 2)
(x 1 2)(x 1 2)x(x 1 4)(x 2 4)
Factor and divide out common
factors.
2
5 }}
Simplify.
(2x2 2 8x)(x 1 2)
5 }}}}}}}}}}
2
3
(x 1 2)(x 1 4)
PRACTICE
EXAMPLES
1 and 2
on pp. 810–811
for Exs. 1–9
Simplify the complex fraction.
22
11x
2. }
4
29x5
7
1. }
2
x2 1 7x
}
2x 2 6
3. }
2
}4
}
212x
x 2 49
18x
4
2
2
1 5x 2 3
6. 2x
}}}}}}
2
224x
4. }
2
1 4x
5. x}}}
x2 2 x 2 20
}
4
7. }}
x25
}
10
x2 2 2x 2 8
}
6x 2 3x2
8. }}
x3 1 4x2
}
x2 2 4
x14
x 2x
8x
24x
x 1 4x 1 3
}
15x
}
2
}3
9.
2x 2 1 5x 2 3
}}
3x2 1 4x 1 1
}}
10x 2 2 5x
}
2x3 2 2x
GEOMETRY Write a rational expression for the ratio of the surface area S
of the given solid to its volume V.
10. Sphere
11. Cone
12. Pyramid with a
square base
l
r
l
h
r
S 5 4πr 2
3
4πr
V5}
3
13. Are the complex fractions
S 5 πr 2 1 πrl
2
πr h
V5}
3
a
}
b and
}
c
h
s
s
S 5 s2 1 2sl
2
sh
V5}
3
a
}
b equivalent? Explain your answer.
}
c
Extension: Simplify Complex Fractions
811