Coburn J.W. Algebra and Trigonometry
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40 CHAPTER R A Review of Basic Concepts and Skills R-40
Sum or Difference of Two Perfect Cubes
Recall that the difference of two perfect squares is factorable, but the sumof two perfect
squares is prime. In contrast, both the sum and difference of two perfect cubes are fac-
torable. For either or we have the following:
1. Each will factor into the product of a binomial ( ) ( )
and a trinomial:
binomial trinomial
2. The terms of the binomial are the quantities
being cubed: (AB) ( )
3. The terms of the trinomial are the square of A,
the product AB, and the square of B, respectively: (AB) (A
2
AB B
2
)
4. The binomial takes the same sign as the original
expression
5. The middle term of the trinomial takes the
opposite sign of the original exercise (the last
term is always positive):
Factoring the Sum or Difference of Two Perfect Cubes:
EXAMPLE 8
Factoring the Sum and Difference of Two Perfect Cubes
Factor completely:
a. b.
Solution
a. write terms as perfect cubes
Use factoring template
and
b.
check for common
write terms as
Use factoring template
and
simplify
factored form
The results for parts (a) and (b) can be checked using multiplication.
Now try Exercises 21 and 22
Using u-Substitution to Factor Quadratic Forms
For any quadratic expression in standard form, the degree of the
leading term is twice the degree of the middle term. Generally, a trinomial is in quad-
ratic form if it can be written as where the parentheses “hold”
the same factors. The equation is in quadratic form since
In many cases, we can factor these expressions using a
placeholder substitution that transforms these expressions into a more recognizable
form. In a study of algebra, the letter “u” often plays this role. If we let u represent
the expression becomes which can be factored
into After “unsubstituting” (replace u with ), we have
1x
2
921x
2
42 1x 321x 321x 221x 22.
x
2
1u 921u 42.
u
2
13u 36,1x
2
2
2
131x
2
2 36
x
2
,
1x
2
2
2
131x
2
2 36 0.
x
4
13x
2
36 0
a1
__
2
2
b1
__
2 c,
ax
2
bx c
1 5m
3
n 40n
4
5n1m 2n21m
2
2mn 4n
2
2.
1m 2n21m
2
2mn 4n
2
2
B S 2nA S m m
3
12n2
3
1m 2n23m
2
m12n2 12n2
2
4
A
3
B
3
1A B21A
2
AB B
2
2
5n3m
3
12n2
3
4
5m
3
n 40n
4
5n1m
3
8n
3
2
B S 5A S x x
3
5
3
1x 521x
2
5x 252
A
3
B
3
1A B21A
2
AB B
2
2
x
3
125 x
3
5
3
5m
3
n 40n
4
x
3
125
A
3
B
3
1A B21A
2
AB B
2
2
A
3
B
3
1A B21A
2
AB B
2
2
A
3
B
3
1A
2
AB B
2
21A B2
1A
2
AB B
2
21A B2
A
3
B
3
A
3
B
3
factors 1GCF 5n2
perfect cubes
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R-41 Section R.4 Factoring Polynomials 41
EXAMPLE 9
Factoring a Quadratic Form
Write in completely factored form:
Solution
Expanding the binomials would produce a fourth-degree polynomial that would
be very difficult to factor. Instead we note the expression is in quadratic form.
Letting u represent (the variable part of the “middle” term),
becomes
factor
To finish up, write the expression in terms of x, substituting for u.
substitute for u
The resulting trinomials can be further factored.
Now try Exercises 23 and 24
It is well known that information is retained longer and used more effectively when
it’s placed in an organized form. The “factoring flowchart” provided in Figure R.5
offers a streamlined and systematic approach to factoring and the concepts involved.
However, with some practice the process tends to “flow” more naturally than follow-
ing a chart, with many of the decisions becoming automatic.
x
2
2x 1 1x 12
2
1x 321x 121x 12
2
x
2
2x 1x
2
2x 321x
2
2x 12
x
2
2x
u
2
2u 3 1u 321u 12
u
2
2u 3.21x
2
2x2 31x
2
2x2
2
x
2
2x
1x
2
2x2
2
21x
2
2x2 3.
• Can an
y
result be factored further? • Pol
y
nomials that cannot be factored are said to be prime.
Two Four
Standard Form:
decreasing order of degree;
positive leading coefficient
Greatest Common Factor
Number of Terms
Factoring
Polynomials
Advanced
methods
(Section 3.2)
Difference
of cubes
Difference
of squares
Factor by
grouping
Sum
of cubes
Trinomials
(a 1)
Trinomials
(a 1)
Three
Figure R.5
D. You’ve just reviewed
how to factor special forms
and quadratic forms
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R.4 EXERCISES
Fill in each blank with the appropriate word or phrase.
Carefully reread the section, if necessary.
1. To factor an expression means to rewrite the
expression as an equivalent .
2. If a polynomial will not factor, it is said to be a(n)
polynomial.
3. The difference of two perfect squares always
factors into the product of a(n) and its
.
4. The expression is said to be a(n)
trinomial, since its
factored form is a perfect (binomial) square.
5. Discuss/Explain why
is not written in completely factored
form, then rewrite it so it is factored completely.
6. Discuss/Explain why is factorable, but
is not. Demonstrate by writing in
factored form, and by exhausting all possibilities
for to show it is prime. x
2
64
x
3
64a
2
b
2
a
3
b
3
12x 62
12x 624x
2
36
x
2
6x 9
42 CHAPTER R A Review of Basic Concepts and Skills R-42
CONCEPTS AND VOCABULARY
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. n
2
20 9n
q
2
4q 45p
2
5p 14
|a| 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
56b17x
2
51
14. a. b.
c.
Trinomial Factoring where a 1
15. a. b.
c.
16. a. b.
c.
Difference of Perfect Squares
17. a. b.
c. d.
e.
18. a. b.
c. d.
e.
Perfect Square Trinomials
19. a. b.
c. d.
20. a. b.
c. d.
Sum/Difference of Perfect Cubes
21. a. b.
c. d. 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
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
v
2
10v 15
x
2
12 13xm
2
13m 42
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22. a. b.
c. d.
u-Substitution
23. a. b.
c.
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 “1” 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.
47. 48.
49. 50. x
3
3x
2
4x 12x
3
5x
2
9x 45
n
2
14n 36m
2
9m 24
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
80z6x
3
39x
2
63x
4p
2
20p 259k
2
24k 16
3k
2
10k 82h
2
7h 6
28 s
2
11s9r r
2
18
25 16n
2
64 9m
2
10z
2
140z 4502x
2
24x 40
b
2
9b 20a
2
7a 10
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
13z 92 54
31n 52
2
12n 102 21
x
6
26x
3
27
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
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
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.
1x 521x 22
12x 3214x
2
6x 92
12x 321x 12
12x 72
2
12x 3212x 32
12x 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
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
R-43 Section R.4 Factoring Polynomials 43
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44 CHAPTER R A Review of Basic Concepts and Skills R-44
WORKING WITH FORMULAS
53. Surface area of a cylinder: 2r
2
2rh
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
54. Volume of a cylindrical shell: R
2
h r
2
h
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
(use
;
3.142.h 10 cm
R 9 cm, r 3 cm
r
R
APPLICATIONS
In many cases, factoring an expression can make it
easier to evaluate as in the following applications.
55. 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.
56. 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
57. 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.
58. 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
right-hand side to determine (a) how many more
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,
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.
59. 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
.
60. 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
,
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R-45 Section R.5 Rational Expressions 45
EXTENDING THE CONCEPT
61. Factor out a constant that leaves integer
coefficients for each term:
a.
b.
62. If is substituted into the result
is zero. What is the value of h?
63. Factor the expression:
64. 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 terms1x 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
and factoring
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.
65. 66.
67. 68.
69. 70. a
4
18a
2
32q
4
28q
2
75
m
6
64p
6
1
16n
4
1x
4
81
x 3.
5x 9H x
3
2x
2
P 27.x 2,
5 x3x1x 32 14 5.
P x3x
2
3x 14
x3x
2
3x 14 5.
x:P 3x
3
3x
2
1x4 5
R.5 Rational Expressions
A rational number is one that can be written as the quotient of two integers. Simi-
larly, a rational expression is one that can be written as the quotient of two polyno-
mials. 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 denom-
inator, we apply the fundamental property of rational expressions.
Fundamental Property of Rational Expressions
If P, Q, and R are polynomials, with Q,
(1) and (2)
In words, the property says (1) a rational expression can be simplified by
canceling common factors in the numerator and denominator, and (2) an equivalent
expression can be formed by multiplying numerator and denominator by the same
nonzero polynomial.
P
Q
P
#
R
Q
#
R
P
#
R
Q
#
R
P
Q
R 0,
Learning Objectives
In Section R.5 you will learn how to:
A. Write a rational expres-
sion in simplest form
B. Multiply and divide
rational expressions
C. Add and subtract
rational expressions
D. Simplify compound
fractions
E. Rewrite formulas and
algebraic models
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46 CHAPTER R A Review of Basic Concepts and Skills R-46
EXAMPLE 1
Simplifying a Rational Expression
Write the expression in simplest form:
Solution
factor numerator and denominator
cancel common factors
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
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)
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. Multiply numerator numerator and denominator denominator.
P
Q
#
R
S
PR
QS
S 0,
2
x 3
13 x2
1x 32
1
122112
x 3
16 2x2
x
2
9
213 x2
1x 321x 32
16 2x2
x
2
9
.
x 1
x 2
1
2
6 423
2
3 423,
11b a2
b a
1.
a b
b a
1
a b
b a
.
x 1
x 2
1x 12
1x 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
A. You’ve just reviewed
how to write a rational
expression in simplest form
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R-47 Section R.5 Rational Expressions 47
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. The quotient of two rational expressions is computed in the same way.
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 72
1
1m 72
#
1m 72
1
1m 32
5m
1
12m 32
1
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,
21a 12
3a13a 22
a 1
1 a
1
21a 12
3a11 a2
1
#
13a
1
221a 1
112
2
13a 2213a 22
1
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
.
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48 CHAPTER R A Review of Basic Concepts and Skills R-48
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
common 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.
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.
EXAMPLE 5
Adding and Subtracting Rational Expressions
Compute as indicated:
a. b.
Solution
a. The LCD for 10x and is find the LCD
write equivalent expressions
simplify
The result is in simplest form.
b. The LCD for and is
find the LCD
write equivalent expressions
distribute
combine like terms
factor and reduce
Now try Exercises 43 through 48
51x 32
1
1x 32
1
1x 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
#
1x 32
1x 32
1x 321x 32.x 3x
2
9
35x 6
50x
2
35x
50x
2
6
50x
2
7
10x
3
25x
2
7
10x
#
15x2
15x2
3
25x
2
#
122
122
50x
2
.25x
2
10x
x
2
9
5
x 3
7
10x
3
25x
2
1w 72
1
1w 72
1
1w 72
1
1w 22
1
#
1w 22
1
1w 72
1
1
1w 721w 72
1w 721w 22
#
1w 22
1w 72
,
add the numerators and write the result
over the LCD
subtract numerators,
write the result over the LCD
B. You’ve just reviewed
how to multiply and divide
rational expressions
College Algebra—
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R-49 Section R.5 Rational Expressions 49
EXAMPLE 6
Adding and Subtracting Rational Expressions
Perform the operations indicated:
a. b.
Solution
a. The LCD for and is
distribute
result
b. The LCD for a and is :
simplify
Now try Exercises 49 through 64
b
2
4ac
4a
2
b
2
4a
2
4ac
4a
2
b
2
4a
2
c
a
b
2
4a
2
c
a
#
14a2
14a2
4a
2
4a
2
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
#
1n 22
1n 22
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
write equivalent
expressions
write equivalent
expressions
subtract numerators, write
the result over the LCD
subtract numerators,
write the result over the
LCD
C. You’ve just reviewed
how to add and subtract
rational expressions
College Algebra—
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.
3
4m
1
3m
2
.
2
3m
3
2
2
3m
3
2
3
4m
1
3m
2
CAUTION
When the second term in a subtraction has a binomial numerator as in Example 6(a), 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!
51n 22
1n 221n 22
n 3
1n 221n 22
5n 10 n 3
1n 221n 22
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.
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