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F. Solving Applications of Basic Geometry

F. Solving Applications of Basic Geometry

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Section R.3 Solving Linear Equations and Inequalities



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







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







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 2␲r

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



2

A ϭ 15 in.2 142 in.2

A ϭ 210 in2





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







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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F. Solving Applications of Basic Geometry

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