F. Solving Applications of Basic Geometry
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Table R.2A
Perimeter Formula
(linear units or units)
Definition and Diagram
Area Formula
(square units or units2)
a three-sided polygon
s1
Triangle
s2
P ϭ s1 ϩ s2 ϩ s3
h
s3
Aϭ
1
bh
2
b
a quadrilateral with four right angles
and opposite sides parallel
Rectangle
W
P ϭ 2L ϩ 2W
A ϭ LW
P ϭ 4S
A ϭ S2
L
a rectangle with four equal sides
Square
S
a quadrilateral with one pair of parallel sides
(called bases b1 and b2)
b1
s2
Trapezoid
s1
s3
s4
Circle
sum of all sides
P ϭ s1 ϩ s2 ϩ s3 ϩ s4
h
Aϭ
h
1b1 ϩ b2 2
2
b2
the set of all points lying in a plane
that are an equal distance
(called the radius r) from a
given point (called the center C).
r
C
C ϭ 2r
or
C ϭ d
A ϭ r2
If an exercise or application uses a formula, begin by stating the formula first.
Using the formula as a template for the values substituted will help to prevent many
careless errors.
EXAMPLE 12A
ᮣ
Computing the Area of a Trapezoidal Window
A basement window is shaped like an isosceles
trapezoid (base angles equal, nonparallel sides
equal in length), with a height of 10 in. and
bases of 1.5 ft and 2 ft. What is the area of the
glass in the window?
1.5 ft
10 in.
2 ft
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Solution
ᮣ
Before applying the area formula, all measures must use the same unit. In inches,
we have 1.5 ft ϭ 18 in. and 2 ft ϭ 24 in.
h
1b1 ϩ b2 2
2
10 in.
118 in. ϩ 24 in.2
Aϭ
2
A ϭ 15 in.2 142 in.2
A ϭ 210 in2
Aϭ
given formula
substitute 10 for h, 18 for b1, and 24 for b2
simplify
result
The area of the glass in the window is 210 in2.
Now try Exercises 101 and 102
ᮣ
Volume
Volume is a measure of the amount of space occupied by a three dimensional object and
is measured in cubic units. Some of the more common formulas are given in Table R.2B.
Table R.2B
Volume Formula
(cubic units or units3)
Definition and Diagram
Rectangular
solid
Cube
a six-sided, solid
figure with opposite
faces congruent and
adjacent faces meeting
at right angles
H
V ϭ LWH
W
L
a rectangular solid
with six congruent,
square faces
V ϭ S3
S
Sphere
the set of all points in space,
an equal distance (called the
radius) from a given point
(called the center)
Right circular
cylinder
union of all line segments
connecting two congruent
circles in parallel planes,
meeting each at a right angle
Right circular
cone
Right
pyramid
union of all line segments
connecting a given point
(vertex) to a given circle
(base) and whose altitude
meets the center of the base
at a right angle
union of all line segments
connecting a given point
(vertex) to a given square
(base) and whose altitude
meets the center of the base
at a right angle
4
V ϭ r3
3
r
C
h
V ϭ r2h
r
1
V ϭ r2h
3
h
r
1
V ϭ s2h
3
h
s
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Section R.3 Solving Linear Equations and Inequalities
EXAMPLE 12B
ᮣ
Computing the Volume of a Composite Figure
Sand at a cement factory is being dumped from a conveyor
belt into a pile shaped like a right circular cone atop a
right circular cylinder (see figure). How many cubic
feet of sand are there at the moment the cone is 6 ft high
with a diameter of 10 ft?
Solution
ᮣ
F. You’ve just seen how
we can solve applications
of basic geometry
Total Volume ϭ volume of cylinder ϩ volume of cone
1
V ϭ r2h1 ϩ r2h2
3
1
ϭ 152 2 132 ϩ 152 2 162
3
ϭ 75 ϩ 50
ϭ 125
6 ft
3 ft
10 ft
verbal model
formula model (note h1
h2 2
substitute 5 for r, 3 for h1, and 6 for h2
simplify
result (exact form)
There are about 392.7 ft3 of sand in the pile.
Now try Exercises 103 and 104
R.3 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section, if necessary.
ᮣ
of sets A and B is written A ʝ B.
of sets A and B is written A ´ B.
1. A(n)
is an equation that is always true,
regardless of the
value while a(n)
is an equation that is always false,
regardless of the
value.
4. The
The
2. For inequalities, the three ways of writing a
solution set are
notation, a number line
graph, and
notation.
5. Discuss/Explain the similarities and differences
between the properties of equality for equations
and those for inequalities.
3. The mathematical sentence 3x ϩ 5 6 7 is a(n)
inequality, while Ϫ2 6 3x ϩ 5 6 7 is
a(n)
inequality.
6. Discuss/Explain the use of the words “and” and
“or” in the statement of compound inequalities.
Include a few examples to illustrate.
DEVELOPING YOUR SKILLS
Solve each equation. Check your answer by substitution.
7. 4x ϩ 31x Ϫ 22 ϭ 18 Ϫ x
8. 15 Ϫ 2x ϭ Ϫ41x ϩ 12 ϩ 9
9. 21 Ϫ 12v ϩ 172 ϭ Ϫ7 Ϫ 3v
10. Ϫ12 Ϫ 5w ϭ Ϫ9 Ϫ 16w ϩ 72
11. 8 Ϫ 13b ϩ 52 ϭ Ϫ5 ϩ 21b ϩ 12
12. 2a ϩ 41a Ϫ 12 ϭ 3 Ϫ 12a ϩ 12
Solve each equation.
13. 15 1b ϩ 102 Ϫ 7 ϭ 13 1b Ϫ 92
14. 61 1n Ϫ 122 ϭ 14 1n ϩ 82 Ϫ 2
15. 32 1m ϩ 62 ϭ Ϫ1
2
16. 45 1n Ϫ 102 ϭ
Ϫ8
9
17. 12x ϩ 5 ϭ 13x ϩ 7
19.
xϩ3
x
ϩ ϭ7
5
3
18. Ϫ4 ϩ 23y ϭ 12y Ϫ 5
20.
zϪ4
z
Ϫ2ϭ
6
2
ᮣ
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21. 15 ϭ Ϫ6 Ϫ
3p
8
22. Ϫ15 Ϫ
2q
ϭ Ϫ21
9
Solve each inequality and write the solution in set
notation.
23. 0.2124 Ϫ 7.5a2 Ϫ 6.1 ϭ 4.1
47. 7 Ϫ 21x ϩ 32 Ն 4x Ϫ 61x Ϫ 32
24. 0.4117 Ϫ 4.25b2 Ϫ 3.15 ϭ 4.16
48. Ϫ3 Ϫ 61x Ϫ 52 Յ 217 Ϫ 3x2 ϩ 1
25. 6.2v Ϫ 12.1v Ϫ 52 ϭ 1.1 Ϫ 3.7v
26. 7.9 Ϫ 2.6w ϭ 1.5w Ϫ 19.1 ϩ 2.1w2
n
2
n
ϩ ϭ
2
5
3
m
2
m
28.
Ϫ ϭ
3
5
4
p
p
29. 3p Ϫ Ϫ 5 ϭ Ϫ 2p ϩ 6
4
6
q
q
30. ϩ 1 Ϫ 3q ϭ 2 Ϫ 4q ϩ
6
8
27.
Identify the following equations as an identity, a
contradiction, or a conditional equation, then state the
solution.
31. Ϫ314z ϩ 52 ϭ Ϫ15z Ϫ 20 ϩ 3z
32. 5x Ϫ 9 Ϫ 2 ϭ Ϫ512 Ϫ x2 Ϫ 1
33. 8 Ϫ 813n ϩ 52 ϭ Ϫ5 ϩ 611 ϩ n2
34. 2a ϩ 41a Ϫ 12 ϭ 1 ϩ 312a ϩ 12
35. Ϫ414x ϩ 52 ϭ Ϫ6 Ϫ 218x ϩ 72
36. Ϫ15x Ϫ 32 ϩ 2x ϭ 11 Ϫ 41x ϩ 22
Write the solution set illustrated on each graph in set
notation and interval notation.
37.
38.
39.
40.
[
Ϫ3 Ϫ2 Ϫ1
0
1
0
[
1
2
51. Ϫ61p Ϫ 12 ϩ 2p Յ Ϫ212p Ϫ 32
52. 91w Ϫ 12 Ϫ 3w Ն Ϫ215 Ϫ 3w2 ϩ 1
Determine the intersection and union of sets A, B, C,
and D as indicated, given A ؍5؊3, ؊2, ؊1, 0, 1, 2, 36,
B ؍52, 4, 6, 86, C ؍5؊ 4, ؊2, 0, 2, 46, and
D ؍54, 5, 6, 76 .
53. A ʝ B and A ´ B 54. A ʝ C and A ´ C
55. A ʝ D and A ´ D 56. B ʝ C and B ´ C
57. B ʝ D and B ´ D 58. C ʝ D and C ´ D
Express the compound inequalities graphically and in
interval notation.
59. x 6 Ϫ2 or x 7 1 60. x 6 Ϫ5 or x 7 5
61. x 6 5 and x Ն Ϫ2 62. x Ն Ϫ4 and x 6 3
63. x Ն 3 and x Յ 1
Solve the compound inequalities and graph the
solution set.
65. 41x Ϫ 12 Յ 20 or x ϩ 6 7 9
66. Ϫ31x ϩ 22 7 15 or x Ϫ 3 Յ Ϫ1
69. 35x ϩ 12 7
0
1
2
0
1
2
)
4
Solve the inequality and write the solution in set
notation. Then graph the solution and write it in
interval notation.
41. 5a Ϫ 11 Ն 2a Ϫ 5
42. Ϫ8n ϩ 5 7 Ϫ2n Ϫ 12
43. 21n ϩ 32 Ϫ 4 Յ 5n Ϫ 1
44. Ϫ51x ϩ 22 Ϫ 3 6 3x ϩ 11
3x
x
45.
ϩ 6 Ϫ4
8
4
3
10
and Ϫ4x 7 1
70. 23x Ϫ 56 Յ 0 and Ϫ3x 6 Ϫ2
3
3
64. x Ն Ϫ5 and x Յ Ϫ7
68. Ϫ3x ϩ 5 Յ 17 and 5x Յ 0
3
[
[
Ϫ3 Ϫ2 Ϫ1
50. 8 Ϫ 16 ϩ 5m2 7 Ϫ9m Ϫ 13 Ϫ 4m2
67. Ϫ2x Ϫ 7 Յ 3 and 2x Յ 0
3
)
Ϫ3 Ϫ2 Ϫ1
Ϫ3 Ϫ2 Ϫ1
2
49. 413x Ϫ 52 ϩ 18 6 215x ϩ 12 ϩ 2x
2y
y
46.
ϩ
6 Ϫ2
5
10
71.
x
3x
ϩ 6 Ϫ3 or x ϩ 1 7 Ϫ5
8
4
72.
x
2x
ϩ
6 Ϫ2 or x Ϫ 3 7 2
5
10
73. Ϫ3 Յ 2x ϩ 5 6 7
74. 2 6 3x Ϫ 4 Յ 19
75. Ϫ0.5 Յ 0.3 Ϫ x Յ 1.7
76. Ϫ8.2 6 1.4 Ϫ x 6 Ϫ0.9
77. Ϫ7 6 Ϫ34x Ϫ 1 Յ 11
78. Ϫ21 Յ Ϫ23x ϩ 9 6 7
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ᮣ
Section R.3 Solving Linear Equations and Inequalities
WORKING WITH FORMULAS
79. Euler’s Polyhedron Formula: V ؉ F ؊ E ؍2
Discovered by Leonhard Euler
in 1752, this simple but
powerful formula states that in
any regular polyhedron, the
number of vertices V and faces
F is always two more than the
number of edges E. (a) Verify
the formula for a simple cube. (b) Verify the
formula for the octahedron shown in the figure.
(c) If a dodecahedron has 12 faces and 30 edges,
how many vertices does it have?
80. Area of a Regular Polygon: A ؍
1
ap
2
The area of any regular
polygon can be found
a
using the formula
shown, where a is the
apothem of the polygon
(perpendicular distance
from center to any edge),
and p is the perimeter.
(a) Verify the formula using a square with sides of
length 6 cm. (b) If the hexagon shown has an area
of 259.8 cm2 with sides 10 cm in length, what is
the length a of the apothem?
ᮣ
37
81. Body mass index: B ؍
704W
H2
The U.S. government publishes a body mass index
formula to help people consider the risk of heart
disease. An index “B” of 27 or more means that a
person is at risk. Here W represents weight in
pounds and H represents height in inches. If your
height is 5¿8– what range of weights will help
ensure you remain safe from the risk of heart
disease?
Source: www.surgeongeneral.gov/topics.
82. Lift capacity: 75S ؉ 125B Յ 750
The capacity in pounds of the lift used by a roofing
company to place roofing shingles and buckets of
roofing nails on rooftops is modeled by the formula
shown, where S represents packs of shingles and B
represents buckets of nails. Use the formula to find
(a) the largest number of shingle packs that can be
lifted, (b) the largest number of nail buckets that
can be lifted, and (c) the largest number of shingle
packs that can be lifted along with three nail
buckets.
APPLICATIONS
Write an equation to model the given information and solve.
83. Celebrity Travel: To avoid paparazzi and
overzealous fans, the arrival gates of planes carrying
celebrities are often kept secret until the last possible
moment. While awaiting the arrival of Angelina
Jolie, a large crowd of fans and photographers had
gathered at Terminal A, Gate 18. However, the
number of fans waiting at Gate 32 was twice that
number increased by 5. If there were 73 fans at
Gate 32, how many were waiting at Gate 18? (See
Section R.1, Example 2a.)
84. Famous Architecture: The Hall of Mirrors is the
central gallery of the Palace of Versailles and is one
of the most famous rooms in the world. The length
of this hall is 11 m less than 8 times the width. If
the hall is 73 m long, what is its width? (See
Section R.1, Example 2b.)
85. Dietary Goals: At the picnic, Mike abandoned his
diet and consumed 13 calories more than twice the
number of calories he normally allots for lunch. If
he consumed 1467 calories, how many calories are
normally allotted for lunch?
86. Marathon Training: While training for the
Chicago marathon, Christina’s longest run of the
week was 5 mi less than double the shortest. If the
longest run was 11.2 mi, how long was the
shortest?
87. Actor’s Ages: At the time of this writing, actor
Will Smith (Enemy of the State, Seven Pounds,
others), was 1 yr older than two-thirds the age of
Samuel Jackson (The Negotiator, Die Hard III,
others). If Will Smith was 41 at this time, how old
was Samuel Jackson?
88. Football versus Fútbol: The area of a regulation
field for American football is about 410 square
meters (m2) less than three-fifths of an Olympicsized soccer field. If an American football field
covers 5350 m2, what is the area of an Olympic
soccer field?
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89. Forensic Studies: In forensic studies, skeletal
remains are analyzed to determine the height,
gender, race, age, and other characteristics of the
decedent. For instance, the height of a male
individual is approximated as 34 in. more than
three and one-third times the length of the radial
bone. If a live individual is 74 in. tall, how long is
his radial bone?
90. Famous Waterways: The Suez Canal and the
Panama Canal are two of the most important
waterways in the world, saving ships thousands of
miles as they journey from port to destination. The
length of the Suez Canal is 39 kilometers (km) less
than three times the length of the Panama Canal. If
the Egyptian canal is 192 km long, how long is the
Central American canal?
Write an inequality to model the given information and solve.
91. Exam scores: Jacques is going to college on an
academic scholarship that requires him to maintain at
least a 75% average in all of his classes. So far he has
scored 82%, 76%, 65%, and 71% on four exams.
What scores are possible on his last exam that will
enable him to keep his scholarship?
92. Timed trials: In the first three trials of the 100-m
butterfly, Johann had times of 50.2, 49.8, and
50.9 sec. How fast must he swim the final timed trial
to have an average time of at most 50 sec?
93. Checking account balance: If the average daily
balance in a certain checking account drops below
$1000, the bank charges the customer a $7.50
service fee. The table
Weekday
Balance
gives the daily balance
Monday
$1125
for one customer. What
Tuesday
$850
must the daily balance be
Wednesday
$625
for Friday to avoid a
service charge?
Thursday
$400
94. Average weight: In the
Lineman
Weight
National Football
Left
tackle
318 lb
League, many consider
an offensive line to be
Left guard
322 lb
“small” if the average
Center
326 lb
weight of the five down
Right guard
315 lb
linemen is less than
Right tackle
?
325 lb. Using the table,
what must the weight of the right tackle be so that
the line will not be considered small?
95. Area of a rectangle: Given the rectangle shown,
what is the range of values for the width, in order
to keep the area less than 150m2?
20 m
w
96. Area of a triangle: Using the triangle shown, find
the height that will guarantee an area equal to or
greater than 48 in2.
h
12 in.
97. Heating and cooling subsidies: As long as the
outside temperature is over 45°F and less than 85°F
145 6 F 6 852, the city does not issue heating or
cooling subsidies for low-income families. What is
the corresponding range of Celsius temperatures
C? Recall that F ϭ 95C ϩ 32.
98. U.S. and European shoe sizes: To convert a
European male shoe size “E” to an American male
shoe size “A,” the formula A ϭ 0.76E Ϫ 23 can be
used. Lillian has five sons in the U.S. military, with
shoe sizes ranging from size 9 to size 14
19 Յ A Յ 142. What is the corresponding range of
European sizes? Round to the nearest half-size.
99. Power tool rentals: Sunshine Equipment Co. rents
its power tools for a $20 fee, plus $4.50/hr.
Kealoha’s Rentals offers the same tools for an $11
fee plus $6.00/hr. How many hours h must a tool
be rented to make the cost at Sunshine a better
deal?
100. Moving van rentals: Stringer Truck Rentals will
rent a moving van for $15.75/day plus $0.35 per
mile. Bertz Van Rentals will rent the same van for
$25/day plus $0.30 per mile. How many miles m
must the van be driven to make the cost at Bertz a
better deal?
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101. Cost of drywall: After
the studs are up, the
3 ft
wall shown in the figure
15 ft
must be covered in
10 ft
7 ft
drywall. (a) How many
square feet of drywall
19 ft
are needed? (b) If drywall
is sold only in 4-ft by 8-ft sheets, approximately how
many sheets are required for this job?
5 in.
103. Trophy bases: The
base of a new trophy
7 in.
has the form of a
cylinder sitting atop
a rectangular solid.
2 in.
If the base is to be
10 in.
cast in a special
10 in.
aluminum, determine the volume of aluminum to
be used.
102. Paving a walkway: Current plans
104. Grain storage: The dimensions of
call for building a circular fountain
a grain silo are shown in the figure.
6m
6 m in diameter with a circular
If the maximum storage capacity of
walkway around it that is 1.5 m
the silo is 95% of the total volume
wide. (a) What is the approximate
of the silo, how many cubic meters
area of the walkway? (b) If the
1.5 m
of corn can be stored?
concrete for the walkway is to be 6 cm deep, what
volume of cement must be used 11 cm ϭ 0.01 m2 ?
ᮣ
39
Section R.4 Factoring Polynomials and Solving Polynomial Equations by Factoring
16 m
6m
EXTENDING THE CONCEPT
105. Solve for x: Ϫ314x2 ϩ 5x Ϫ 22 ϩ 7x ϭ
614 Ϫ x Ϫ 2x2 2 Ϫ 19
106. Solve for n: 553 Ϫ 34 Ϫ 215 Ϫ 9n2 4 6 ϩ 15 ϭ
Ϫ655 ϩ 2 3n Ϫ 1019 ϩ n2 4 6
107. Use your local library, the Internet, or another
resource to find the highest and lowest point on each
of the seven continents. Express the range of altitudes
for each continent as a joint inequality. Which
continent has the greatest range?
108. The sum of two consecutive even integers is greater
than or equal to 12 and less than or equal to 22. List
all possible values for the two integers.
R.4
Place the correct inequality symbol in the blank to make
the statement true.
109. If m 7 0 and n 6 0, then mn
0.
110. If m 7 n and p 7 0, then mp
np.
111. If m 6 n and p 7 0, then mp
np.
112. If m Յ n and p 6 0, then mp
np.
113. If m 7 n, then Ϫm
Ϫn.
114. If 0 6 m 6 n, then m1
1
n.
115. If m 7 0 and n 6 0, then m2
116. If m 6 0, then m3
n.
0.
Factoring Polynomials and Solving Polynomial
Equations by Factoring
LEARNING OBJECTIVES
In Section R.4 you will review:
A. Factoring out the greatest
common factor
It is often said that knowing which tool to use is just as important as knowing how to use
the tool. In this section, we review the tools needed to factor an expression, an important
part of solving polynomial equations. This section will also help us decide which factoring tool is appropriate when many different factorable expressions are presented.
B. Common binomial factors
and factoring by grouping
C. Factoring quadratic
polynomials
D. Factoring special forms
and quadratic forms
E. Solving Polynomial
Equations by Factoring
A. The Greatest Common Factor
To factor an expression means to rewrite the expression as an equivalent product. The
distributive property is an example of factoring in action. To factor 2x2 ϩ 6x, we might
first rewrite each term using the common factor 2x: 2x2 ϩ 6x ϭ 2x # x ϩ 2x # 3, then
apply the distributive property to obtain 2x1x ϩ 32. We commonly say that we have
factored out 2x. The greatest common factor (or GCF) is the largest factor common
to all terms in the polynomial.
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EXAMPLE 1
ᮣ
Factoring Polynomials
Factor each polynomial:
a. 12x2 ϩ 18xy Ϫ 30y
Solution
ᮣ
b. x5 ϩ x2
a. 6 is common to all three terms:
12x2 ϩ 18xy Ϫ 30y
ϭ 612x2 ϩ 3xy Ϫ 5y2
mentally: 6 # 2x2 ϩ 6 # 3xy Ϫ 6 # 5y
b. x2 is common to both terms:
A. You’ve just seen how
we can factor out the greatest
common factor
x 5 ϩ x2
ϭ x2 1x3 ϩ 12
mentally: x 2 # x 3 ϩ x 2 # 1
Now try Exercises 7 and 8
ᮣ
B. Common Binomial Factors and Factoring by Grouping
If the terms of a polynomial have a common binomial factor, it can also be factored
out using the distributive property.
EXAMPLE 2
ᮣ
Factoring Out a Common Binomial Factor
Factor:
a. 1x ϩ 32x2 ϩ 1x ϩ 325
Solution
ᮣ
b. x2 1x Ϫ 22 Ϫ 31x Ϫ 22
a. 1x ϩ 32x2 ϩ 1x ϩ 325
ϭ 1x ϩ 32 1x2 ϩ 52
b. x2 1x Ϫ 22 Ϫ 31x Ϫ 22
ϭ 1x Ϫ 22 1x2 Ϫ 32
Now try Exercises 9 and 10
ᮣ
One application of removing a binomial factor involves factoring by grouping.
At first glance, the expression x3 ϩ 2x2 ϩ 3x ϩ 6 appears unfactorable. But by grouping the terms (applying the associative property), we can remove a monomial factor
from each subgroup, which then reveals a common binomial factor.
x3 ϩ 2x2 ϩ 3x ϩ 6 ϭ x2 1x ϩ 22 ϩ 31x ϩ 22
ϭ 1x ϩ 22 1x2 ϩ 32
This grouping of terms must take into account any sign changes and common factors, as seen in Example 3. Also, it will be helpful to note that a general four-term polynomial A ϩ B ϩ C ϩ D is factorable by grouping only if AD ϭ BC.
EXAMPLE 3
ᮣ
Factoring by Grouping
Factor 3t3 ϩ 15t2 Ϫ 6t Ϫ 30.
Solution
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Notice that all four terms have a common factor of 3. Begin by factoring it out.
3t3 ϩ 15t2 Ϫ 6t Ϫ 30
ϭ 31t3 ϩ 5t2 Ϫ 2t Ϫ 102
ϭ 31t3 ϩ 5t2 Ϫ 2t Ϫ 102
ϭ 3 3 t2 1t ϩ 52 Ϫ 21t ϩ 52 4
ϭ 31t ϩ 52 1t2 Ϫ 22
original polynomial
factor out 3
group remaining terms
factor common monomial
factor common binomial
Now try Exercises 11 and 12
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