Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models
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50 CHAPTER R A Review of Basic Concepts and Skills R–50
College Algebra Graphs & Models—
Perfect Square Trinomials
19. a. b.
c. d.
20. a. b.
c. d.
Sum/Difference of Perfect Cubes
21. a. b.
c. d.
22. a. b.
c. d.
u-Substitution
23. a. b.
c. x
6
⫺ 7x
3
⫺ 8
x
4
⫹ 13x
2
⫹ 36x
4
⫺ 10x
2
⫹ 9
3r
4
⫺ 24rb
3
⫺ 0.125
n
3
⫹
8
27
27q
3
⫺ 125
⫺2t
4
⫹ 54tg
3
⫺ 0.027
m
3
⫹
1
8
8p
3
⫺ 27
16q
2
⫹ 40q ⫹ 2525p
2
⫺ 60p ⫹ 36
z
2
⫺ 18z ⫹ 81x
2
⫹ 12x ⫹ 36
9n
2
⫺ 42n ⫹ 494m
2
⫺ 20m ⫹ 25
b
2
⫹ 10b ⫹ 25a
2
⫺ 6a ⫹ 9
24. a.
b.
c.
25. Completely factor each of the following (recall that
“1” is its own perfect square and perfect cube).
a. b.
c. d.
26. Carefully factor each of the following trinomials, if
possible. Note differences and similarities.
a. b.
c. d.
e. f.
Factor each expression completely, if possible. Rewrite
the expression in standard form (factor out “ ” if
needed) and factor out the GCF if one exists. If you
believe the expression will not factor, write “prime.”
27. 28.
29. 30.
31. 32.
33. 34.
35. 36.
37. 38.
39. 40.
41. 42.
43. 44.
45. 46. 27r
3
⫹ 648x
3
⫺ 125
b
2
⫺ 9b ⫺ 36a
2
⫺ 7a ⫺ 60
⫺30n ⫺ 4n
2
⫹ 2n
3
12m
2
⫺ 40m ⫹ 4m
3
⫺28z
3
⫹ 16z
2
⫹ 80z⫺6x
3
⫹ 39x
2
⫺ 63x
4p
2
⫺ 20p ⫹ 259k
2
⫺ 24k ⫹ 16
3k
2
⫹ 10k ⫹ 82h
2
⫹ 7h ⫹ 6
28 ⫹ s
2
⫺ 11s⫺9r ⫹ r
2
⫹ 18
25 ⫺ 16n
2
64 ⫺ 9m
2
10z
2
⫺ 140z ⫹ 4502x
2
⫺ 24x ⫹ 40
b
2
⫹ 9b ⫹ 20a
2
⫹ 7a ⫹ 10
⫺1
x
2
⫹ 5x ⫺ 6x
2
⫺ 5x ⫹ 6
x
2
⫺ x ⫺ 6x
2
⫹ x ⫹ 6
x
2
⫹ x ⫺ 6x
2
⫺ x ⫹ 6
28x
3
⫺ 7xn
3
⫹ 1
n
3
⫺ 1n
2
⫺ 1
21z ⫹ 32
2
⫹ 31z ⫹ 32⫺ 54
31n ⫹ 52
2
⫹ 21n ⫹ 52⫺ 21
x
6
⫺ 26x
3
⫺ 27
䊳
DEVELOPING YOUR SKILLS
Factor each expression using the method indicated.
Greatest Common Factor
7. a. b.
c.
8. a. b.
c.
Common Binomial Factor
9. a.
b.
c.
10. a.
b.
c.
Grouping
11. a.
b.
c.
12. a.
b.
c.
Trinomial Factoring where
13. a. b.
c.
14. a. b.
c.
Trinomial Factoring where
15. a. b.
c.
16. a. b.
c.
Difference of Perfect Squares
17. a. b.
c. d.
e.
18. a. b.
c. d.
e. x
2
⫺ 17
16z
4
⫺ 81v
4
⫺ 1
25w
2
⫺
1
49
9v
2
⫺
1
25
b
2
⫺ 5
121h
2
⫺ 14450x
2
⫺ 72
9x
2
⫺ 494s
2
⫺ 25
15z
2
⫺ 22z ⫺ 48
20x
2
⫹ 53x ⫹ 186v
2
⫹ v ⫺ 35
10u
2
⫺ 19u ⫺ 15
4q
2
⫹ 7q ⫺ 153p
2
⫺ 13p ⫺ 10
a ⴝ 1
v
2
⫹ 10v ⫹ 15
x
2
⫹ 12 ⫹ 13x⫺m
2
⫹ 13m ⫺ 42
n
2
⫹ 20 ⫺ 9n
q
2
⫺ 4q ⫹ 12⫺p
2
⫹ 5p ⫹ 14
円a円 ⴝ 1
3x
2
⫺ xy ⫺ 6x ⫹ 2y
4k
3
⫹ 6k
2
⫺ 2k ⫺ 3
6h
3
⫺ 9h
2
⫺ 2h ⫹ 3
k
5
⫺ 7k
3
⫺ 5k
2
⫹ 35
h
5
⫺ 12h
4
⫺ 3h ⫹ 36
9q
3
⫹ 6q
2
⫹ 15q ⫹ 10
3p1q
2
⫹ 52⫹ 71q
2
⫹ 52
1v ⫺ 522v ⫹ 1v ⫺ 523
5x1x ⫺ 32⫺ 21x ⫺ 32
4m1n ⫹ 72⫺ 111n ⫹ 72
1b
2
⫹ 323b ⫹ 1b
2
⫹ 322
2a1a ⫹ 22⫹ 31a ⫹ 22
⫺6g
5
⫹ 12g
4
⫺ 9g
3
9p
2
⫹ 27p
3
⫺ 18p
4
⫺13n
2
⫺ 52
⫺3a
4
⫹ 9a
2
⫺ 6a
3
21b
3
⫺ 14b
2
⫹ 56b⫺17x
2
⫹ 51
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52. Match each polynomial to its factored form. Two
of them are prime.
a.
b.
c.
d.
e.
f.
g.
h.
i.
A.
B. C.
D. E. prime trinomial
F. prime binomial G.
H.
I.
Determine whether each equation is quadratic. If so,
identify the coefficients a, b, and c. If not, discuss why.
53. 54.
55. 56.
57. 58.
59. 60.
61. ⫺3x
2
⫹ 9x ⫺ 5 ⫹ 2x
3
⫽ 0
5 ⫽⫺4x
2
2x
2
⫹ 7 ⫽ 0
0.5x ⫽ 0.25x
2
1
4
x
2
⫽ 6x
12 ⫺ 4x ⫽ 9
2
3
x ⫺ 7 ⫽ 0
21 ⫹ x
2
⫺ 4x ⫽ 02x ⫺ 15 ⫺ x
2
⫽ 0
1x ⫺ 521x ⫹ 22
12x ⫹ 3214x
2
⫺ 6x ⫹ 92
12x ⫹ 321x ⫺ 12
12x ⫺ 72
2
12x ⫹ 3212x ⫺ 3212x ⫺ 321x ⫹ 12
1x ⫺ 521x
2
⫹ 5x ⫹ 252
x
2
⫹ 25
2x
2
⫹ x ⫺ 3
2x
2
⫺ x ⫺ 3
x
2
⫹ 3x ⫹ 10
x
2
⫺ 3x ⫺ 10
8x
3
⫹ 27
x
3
⫺ 125
4x
2
⫺ 28x ⫹ 49
4x
2
⫺ 9
R–51 Section R.4 Factoring Polynomials and Solving Polynomial Equations by Factoring 51
College Algebra Graphs & Models—
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
93.
94.
95.
96.
97.
98.
99.
100. 1d
2
⫺ d2
2
⫺ 81d
2
⫺ d2⫹ 12 ⫽ 0
1b
2
⫺ 3b2
2
⫺ 141b
2
⫺ 3b2⫹ 40 ⫽ 0
z
4
⫺ 20z
2
⫹ 64 ⫽ 0
x
4
⫺ 29x
2
⫹ 100 ⫽ 0
3c
2
⫹ c ⫽ c
3
⫹ 3
x
3
⫺ 25x ⫽ 2x
2
⫺ 50
q
3
⫺ 4q ⫹ 24 ⫽ 6q
2
p
3
⫹ 7p
2
⫺ 63 ⫽ 9p
7x
2
⫹ 15x ⫽ 2x
3
3x
3
⫽⫺7x
2
⫹ 6x
x
3
⫽ 13x
2
⫺ 42x
22x ⫽ x
3
⫺ 9x
2
⫺3v
2
⫽⫺v ⫺ 2
2w
2
⫺ 5w ⫽ 3
⫺3z
2
⫹ 12z ⫹ 36 ⫽ 0
2x
2
⫺ 4x ⫺ 30 ⫽ 0
1g ⫹ 1721g ⫺ 22⫽ 20
1t ⫹ 421t ⫹ 72⫽ 54
7 ⫹ 1s ⫺ 42s ⫽ 28
9 ⫹ 1r ⫺ 52r ⫽ 33
1d ⫺ 102d ⫹ 10 ⫽⫺6
1c ⫺ 122c ⫺ 15 ⫽ 30
62.
63.
64.
Solve using the zero factor property. Be sure each
equation is in standard form and factor out any
common factors before attempting to solve. Check all
answers in the original equation.
65. 66.
67. 68.
69. 70.
71. 72.
73. 74.
75.
76.
77.
78. n
3
⫺ 3n
2
⫺ 4n ⫹ 12 ⫽ 0
m
3
⫹ 5m
2
⫺ 9m ⫺ 45 ⫽ 0
h
2
⫹ 14h ⫺ 2 ⫽⫺51
g
2
⫹ 18g ⫹ 70 ⫽⫺11
b
2
⫹ 8 ⫽ 12a
2
⫺ 17 ⫽⫺8
9w ⫽⫺6w
2
⫺14h
2
⫽ 7h
6q
2
⫺ 18q ⫽ 05p
2
⫺ 10p ⫽ 0
⫺10n ⫽ n
2
⫹ 25m
2
⫽ 8m ⫺ 16
z
2
⫺ 10z ⫽⫺21x
2
⫺ 15 ⫽ 2x
1x ⫹ 52
2
⫺ 1x ⫹ 52⫹ 4 ⫽ 17
1x ⫺ 12
2
⫹ 1x ⫺ 12⫹ 4 ⫽ 9
z
2
⫺ 6z ⫹ 9 ⫺ z
3
⫽ 047. 48.
49. 50.
51. Match each expression with the description that fits best.
a. prime polynomial
b. standard trinomial
c. perfect square trinomial
d. difference of cubes
e. binomial square
f. sum of cubes
g. binomial conjugates
h. difference of squares
i. standard trinomial
A. B.
C. D.
E. F.
G. H.
I. and 1x ⫹ 721x ⫺ 72
x
2
⫹ 92x
2
⫺ x ⫺ 3
8s
3
⫺ 125t
3
x
2
⫺ 3x ⫺ 10
x
2
⫺ 144x
2
⫺ 10x ⫹ 25
1x ⫹ 32
2
x
3
⫹ 27
a ⫽ 1
a ⫽ 1
x
3
⫹ 3x
2
⫺ 4x ⫺ 12x
3
⫺ 5x
2
⫺ 9x ⫹ 45
n
2
⫺ 14n ⫺ 36m
2
⫹ 9m ⫺ 24
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52 CHAPTER R A Review of Basic Concepts and Skills R–52
College Algebra Graphs & Models—
102. Volume of a cylindrical shell:
The volume of a cylindrical shell (a
larger cylinder with a smaller cylinder
removed) can be found using the
formula shown, where R is the radius of
the larger cylinder and r is the radius of
the smaller. Factor the expression
completely and use the result to find the
volume of a shell where
, and . Answer in exact
form and in approximate form rounded to the nearest
whole number.
h ⫽ 10 cmR ⫽ 9 cm, r ⫽ 3 cm
R
2
h ⴚ r
2
h
䊳
WORKING WITH FORMULAS
101. Surface area of a cylinder:
The surface area of a cylinder is given by the formula
shown, where h is the height of the cylinder and r is the
radius. Factor out the GCF and use the result to find the
surface area of a cylinder where cm and cm.
Answer in exact form and in approximate form rounded to
the nearest whole number.
h ⫽ 65r ⫽ 35
2r
2
ⴙ 2rh
right-hand side to determine (a) how many more or
fewer books fit the width of the box (than the
height), (b) how many more or fewer books fit the
length of the box (than the height), and (c) the
number of books shipped per box if they are
stacked 10 high in the box.
107. Space-Time relationships: Due to the work of
Albert Einstein and other physicists who labored
on space-time relationships, it is known that the
faster an object moves the shorter it appears to
become. This phenomenon is modeled by the
Lorentz transformation
where is the length of the object at rest, L is the
relative length when the object is moving at velocity
v, and c is the speed of light. Factor the radicand and
use the result to determine the relative length of a
12-in. ruler if it is shot past a stationary observer at
0.75 times the speed of light .
108. Tubular fluid flow: As a fluid flows through a
tube, it is flowing faster at the center of the tube
than at the sides, where the tube exerts a backward
drag. Poiseuille’s law gives the velocity of the flow
at any point of the cross section: ,
where R is the inner radius of the tube, r is the
distance from the center of the tube to a point in
the flow, G represents what is called the pressure
gradient, and is a constant that depends on the
viscosity of the fluid. Factor the right-hand side
and find v given
and
⫽ 0.25.
G ⫽ 15,r ⫽ 0.3 cm,R ⫽ 0.5 cm,
v ⫽
G
4
1R
2
⫺ r
2
2
1v ⫽ 0.75c2
L
0
L ⫽ L
0
B
1 ⫺ a
v
c
b
2
,
䊳
APPLICATIONS
103. Conical shells: The volume of a conical shell (like
the shell of an ice cream cone) is given by the
formula where R is the outer
radius and r is the inner radius of the cone. Write
the formula in completely factored form, then find
the volume of a shell when
and Answer in exact form
and in approximate form rounded to the nearest
tenth.
104. Spherical shells: The volume of
a spherical shell (like the outer
shell of a cherry cordial) is given
by the formula
where R is the outer radius and r
is the inner radius of the shell.
Write the right-hand side in
completely factored form, then find the volume of
a shell where and Answer
in exact form and in approximate form rounded to
the nearest tenth.
105. Volume of a box: The volume of a rectangular box
x inches in height is given by the relationship
Factor the right-hand side to
determine: (a) The number of inches that the width
exceeds the height, (b) the number of inches the
length exceeds the height, and (c) the volume given
the height is 2 ft.
106. Shipping textbooks: A publisher ships paperback
books stacked x copies high in a box. The total
number of books shipped per box is given by the
relationship Factor the
B ⫽ x
3
⫺ 13x
2
⫹ 42x.
V ⫽ x
3
⫹ 8x
2
⫹ 15x.
r ⫽ 1.5 cm.R ⫽ 1.8 cm
V ⫽
4
3
R
3
⫺
4
3
r
3
,
h ⫽ 9 cm.r ⫽ 4.9 cm,
R ⫽ 5.1 cm,
V ⫽
1
3
R
2
h ⫺
1
3
r
2
h,
r
R
r
R
In many cases, factoring an expression can make it easier to evaluate as in the following applications.
cob19545_chR_040-053.qxd 11/23/10 7:31 PM Page 52
Solve by factoring.
109. Envelope sizes: Large mailing envelopes often come
in standard sizes, with 5- by 7-in. and 9- by 12-in.
envelopes being the most common. The next larger
size envelope has an area of 143 in
2
, with a length
that is 2 in. longer than the width. What are the
dimensions of the larger envelope?
110. Paper sizes: Letter size paper is 8.5 in. by 11 in.
Legal size paper is in. by 14 in. The next larger
(common) size of paper has an area of 187 in
2
, with
8
1
2
a length that is 6 in. longer than the width. What are
the dimensions of the Ledger size paper?
Letter Legal
Ledger
R–53 Section R.5 Rational Expressions and Equations 53
College Algebra Graphs & Models—
Then we group the inner
terms with x and factor again:
The expression can now be evaluated using any input
and the order of operations. If we quickly
find that Use this method to evaluate
for
Factor each expression completely.
115. 116.
117. 118.
119. 120. a
4
⫺ 18a
2
⫹ 32q
4
⫺ 28q
2
⫹ 75
m
6
⫺ 64p
6
⫺ 1
16n
4
⫺ 1x
4
⫺ 81
x ⫽⫺3.H ⫽ x
3
⫹ 2x
2
⫹ 5x ⫺ 9
P ⫽ 27.
x ⫽ 2,
P ⫽ x3x
2
⫹ 3x ⫹ 14⫹ 5 ⫽ x3x1x ⫹ 32⫹ 14⫹ 5.
x3x
2
⫹ 3x ⫹ 14⫹ 5.
䊳
EXTENDING THE CONCEPT
111. Factor out a constant that leaves integer
coefficients for each term:
a.
b.
112. If is substituted into the result
is zero. What is the value of h?
113. Factor the expression:
114. As an alternative to evaluating polynomials by direct
substitution, nested factoring can be used. The
method has the advantage of using only products
and sums—no powers. For
we begin by grouping all variable terms
and factoring x: P ⫽ 3x
3
⫹ 3x
2
⫹ 1x4⫹ 5 ⫽
1x ⫹ 5,
P ⫽ x
3
⫹ 3x
2
⫹
192x
3
⫺ 164x
2
⫺ 270x.
2x
3
⫹ hx ⫹ 8,x ⫽ 2
2
3
b
5
⫺
1
6
b
3
⫹
4
9
b
2
⫺ 1
1
2
x
4
⫹
1
8
x
3
⫺
3
4
x
2
⫹ 4
LEARNING OBJECTIVES
In Section R.5 you will review how to:
A. Write a rational
expression in simplest
form
B. Multiply and divide
rational expressions
C. Add and subtract rational
expressions
D. Simplify compound
fractions
E. Solve rational equations
A rational number is one that can be written as the quotient of two integers. Similarly,
a rational expression is one that can be written as the quotient of two polynomials. We
can apply the skills developed in a study of fractions (how to reduce, add, subtract,
multiply, and divide) to rational expressions, sometimes called algebraic fractions.
A. Writing a Rational Expression in Simplest Form
A rational expression is in simplest form when the numerator and denominator have
no common factors (other than 1). After factoring the numerator and denominator, we
apply the fundamental property of rational expressions.
Fundamental Property of Rational Expressions
If P, Q, and R are polynomials, with Q,
112
P
#
R
Q
#
R
⫽
P
Q
and
122
P
Q
⫽
P
#
R
Q
#
R
R ⫽ 0,
R.5 Rational Expressions and Equations
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54 CHAPTER R A Review of Basic Concepts and Skills R–54
College Algebra Graphs & Models—
In words, the property says (1) a rational expression can be simplified by cancel-
ing common factors in the numerator and denominator, and (2) an equivalent expres-
sion can be formed by multiplying numerator and denominator by the same nonzero
polynomial.
EXAMPLE 1
Simplifying a Rational Expression
Write the expression in simplest form:
Solution
factor numerator and denominator
common factors reduce to 1
simplest form
Now try Exercises 7 through 10
When simplifying rational expressions, we sometimes encounter expressions of
the form If we factor from the numerator, we see that
⫺11b ⫺ a2
b ⫺ a
⫽⫺1.
a ⫺ b
b ⫺ a
⫽⫺1
a ⫺ b
b ⫺ a
.
⫽
x ⫹ 1
x ⫺ 2
⫽
1x
1
⫺ 121x ⫹ 12
1x ⫺ 121x ⫺ 22
x
2
⫺ 1
x
2
⫺ 3x ⫹ 2
⫽
1x ⫺ 121x ⫹ 12
1x ⫺ 121x ⫺ 22
x
2
⫺ 1
x
2
⫺ 3x ⫹ 2
.
WORTHY OF NOTE
If we view a and b as two points on
the number line, we note that they
are the same distance apart,
regardless of the order they are
subtracted. This tells us the
numerator and denominator will
have the same absolute value but
be opposite in sign, giving a value of
(check using a few test values).⫺1
CAUTION
When reducing rational numbers or expressions, only common factors can be reduced.
It is incorrect to reduce (or divide out) individual terms: and
(except for )x ⫽ 0
x ⫹ 1
x ⫹ 2
⫽
1
2
⫺6 ⫹ 423
2
⫽ ⫺3 ⫹ 423,
Note that after simplifying an expression, we
are actually saying the resulting (simpler)
expression is equivalent to the original expres-
sion for all values where both are defined. The
first expression is not defined when or
, the second when (since the
denominators would be zero). The calculator
screens shown in Figure R.5 help to illustrate
this fact, and it appears that we would very
much prefer to be working with the simpler
expression!
x ⫽ 2x ⫽ 2
x ⫽ 1
Figure R.5
cob19545_chR_054-064.qxd 7/28/10 2:43 PM Page 54
EXAMPLE 2
Simplifying a Rational Expression
Write the expression in simplest form:
Solution
factor numerator and denominator
reduce:
simplest form
Now try Exercises 11 through 16
B. Multiplication and Division of Rational Expressions
Operations on rational expressions use the factoring skills reviewed earlier, along with
much of what we know about rational numbers.
Multiplying Rational Expressions
Given that P, Q, R, and S are polynomials with Q,
1. Factor all numerators and denominators completely.
2. Reduce common factors.
3.
EXAMPLE 3
Multiplying Rational Expressions
Compute the product:
Solution
factor
reduce:
simplest form
Now try Exercises 17 through 20
To divide fractions, we multiply the first expression by the reciprocal of the
second (we sometimes say, “invert the divisor and multiply”). The quotient of two
rational expressions is computed in the same way.
⫽
⫺21a ⫹ 12
3a13a ⫺ 22
a ⫺ 1
1 ⫺ a
⫽⫺1 ⫽
21a ⫹ 12
3a11 ⫺ a2
#
13a ⫹
1
221a ⫺ 1
1⫺12
2
13a ⫺ 2213a ⫹ 22
2a ⫹ 2
3a ⫺ 3a
2
#
3a
2
⫺ a ⫺ 2
9a
2
⫺ 4
⫽
21a ⫹ 12
3a11 ⫺ a2
#
13a ⫹ 221a ⫺ 12
13a ⫺ 2213a ⫹ 22
2a ⫹ 2
3a ⫺ 3a
2
#
3a
2
⫺ a ⫺ 2
9a
2
⫺ 4
.
Multiply numerator ⫻ numerator and denominator ⫻ denominator.
P
Q
#
R
S
⫽
PR
QS
S ⫽ 0,
⫽
⫺2
x ⫹ 3
3 ⫺ x
x ⫺ 3
⫽⫺1 ⫽
1221⫺12
x ⫹ 3
6 ⫺ 2x
x
2
⫺ 9
⫽
213 ⫺ x2
1x ⫺ 321x ⫹ 32
6 ⫺ 2x
x
2
⫺ 9
.
R–55 Section R.5 Rational Expressions and Equations 55
College Algebra Graphs & Models—
A. You’ve just seen how
we can write a rational
expression in simplest form
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56 CHAPTER R A Review of Basic Concepts and Skills R–56
College Algebra Graphs & Models—
Dividing Rational Expressions
Given that P, Q, R, and S are polynomials with Q, R,
Invert the divisor and multiply.
EXAMPLE 4
Dividing Rational Expressions
Compute the quotient
Solution
invert and multiply
factor
factor and reduce
lowest terms
Note that we sometimes refer to simplest form as lowest terms.
Now try Exercises 21 through 42
⫽
12m ⫺ 321m ⫺ 32
51m ⫺ 72
⫽
m
1
12m ⫺ 3212m ⫺ 32
1
1m ⫹ 721m ⫺ 72
#
1m ⫹
1
721m ⫺ 32
5m12m ⫺ 32
⫽
m14m
2
⫺ 12m ⫹ 92
1m ⫹ 721m ⫺ 72
#
1m ⫹ 721m ⫺ 32
5m12m ⫺ 32
⫽
4m
3
⫺ 12m
2
⫹ 9m
m
2
⫺ 49
#
m
2
⫹ 4m ⫺ 21
10m
2
⫺ 15m
4m
3
⫺ 12m
2
⫹ 9m
m
2
⫺ 49
⫼
10m
2
⫺ 15m
m
2
⫹ 4m ⫺ 21
4m
3
⫺ 12m
2
⫹ 9m
m
2
⫺ 49
⫼
10m
2
⫺ 15m
m
2
⫹ 4m ⫺ 21
.
P
Q
⫼
R
S
⫽
P
Q
#
S
R
⫽
PS
QR
S ⫽ 0,
CAUTION
For products like it is a common mistake to think that all factors
“cancel,” leaving an answer of zero. Actually, all factors reduce to 1, and the result is a
value of 1 for all inputs where the product is defined.
C. Addition and Subtraction of Rational Expressions
Recall that the addition and subtraction of fractions requires finding the lowest com-
mon denominator (LCD) and building equivalent fractions. The sum or difference of
the numerators is then placed over this denominator. The procedure for the addition
and subtraction of rational expressions is very much the same. Note that the LCD can
also be described as the least common multiple (LCM) of all denominators.
Addition and Subtraction of Rational Expressions
1. Find the LCD of all rational expressions.
2. Build equivalent expressions using the LCD.
3. Add or subtract numerators as indicated.
4. Write the result in lowest terms.
1w ⫹ 72
1
1w ⫺ 72
1
1w ⫺ 721w ⫺ 22
#
w ⫺ 2
1
w ⫹ 7
⫽ 1
1w ⫹ 721w ⫺ 72
1w ⫺ 721w ⫺ 22
#
w ⫺ 2
w ⫹ 7
,
B. You’ve just seen how
we can multiply and divide
rational expressions
cob19545_chR_054-064.qxd 7/28/10 2:44 PM Page 56
EXAMPLE 5
Adding and Subtracting Rational Expressions
Compute as indicated:
a. b.
Solution
a. The LCM for 10x and 25x
2
is 50x
2
. find the LCD
write equivalent expressions
simplify
add the numerators and write the result over the LCD
The result is in simplest form.
b. The LCM for and is
find the LCD
write equivalent expressions
distribute
combine like terms
factor and reduce
Now try Exercises 43 through 48
EXAMPLE 6
Adding and Subtracting Rational Expressions
Perform the operations indicated:
a. b.
Solution
a. The LCM for and is
distribute
result
⫽
4n ⫺ 7
1n ⫹ 221n ⫺ 22
⫽
5n ⫺ 10 ⫺ n ⫹ 3
1n ⫹ 221n ⫺ 22
⫽
51n ⫺ 22⫺ 1n ⫺ 32
1n ⫹ 221n ⫺ 22
5
n ⫹ 2
⫺
n ⫺ 3
n
2
⫺ 4
⫽
5
1n ⫹ 22
#
n ⫺ 2
n ⫺ 2
⫺
n ⫺ 3
1n ⫹ 221n ⫺ 22
1n ⫹ 221n ⫺ 22.n
2
⫺ 4n ⫹ 2
b
2
4a
2
⫺
c
a
5
n ⫹ 2
⫺
n ⫺ 3
n
2
⫺ 4
⫽
51x ⫺ 32
1
1x ⫺ 321x ⫹ 32
⫽
5
x ⫹ 3
⫽
5x ⫺ 15
1x ⫺ 321x ⫹ 32
⫽
10x ⫺ 5x ⫺ 15
1x ⫺ 321x ⫹ 32
⫽
10x ⫺ 51x ⫹ 32
1x ⫺ 321x ⫹ 32
10x
x
2
⫺ 9
⫺
5
x ⫺ 3
⫽
10x
1x ⫺ 321x ⫹ 32
⫺
5
x ⫺ 3
#
x ⫹ 3
x ⫹ 3
1x ⫺ 321x ⫹ 32.x ⫺ 3x
2
⫺ 9
⫽
35x ⫹ 6
50x
2
⫽
35x
50x
2
⫹
6
50x
2
7
10x
⫹
3
25x
2
⫽
7
10x
#
5x
5x
⫹
3
25x
2
#
2
2
10x
x
2
⫺ 9
⫺
5
x ⫺ 3
7
10x
⫹
3
25x
2
R–57 Section R.5 Rational Expressions and Equations 57
College Algebra Graphs & Models—
subtract numerators, write
the result over the LCD
write equivalent
expressions
subtract numerators, write
the result over the LCD
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58 CHAPTER R A Review of Basic Concepts and Skills R–58
College Algebra Graphs & Models—
b. The LCM for a and is :
simplify
Now try Exercises 49 through 64
CAUTION
When the second term in a subtraction has a binomial numerator as in Example 6a, be
sure the subtraction is applied to both terms. It is a common error to write
in which the subtraction is applied
to the first term only. This is incorrect!
D. Simplifying Compound Fractions
Rational expressions whose numerator or denominator contain a fraction are called
compound fractions. The expression is a compound fraction with a
numerator of and a denominator of The two methods commonly
used to simplify compound fractions are summarized in the following boxes.
Simplifying Compound Fractions (Method I)
1. Add/subtract fractions in the numerator, writing them as a
single expression.
2. Add/subtract fractions in the denominator, also writing them
as a single expression.
3. Multiply the numerator by the reciprocal of the denominator
and simplify if possible.
Simplifying Compound Fractions (Method II)
1. Find the LCD of all fractions in the numerator and denominator.
2. Multiply the numerator and denominator by this LCD and simplify.
3. Simplify further if possible.
Method II is illustrated in Example 7.
EXAMPLE 7
Simplifying a Compound Fraction
Simplify the compound fraction:
2
3m
⫺
3
2
3
4m
⫺
1
3m
2
3
4m
⫺
1
3m
2
.
2
3m
⫺
3
2
2
3m
⫺
3
2
3
4m
⫺
1
3m
2
51n ⫺ 22
1n ⫹ 221n ⫺ 22
⫺
n ⫺ 3
1n ⫹ 221n ⫺ 22
⫽
5n ⫺ 10 ⫺ n ⫺ 3
1n ⫹ 221n ⫺ 22
⫽
b
2
⫺ 4ac
4a
2
⫽
b
2
4a
2
⫺
4ac
4a
2
b
2
4a
2
⫺
c
a
⫽
b
2
4a
2
⫺
c
a
#
4a
4a
4a
2
4a
2
write equivalent
expressions
subtract numerators, write
the result over the LCD
C. You’ve just seen how
we can add and subtract
rational expressions
X
cob19545_chR_054-064.qxd 11/22/10 10:27 AM Page 58
Solution
The LCD for all fractions is
distribute
simplify
factor and write in lowest terms
Now try Exercises 65 through 74
E. Solving Rational Equations
In Section R.3 we solved linear equations using basic properties of equality. If any
equation contained fractional terms, we “cleared the fractions” using the least common
denominator (LCD). We can also use this idea to solve rational equations, or equa-
tions that contain rational expressions.
Solving Rational Equations
1. Identify and exclude any values that cause
a zero denominator.
2. Multiply both sides by the LCD and simplify
(this will eliminate all denominators).
3. Solve the resulting equation.
4. Check all solutions in the original equation.
EXAMPLE 8
Solving a Rational Equation
Solve for m:
Solution
Since the LCD is where and
multiply by LCD
distribute and simplify
denominators are eliminated
distribute
solve for
m
m ⫽ 6
2 m ⫺ 2 ⫺ m ⫽ 4
2 1m ⫺ 12⫺ m ⫽ 4
m
1m ⫺ 12
1
a
2
m
b⫺
m1m ⫺ 12
1
a
1
m ⫺ 1
b⫽
m
1m ⫺ 12
1
a
4
m1m ⫺ 12
b
m1m ⫺ 12a
2
m
⫺
1
m ⫺ 1
b⫽ m1m ⫺ 12c
4
m1m ⫺ 12
d
m ⫽ 1.m ⫽ 0m1m ⫺ 12,m
2
⫺ m ⫽ m1m ⫺ 12,
2
m
⫺
1
m ⫺ 1
⫽
4
m
2
⫺ m
.
⫽
2m14 ⫺
⫺1
9m2
9m ⫺ 4
⫽⫺2m
⫽
8m ⫺ 18m
2
9m ⫺ 4
⫽
a
2
3m
ba
12m
2
1
b⫺ a
3
2
ba
12m
2
1
b
a
3
4m
ba
12m
2
1
b⫺ a
1
3m
2
ba
12m
2
1
b
2
3m
⫺
3
2
3
4m
⫺
1
3m
2
⫽
a
2
3m
⫺
3
2
ba
12m
2
1
b
a
3
4m
⫺
1
3m
2
ba
12m
2
1
b
12m
2
.
R–59 Section R.5 Rational Expressions and Equations 59
College Algebra Graphs & Models—
multiply numerator and
denominator by 12m
2
⫽
12m
2
1
D. You’ve just seen how
we can simplify compound
fractions
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