03 Algebra and Trigonometry 3rd Edition

 

Chapter R  Basic Concepts of Algebra 

R.1

  

The

 

Real

‐

Number

 

System

 

R.2  Integer Exponents, Scientific Notation, and Order of Operations 

R.3  Addition, Subtraction, and Multiplication of Polynomials 

R.4

  

Factoring

 

R.5  Rational Expressions 

R.6  Radical Notation and Rational Exponents 

R.7  The Basics of Equation Solving 

 

Chapter 1  Graphs, Functions, and Models 

1.1  Introduction to Graphing 

1.2  Functions and Graphs 

1.3

  

Linear

 

Functions,

 

Slope,

 

and

 

Applications

 

1.4  Equations of Lines and Modeling 

1.5  More on Functions 

1.6  The Algebra of Functions 

1.7  Symmetry and Transformations 

 

Chapter 2  Functions, Equations, and Inequalities 

2.5  More Equation Solving 

2.6  Solving Linear Inequalities 

 

Chapter 3  Polynomial And Rational Functions 

3.1  Polynomial Functions and Models 

3.2

  

Graphing

 

Polynomial

 

Functions

 

3.3  Polynomial Division; The Remainder and Factor Theorems 

3.4  Theorems about Zeros of Polynomial Functions 

3.5  Rational Functions 

3.6

  

Polynomial

 

and

 

Rational

 

Inequalities

 

3.7  Variation and Applications 

 

Chapter 4  Exponential and Logarithmic Functions 

4.1

  

Inverse

 

Functions

 

4.2  Exponential Functions and Graphs 

4.3  Logarithmic Functions and Graphs 

4.4  Properties of Logarithmic Functions 

4.5  Solving Exponential and Logarithmic Equations 

4.6  Applications and Models:  Growth and Decay, and Compound Interest 

 

5.4  Radians, Arc Length, and Angular Speed 

5.5  Circular Functions:  Graphs and Properties 

5.6

  

Graphs

 

of

 

Transformed

 

Sine

 

and

 

Cosine

 

Functions

 

 

Chapter 6  Trigonometric Identities, Inverse Functions, and Equations 

6.1

  

Identities:

  

Pythagorean

 

and

 

Sum

 

and

 

Difference

 

6.2  Identities:  Cofunction, Double‐Angle, and Half‐Angle 

6.3  Proving Trigonometric Identities 

6.4  Inverses of the Trigonometric Functions 

6.5

  

Solving

 

Trigonometric

 

Equations

 

 

Chapter 7  Applications of Trigonometry 

7.1  The Law of Sines 

7.2

  

The

 

Law

 

of

 

Cosines

 

7.3  Complex Numbers:  Trigonometric Form 

7.4  Polar Coordinates and Graphs 

7.5  Vectors and Applications 

7.6  Vector Operations 

 

Chapter 8  Systems of Equations and Matrices 

8.5  Inverses of Matrices 

8.6  Determinants and Cramer's Rule 

8.7

  

Systems

 

of

 

Inequalities

 

and

 

Linear

 

Programming

 

8.8  Partial Fractions 

 

Chapter

 

9

  

Analytic

 

Geometry

 

Topics

       

9.1  The Parabola 

9.2  The Circle and the Eclipse 

9.3  The Hyperbola 

9.4

  

Nonlinear

 

Systems

 

of

 

Equations

 

and

 

Inequalities

 

9.5  Rotation of Axes 

9.6  Polar Equations of Conics 

9.7  Parametric Equations 

 

Chapter 10  Sequences, Series, and Combinatorics 

10.1  Sequences and Series 

10.2  Arithmetic Sequences and Series 

10.3  Geometric Sequences and Series 

10.4  Mathematical Induction 

10.5  Combinatorics:  Permutations 

R

ina wants to establish a college fund for her newborn daughter that will have accumulated $120,000 at the end of 18 yr. If she can count on an interest rate of 6%, compounded monthly, how much should she deposit each month to accomplish this?

This problem appears as Exercise 95 in Section R.2.

G

Basic Concepts

of Algebra

R.1 The Real-Number System

R.2 _{Integer Exponents, Scientific Notation,} and Order of Operations

R.3 _{Addition, Subtraction, and} Multiplication of Polynomials R.4 _Factoring

R.5 _{Rational Expressions}

R.6 _{Radical Notation and Rational Exponents} R.7 _{The Basics of Equation Solving}

SUMMARY AND REVIEW

TEST

2.1

Polynomial

Functions and

Modeling

R

.1

The Real-Number

System

Identify various kinds of real numbers.

Use interval notation to write a set of numbers. Identify the properties of real numbers.

Find the absolute value of a real number.

Real Numbers

In applications of algebraic concepts, we use real numbers to represent quantities such as distance, time, speed, area, profit, loss, and tempera-ture. Some frequently used sets of real numbers and the relationships among them are shown below.

Real numbers

Rational numbers

Negative integers: −1, −2, −3, …

Natural numbers (positive integers): 1, 2, 3, …

Zero: 0

−, − −, −−, −−, 8.3, 0.56, …

2 3

4 5

19

−5

−7 8

−

Whole numbers: 0, 1, 2, 3, …

Rational numbers that are not integers: Integers:

…, −3, −2, −1, 0, 1, 2, 3, …

Irrational numbers:

−4.030030003…, … √2, p, −√3, √27,

5 4

Numbers that can be expressed in the form , where pand qare in-tegers and , are rational numbers. Decimal notation for rational numbers either terminates (ends) or repeats. Each of the following is a rational number.

a)0 for any nonzero integer a

b)⫺_{7 , or}

c) Terminating decimal

d)⫺5 Repeating decimal

11苷⫺0.45 1

4 苷0.25

7 ⴚ1 ⴚ7ⴝⴚ7

1 0ⴝ 0

a

q苷₀

The real numbers that are not rational are irrational numbers. Decimal notation for irrational numbers neither terminates nor repeats. Each of the following is an irrational number.

a) There is no repeating block of digits.

and 3.14 are rational approximationsof the irrational number

b) There is no repeating block of digits.

c) Although there is a pattern, there is no

repeating block of digits.

The set of all rational numbers combined with the set of all irrational numbers gives us the set of real numbers. The real numbers are modeled

using a number line, as shown below.

Each point on the line represents a real number, and every real number is represented by a point on the line.

The order of the real numbers can be determined from the number line. If a number a is to the left of a number b, then a is less than b

. Similarly,ais greater thanb if ais to the right ofbon the number line. For example, we see from the number line above that , because ⫺_{2.9 is to the left of . Also, , because} is to the right of .

The statement , read “ais less than or equal to b,” is true if either is true or is true.

The symbol 僆is used to indicate that a member, or element, belongs to a set. Thus if we let represent the set of rational numbers, we can see from the diagram on page 2 that . We can also write to indi-cate that is notan element of the set of rational numbers.

When allthe elements of one set are elements of a second set, we say that the first set is a subsetof the second set. The symbol 債is used to denote this. For instance, if we let represent the set of real numbers, we can see from the diagram that (read “ is a subset of ”).

Interval Notation

Sets of real numbers can be expressed using interval notation. For example, for real numbers aand bsuch that , the open interval is the set of real numbers between, but not including,aand b. That is,

.

The points aand bare endpointsof the interval. The parentheses indicate that the endpoints are not included in the interval.

Some intervals extend without bound in one or both directions. The interval , for example, begins at a and extends to the right without bound. That is,

. 关a,

⬁

兲苷兵_x兩_xⱖ_a其 关a,

⬁

兲

共a,b兲苷兵_x兩_a⬍_x⬍_b其

共a,b兲 a ⬍_b

⺢ ⺡

⺡債⺢ ⺢

兹2

兹2僆⺡ 0.56 僆⺡

⺡

a 苷_b a⬍_b

aⱕ_b 兹3

17 4 17

4 ⬎兹3

⫺3

5

⫺_2.9⬍ ⫺3

5

共a⬎_b兲 共a⬍_b兲

⫺_2.9 ⫺E 兹₃ p *

⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 1 2 3 4 5 ⫺_{6.12122122212222 . . .}

兹2苷_{1.414213562 . . .}

␲_.

兲

共

22

7

␲苷_{3.1415926535 . . .}

(

)

a (a, b) b

[

The various types of intervals are listed below.

The interval , graphed below, names the set of all real num-bers, .

EXAMPLE 1 _{Write interval notation for each set and graph the set.}

a) b)

c) d)

Solution

a) ;

b) ;

c) ;

0 1 ⫺4

⫺5 ⫺3⫺2⫺1 2 3 4 5 兵x兩⫺₅⬍_xⱕ ⫺₂其苷共⫺_5,⫺₂兴

0 1 ⫺4

⫺5 ⫺3⫺2⫺1 2 3 4 5 兵x兩xⱖ_1.7其苷关_1.7,

⬁

兲

0 1 ⫺4

⫺5 ⫺3⫺2⫺1 2 3 4 5 兵x兩⫺₄⬍_x⬍₅其苷共⫺_{4, 5}兲

兵

x兩x⬍兹₅

其

兵x兩⫺₅⬍_xⱕ ⫺₂其

兵x兩xⱖ_1.7其 兵x兩⫺₄⬍_x⬍₅其

⺢

共⫺

⬁

_,

⬁

兲

Intervals: Types, Notation, and Graphs

INTERVAL SET

TYPE NOTATION NOTATION GRAPH

Open

Closed

Half-open

Half-open

Open

Half-open

Open

Half-open

]

b 兵x兩xⱕ_b其

共⫺

⬁

_,b兴

)

b 兵x兩x⬍_b其

共⫺

⬁

_,b兲

[

a 兵x兩xⱖ_a其

关a,

⬁

兲

(

a 兵x兩x⬎_a其

共a,

⬁

兲

(

]

a b

兵x兩a⬍_xⱕ_b其 共a,b兴

[

)

a b

兵x兩aⱕ_x⬍_b其 关a,b兲

[

]

a b

兵x兩aⱕ_xⱕ_b其 关a,b兴

(

)

a b

d) ;

Properties of the Real Numbers

The following properties can be used to manipulate algebraic expressions as well as real numbers.

Properties of the Real Numbers

For any real numbers a,b, and c:

and Commutative properties of

addition and multiplication

and Associative properties of addition and multiplication

Additive identity property

Additive inverse property

Multiplicative identity property

Multiplicative inverse property

Distributive property

Note that the distributive property is also true for subtraction since .

EXAMPLE 2 _{State the property being illustrated in each sentence.}

a) b)

c) d)

e)

Solution

SENTENCE PROPERTY

a) Commutative property of multiplication:

b) Associative property of addition:

c) Additive inverse property:

d) Multiplicative identity property:

e) Distributive property:

a共b⫹_c兲苷_ab⫹_ac 2共a⫺_b兲苷_2a ⫺_2b

a ⭈₁苷₁⭈_a苷_a

6⭈₁苷₁⭈₆苷₆

a⫹共⫺_a兲苷₀

14⫹共⫺14兲苷0

a ⫹共_b ⫹_c兲苷共_a⫹_b兲⫹_c 5⫹共_m⫹_n兲苷共₅⫹_m兲⫹_n

ab苷_ba

8⭈₅苷₅⭈₈ 2共a⫺_b兲苷_2a ⫺_2b

6⭈₁苷₁⭈₆苷₆

14⫹共⫺₁₄兲苷₀

5⫹共_m⫹_n兲苷共₅⫹_m兲⫹_n 8⭈₅苷₅⭈₈

a共b⫺_c兲苷_a关_b⫹共⫺_c兲兴苷_ab⫹_a共⫺_c兲苷_ab⫺_ac a共b⫹_c兲苷_ab⫹_ac

共a苷₀兲 a ⭈ 1

a 苷

1

a ⭈a 苷1 a⭈₁苷₁⭈_a苷_a

⫺_a ⫹_a 苷_a ⫹共⫺_a兲苷₀ a⫹₀苷₀⫹_a苷_a a共bc兲苷共_ab兲_c

a⫹共_b⫹_c兲苷共_a⫹_b兲⫹_c ab苷_ba

a⫹_b苷_b⫹_a

0 1 ⫺4

a b

兩a ⫺ _b兩⫽ 兩_b ⫺ _a兩

Absolute Value

The number line can be used to provide a geometric interpretation of

absolute value. The absolute value of a number a, denoted , is its dis-tance from 0 on the number line. For example, , because the distance of ⫺_{5 from 0 is 5. Similarly, , because the distance of} from 0 is .

Absolute Value

For any real number a,

When ais nonnegative, the absolute value ofais a. When ais negative, the absolute value ofais the opposite, or additive inverse, ofa. Thus,

is never negative; that is, for any real number a, .

Absolute value can be used to find the distance between two points on the number line.

Distance Between Two Points on the Number Line

For any real numbers aand b, the distance between aand bis , or equivalently, .

EXAMPLE 3 _{Find the distance between} ⫺_{2 and 3.}

Solution The distance is

, or equivalently, .

We can also use the absolute-value operation on a graphing calculator to find the distance between two points. On many graphing calculators, ab-solute value is denoted “abs” and is found in the MATH NUMmenu and also in the CATALOG.

5 abs (3⫺(⫺₂₎₎

5 abs (⫺₂⫺3)

兩3⫺共⫺2兲兩苷兩3⫹2兩 苷兩5兩 苷5 兩⫺2⫺3兩苷兩⫺5兩 苷5

兩b⫺_a兩 兩a⫺_b兩

兩a兩 ⱖ₀ 兩a兩

兩a兩 苷

再

a, ⫺_a,

if aⱖ_0, if a⬍_0.

3 4

3 4

ⱍ

3 4

ⱍ

苷

3 4

兩⫺₅兩苷₅ 兩a兩

In Exercises 1– 10, consider the numbers ⫺_{12, , ,}

, ,0, , , ,⫺_{1.96, 9,}

, , , .

1.Which are whole numbers? , 0, 9,

2.Which are integers? ⫺12, , 0, 9,

3.Which are irrational numbers?

4.Which are natural numbers? , 9,

5.Which are rational numbers?

6.Which are real numbers? All of them

7.Which are rational numbers but not integers?

8.Which are integers but not whole numbers? ⫺₁₂

9.Which are integers but not natural numbers? ⫺_{12, 0}

10.Which are real numbers but not integers? ⱓ

Write interval notation. Then graph the interval.

11. ⱓ _12. ⱓ

13. ⱓ _14. ⱓ

15. ⱓ _16. ⱓ

17. ⱓ _18. ⱓ

19. ⱓ _20. ⱓ

Write interval notation for the graph. 21. 22. 23. 24. 25. 26. 27. 28.

In Exercises 29 – 46, the following notation is used: the set of natural numbers, the set of whole numbers, the set of integers, the set of rational numbers, the set of irrational numbers, and

the set of real numbers. Classify the statement as true or false.

29. True 30. True

31. False 32. True

33. True 34. False

35. False 36. False

37. False 38. True

39. True 40. True

41. True 42. False

43. True 44. True

45. False 46. False

Name the property illustrated by the sentence.

47. Commutative property of

multiplication

48. Associative property

of addition

49. 50. ⱓ

Multiplicative identity property

51. ⱓ 52.

Distributive property

4共y⫺_z兲苷_4y⫺_4z 5共ab兲苷共_5a兲_b

x⫹₄苷₄⫹_x ⫺₃⭈₁苷⫺₃

3⫹共_x⫹_y兲苷共₃⫹_x兲⫹_y 6⭈_x苷_x⭈₆

⺡債⺙ ⺢債⺪ ⺪債⺡ ⺡債⺢ ⺪債⺞ ⺧債⺪ ⺞債⺧ 1.089僆⺙

1僆⺪ 24僆⺧

⫺_1僆⺧

兹11僆⺢

⫺兹₆僆⺡

⫺11 5 僆⺡

⫺_10.1僆⺢ 3.2僆⺪

0僆⺞ 6僆⺞

⺢苷 ⺙苷 ⺡苷 ⺪苷 ⺧苷 ⺞苷 q

]

共⫺⬁_,q兴

p

(

共p,

⬁

兲

(

]

x x⫹_h

共x,x⫹_h兴

[

]

x x⫹_h

关x,x⫹_h兴

⫺₁₀ ⫺

(

₉ ⫺₈ ⫺₇ ⫺₆ ⫺

]

₅ ⫺₄ ⫺₃ ⫺₂ ⫺_{1 0 1 2}

共⫺_9,⫺₅兴

⫺10 ⫺

[

9 ⫺8 ⫺7 ⫺6 ⫺5 ⫺

)

4 ⫺3 ⫺2 ⫺1 0 1 2

关⫺_9,⫺₄兲

⫺₆ ⫺₅ ⫺₄ ⫺₃ ⫺₂ ⫺

[

_{1 0 1}

]

_{2 3 4 5 6}

关⫺_{1, 2}兴

⫺₆ ⫺₅ ⫺₄ ⫺₃ ⫺₂ ⫺_{1 0}

(

_{1 2 3 4}

)

_{5 6}

共0, 5兲

兵x兩⫺₃⬎_x其 兵x兩7⬍_x其

兵

x兩xⱖ兹₃

其

兵x兩x⬎_3.8其

兵x兩x⬎ ⫺₅其 兵x兩xⱕ ⫺₂其

兵x兩1⬍_xⱕ₆其 兵x兩⫺₄ⱕ_x⬍ ⫺₁其

兵x兩⫺₄⬍_x⬍₄其 兵x兩⫺₃ⱕ_xⱕ₃其

兹25 兹3 8 兹25 兹3 8 兹25 兹3 8 5 7 兹3 4 兹25 423

兹5 5 ⫺兹₁₄ 5.242242224 . . .

兹3 8 ⫺7 3 5.3 兹7

Exercise Set

R

.1

, ,

,兹5 5,兹3 4

⫺兹₁₄

5.242242224 . . .

兹7

, ,⫺_1.96,

,57

423

⫺7

3

5.3

⫺_{12, , , , 0,}

⫺_{1.96, 9, , ,}5

7 兹25 423

兹3

8

⫺7

3

53. 54.

Commutative property of multiplication

55. Commutative property

of addition

56. Additive identity property

57. Multiplicative inverse property

58. Distributive property

Simplify.

59. 7.1 60. 86.2

61. 347 62. 54

63. 64.

65. 0 66. 15

67. 68.

Find the distance between the given pair of points on the number line.

69.⫺_{5, 6 11 70.}⫺_{2.5, 0 2.5}

71.⫺_8, ⫺_{2 6 72. ,}

73.6.7, 12.1 5.4 74.⫺_14, ⫺_{3 11}

75. , 76.⫺_{3.4, 10.2 13.6}

77.⫺_{7, 0 7 78.3, 19 16}

Collaborative Discussion and Writing

To the student and the instructor: The Collaborative Discussion and Writing exercises are meant to be answered with one or more sentences. These exercises can also be discussed and answered collaboratively by the entire class or by small groups. Because of their open-ended nature, the answers to these exercises do

21 8

15 8 ⫺3

4

1 24

23 12 15

8

兹3

ⱍ

⫺兹₃

ⱍ

5 4

兩

5 4

兩

兩15兩 兩0兩

12 19

兩

12 19

兩

兹97

ⱍ

⫺兹₉₇

ⱍ

兩⫺₅₄兩 兩347兩

兩⫺_86.2兩 兩⫺_7.1兩

9x⫹_9y苷₉共_x⫹_y兲 8⭈ 1

8 苷1

t⫹₀苷_t

⫺₆共_m⫹_n兲苷⫺₆共_n⫹_m兲

⫺₇⫹₇苷₀

2共a⫹_b兲苷共_a⫹_b兲_{2 not appear at the back of the book. They are denoted} by the words “Discussion and Writing.”

79.How would you convince a classmate that division is not associative?

80.Under what circumstances is a rational number?

Synthesis

To the student and the instructor: The Synthesis exercises found at the end of every exercise set challenge students to combine concepts or skills studied in that section or in preceding parts of the text.

Between any two (different) real numbers there are many other real numbers. Find each of the following. Answers may vary.

81.An irrational number between 0.124 and 0.125

Answers may vary;

82.A rational number between and

Answers may vary;⫺_1.415

83.A rational number between and

Answers may vary;⫺_0.00999

84.An irrational number between and

Answers may vary;

85.The hypotenuse of an isosceles right triangle with legs of length 1 unit can be used to “measure” a value for by using the Pythagorean theorem, as shown.

c ₁

1 兹2

兹5.995

兹6 兹5.99

⫺ 1

100

⫺ 1

101

⫺兹₂ ⫺兹_2.01

0.124124412444 . . .

兹a

c苷兹₂ c2苷₂ c2苷₁2⫹₁2

Draw a right triangle that could be used to “measure”兹₁₀units. ⱓ

R

.2

Integer

Exponents,

Scientific

Notation, and

Order of

Operations

Simplify expressions with integer exponents. Solve problems using scientific notation. Use the rules for order of operations.

Integers as Exponents

When a positive integer is used as an exponent, it indicates the number of times a factor appears in a product. For example, means and means 5.

For any positive integer n,

,

nfactors

where ais the baseand nis the exponent.

Zero and negative-integer exponents are defined as follows.

For any nonzero real number aand any integer m,

and .

EXAMPLE 1 _{Simplify each of the following.}

a) b)

Solution

a) b)

EXAMPLE 2 _{Write each of the following with positive exponents.}

a) b) c)

Solution

a)

b)

c) x

⫺₃

y⫺₈苷x ⫺₃

⭈ 1 y⫺₈苷

1

x3⭈y

8_苷 y8

x3

1

共0.82兲⫺7苷共0.82兲

⫺共⫺₇兲

苷共_0.82兲7

4⫺5苷 1

45

x⫺3

y⫺₈

1 共0.82兲⫺₇

4⫺5

共⫺_3.4兲0苷₁

60苷₁

共⫺_3.4兲0 60

a⫺m_苷 1 am a0苷₁

an苷_a⭈_a⭈_a⭈ ⭈ ⭈_a

51 7⭈₇⭈₇ 73













The results in Example 2 can be generalized as follows.

For any nonzero numbers aand band any integers mand n,

.

(A factor can be moved to the other side of the fraction bar if the sign of the exponent is changed.)

EXAMPLE 3 _{Write an equivalent expression without negative exponents:}

.

Solution Since each exponent is negative, we move each factor to the other side of the fraction bar and change the sign of each exponent:

.

The following properties of exponents can be used to simplify expressions.

Properties of Exponents

For any real numbers aand band any integers mand n, assuming 0 is not raised to a nonpositive power:

Product rule

Quotient rule

Power rule

Raising a product to a power

Raising a quotient to a power

EXAMPLE 4 _{Simplify each of the following.}

a) b)

c) d)

e)

冉

45x

⫺₄

y2

9z⫺8

冊

⫺₃

共2s⫺2兲5 共t⫺3兲5

48x12

16x4 y⫺₅

⭈_y3

共b苷₀兲

冉

a

b

冊

m

苷a m

bm 共ab兲m苷_am_bm 共am兲n苷_amn

共a苷₀兲 am

an 苷a m⫺_n

am⭈_an苷_am⫹_n

x⫺3y⫺8 z⫺10 苷

z10 x3y8 x⫺3y⫺8

z⫺10 a⫺m b⫺n 苷

Solution

a) , or

b)

c) , or

d) , or

e)

, or

Scientific Notation

We can use scientific notation to name very large and very small positive numbers and to perform computations.

Scientific Notation

Scientific notationfor a number is an expression of the type

,

where ,Nis in decimal notation, and mis an integer.

Keep in mind that in scientific notation positive exponents are used for numbers greater than or equal to 10 and negative exponents for numbers between 0 and 1.

EXAMPLE 5 _{Undergraduate Enrollment. In a recent year, there were} 16,539,000 undergraduate students enrolled in post-secondary institutions in the United States (Source: U.S. National Center for Education Statistics). Convert this number to scientific notation.

Solution We want the decimal point to be positioned between the 1 and the 6, so we move it 7 places to the left. Since the number to be converted is greater than 10, the exponent must be positive.

16,539,000苷_1.6539⫻₁₀7 1ⱕ_N⬍₁₀

N⫻₁₀m

x12

125y6z24 苷5

⫺3_x12_y⫺6

z24 苷 x12

53y6z24

冉

45x⫺₄

y2

9z⫺₈

冊

⫺₃

苷

冉

5x

⫺₄

y2 z⫺₈

冊

⫺₃

32

s10 共2s⫺2兲5苷₂5共_s⫺₂

兲5_苷32s⫺₁₀

1

t15 共t⫺₃

兲5_苷t⫺₃⭈₅

苷_t⫺₁₅

48x12

16x4 苷

48 16x

12⫺₄

苷_3x8

1

y2 y⫺₅

⭈_y3苷_y⫺₅⫹₃

EXAMPLE 6 _{Mass of a Neutron. The mass of a neutron is about} 0.00000000000000000000000000167 kg. Convert this number to scien-tific notation.

Solution We want the decimal point to be positioned between the 1 and the 6, so we move it 27 places to the right. Since the number to be converted is between 0 and 1, the exponent must be negative.

EXAMPLE 7 _{Convert each of the following to decimal notation.}

a) b)

Solution

a) The exponent is negative, so the number is between 0 and 1. We move the decimal point 4 places to the left.

b) The exponent is positive, so the number is greater than 10. We move the decimal point 5 places to the right.

Most calculators make use of scientific notation. For example, the num-ber 48,000,000,000,000 might be expressed in one of the ways shown below.

EXAMPLE 8 _{Distance to a Star. The nearest star, Alpha Centauri C, is} about 4.22 light-years from Earth. One light-yearis the distance that light travels in one year and is about miles. How many miles is it from Earth to Alpha Centauri C? Express your answer in scientific notation.

Solution

This is not scientific notation because

.

miles Writing scientific notation

苷_2.48136⫻₁₀13

苷_2.48136⫻共₁₀1⫻₁₀12兲 苷共_2.48136⫻₁₀1兲⫻₁₀12

24.8136w₁₀

苷_24.8136⫻₁₀12 4.22⫻共_5.88⫻₁₀12兲苷共_4.22⫻_5.88兲⫻₁₀12

5.88⫻₁₀12

4.8 13 4.8E13

9.4⫻₁₀5苷_940,000

7.632⫻₁₀⫺4苷_0.0007632

9.4⫻₁₀5 7.632⫻₁₀⫺4

0.00000000000000000000000000167苷_1.67⫻₁₀⫺27

2.48136E13 4.22ⴱ5.88E12

Order of Operations

Recall that to simplify the expression , first we multiply 4 and 5 to get 20 and then add 3 to get 23. Mathematicians have agreed on the follow-ing procedure, or rules for order of operations.

Rules for Order of Operations

1. Do all calculations within grouping symbols before operations outside. When nested grouping symbols are present, work from the inside out.

2. Evaluate all exponential expressions.

3. Do all multiplications and divisions in order from left to right.

4. Do all additions and subtractions in order from left to right.

EXAMPLE 9 _{Calculate each of the following.}

a) b)

Solution

a) Doing the calculation within

parentheses

Evaluating the exponential expression Multiplying

Subtracting

b)

Note that fraction bars act as grouping symbols. That is, the given

ex-pression is equivalent to .

We can also enter these computations on a graphing calculator as shown below.

To confirm that it is essential to include parentheses around the numer-ator and around the denominnumer-ator when the computation in Example 9(b) is entered in a calculator, enter the computation without using these parenthe-ses. What is the result?

1 (10/(8⫺6)⫹9ⴱ4)/(2_ˆ5⫹32₎

44 8(5⫺3)_ˆ3⫺20

关10⫼共8⫺6兲⫹9⭈4兴⫼共25⫹32兲 苷5⫹36

41 苷

41 41苷1 10⫼共₈⫺₆兲⫹₉⭈₄

25⫹₃2

苷10⫼2⫹9⭈4

32⫹₉ 苷₄₄

苷₆₄⫺₂₀ 苷₈⭈₈⫺₂₀ 8共5⫺₃兲3⫺₂₀苷₈⭈₂3⫺₂₀

10⫼共₈⫺₆兲⫹₉⭈₄ 25⫹₃2 8共5⫺₃兲3⫺₂₀

3⫹₄⭈₅

EXAMPLE 10 _{Compound Interest. If a principal P is invested at an} interest rate r, compounded n times per year, in tyears it will grow to an amount Agiven by

.

Suppose that $1250 is invested at 4.6% interest, compounded quarterly. How much is in the account at the end of 8 years?

Solution We have , , or 0.046, , and .

Sub-stituting, we find that the amount in the account at the end of 8 years is given by

.

Next, we evaluate this expression:

Dividing Adding

Multiplying in the exponent

Evaluating the exponential expression Multiplying

. Rounding to the nearest cent The amount in the account at the end of 8 years is $1802.26.

⬇1802.26 ⬇1802.263969 ⬇1250共1.441811175兲

苷₁₂₅₀共_1.0115兲32 苷₁₂₅₀共1.0115兲4⭈₈

A苷₁₂₅₀共₁⫹0.0115兲4⭈8 A苷₁₂₅₀

冉

₁⫹0.046

4

冊

4⭈₈

t苷₈ n苷₄

r苷_4.6% P苷₁₂₅₀

A苷_P

冉

₁⫹ r n

冊

R

.2

Exercise Set

Simplify.

1. 1 2. 1

3. 4.

5. , or 25 6. , or

7. 1 8. 1

9. , or 10.

11. , or 12.

13. 14.

15. ⱓ _16. ⱓ

17. ⱓ _18. ⱓ

19. ⱓ_20. ⱓ

21. 22.

23. 24.

25. 26.

27. , or 28. , or

29. , or 30. , or

31. , or 32. , or4b

a2

4a⫺2_b

20a5b⫺₂

5a7b⫺₃

8x y5

8xy⫺5

32x⫺₄

y3

4x⫺₅

y8

x4 y5 x4y⫺5

x3y⫺₃

x⫺₁

y2 x3

y3 x3y⫺3

x2y⫺₂

x⫺₁

y

1

y3 y⫺3

y⫺₂₄

y⫺₂₁

1

x21 x⫺21

x⫺₅

x16

a7 a39 a32 b3

b40 b37

288x7 共2x兲5共3x兲2 ⫺_200n5

共⫺_2n兲3共_5n兲2

432y5 共4y兲2共3y兲3

72x5 共2x兲3共3x兲2

共8ab7兲 共⫺_7a⫺5_b2兲 共6x⫺3y5兲 共⫺_7x2_y⫺9兲

共4xy2兲 共3x⫺₄

y5兲 共5a2b兲 共3a⫺₃

b4兲

共⫺_6b⫺4兲 共_2b⫺7兲 共⫺_3a⫺5兲 共_5a⫺7兲

12y7

3y4⭈_4y3

6x5

2x3⭈_3x2

35

36⭈₃⫺5⭈₃4

1 7 7⫺1

73⭈₇⫺5⭈₇

b8 b⫺₄

⭈_b12

1

y4 y⫺4

y3⭈_y⫺7

n9⭈_n⫺9 m⫺5⭈_m5

1 65 6⫺5

62⭈₆⫺7

52

58⭈₅⫺6

a4 a0⭈_a4 x9

x9⭈_x0

共

⫺43兲0 180

33. 34.

35. 36.

37. 38.

39. ⱓ _40.

41. ⱓ 42. ⱓ

43. ⱓ 44. ⱓ

Convert to scientific notation.

45.405,000 46.1,670,000

47.0.00000039 48.0.00092

49.234,600,000,000 50.8,904,000,000

51.0.00104 52.0.00000000514

53.One cubic inch is approximately equal to .

54.The United States government collected

$1,137,000,000,000 in individual income taxes in a recent year (Source: U.S. Internal Revenue Service).

Convert to decimal notation.

55. 0.000083 56. 4,100,000

57. 20,700,000 58.

0.00000315

59. 60.

34,960,000,000 840,900,000,000

61. 62.

0.0000000541 0.000000000627

63.The amount of solid waste generated in the United States in a recent year was tons (Source: Franklin Associates, Ltd.). 231,900,000

64.The mass of a proton is about g.

0.00000000000000000000000167

Compute. Write the answer using scientific notation. 65. 66. 67. 68. 69. 70. 71. 72.

Solve. Write the answer using scientific notation. 73.Distance to Pluto. The distance from Earth to the

sun is defined as 1 astronomical unit, or AU. It is about 93 million miles. The average distance from Earth to the planet Pluto is 39 AUs. Find this distance in miles.

74.Parsecs. One parsec is about 3.26 light-years and 1 light-year is about Find the number of miles in 1 parsec.

75.Nanowires. A nanometeris 0.000000001 m. Scientists have developed optical nanowires to transmit light waves short distances. A nanowire with a diameter of 360 nanometers has been used in experiments on :the transmission of light (Source:

New York Times,January 29, 2004). Find the diameter of such a wire in meters.

76.iTunes. In the first quarter of 2004, Apple

Computer was selling 2.7 million songs per week on iTunes, its online music service (Source: Apple Computer). At $0.99 per song, what is the revenue during a 13-week period?

77.Chesapeake Bay Bridge-Tunnel. The 17.6-mile-long Chesapeake Bay Bridge-Tunnel was completed in 1964. Construction costs were $210 million. Find the average cost per mile.

78.Personal Space in Hong Kong. The area of Hong Kong is 412 square miles. It is estimated that the population of Hong Kong will be 9,600,000 in 2050. Find the number of square miles of land per person

in 2050. sq mi

79.Nuclear Disintegration. One gram of radium produces 37 billion disintegrations per second. How many disintegrations are produced in 1 hr?

disintegrations

80.Length of Earth’s Orbit. The average distance from the earth to the sun is 93 million mi. About how far does the earth travel in a yearly orbit? (Assume a circular orbit.) 5.8⫻₁₀8mi

1.332⫻₁₀14

4.3⫻₁₀⫺5

$1.19⫻₁₀7

$3.4749⫻₁₀7

3.6⫻₁₀⫺7 _m

1.91688⫻₁₀13 _mi

5.88⫻₁₀12 _mi.

3.627⫻₁₀9 _mi

2.5⫻₁₀⫺7

1.3⫻₁₀4 5.2⫻₁₀10

2.5⫻₁₀5

1.8⫻₁₀⫺3 7.2⫻₁₀⫺9

5.5⫻₁₀30

1.1⫻₁₀⫺40 2.0⫻₁₀⫺71

8⫻₁₀⫺14

6.4⫻₁₀⫺7 8.0⫻₁₀6

2.368⫻108 共6.4⫻1012兲 共3.7⫻10⫺5兲

2.21⫻₁₀⫺10 共2.6⫻₁₀⫺18兲 共_8.5⫻₁₀7兲

7.462⫻₁₀⫺13 共9.1⫻₁₀⫺17兲 共_8.2⫻₁₀3兲

1.395⫻₁₀3 共3.1⫻₁₀5兲 共_4.5⫻₁₀⫺3兲

1.67⫻₁₀⫺24 2.319⫻₁₀8

6.27⫻₁₀⫺10 5.41⫻₁₀⫺8

8.409⫻₁₀11 3.496⫻₁₀10

3.15⫻₁₀⫺6 2.07⫻₁₀7

4.1⫻₁₀6 8.3⫻₁₀⫺5

1.137⫻₁₀12

1.6⫻₁₀⫺5

0.000016 m3

5.14⫻₁₀⫺9

1.04⫻₁₀⫺3

8.904⫻₁₀9

2.346⫻₁₀11

9.2⫻₁₀⫺4

3.9⫻₁₀⫺7

1.67⫻₁₀6

4.05⫻₁₀5

冉

125p12q⫺14_r22

25p8q6r⫺15

冊

⫺4

冉

24a10b⫺8_c7

12a6b⫺3_c5

冊

⫺5

冉

3x5y⫺8

z⫺2

冊

4

冉

2x⫺3_y7

z⫺1

冊

3

128n7 共4n⫺1_兲2_共2n3_兲3

共3m4兲3共2m⫺5_兲4

x15z6 ⫺₆₄ 共⫺_4x⫺5_z⫺2兲⫺3 c2d4

25

共⫺_5c⫺1_d⫺2兲⫺2

81x8 共⫺_3x2兲4 ⫺_32x15

共⫺_2x3兲5

16x2y6 共4xy3兲2

Calculate.

81. 2

82. 18

83. 2048

84. 2

85. 5

86. ⫺₅

Compound Interest. Use the compound interest formula from Example 10 in Exercises 87– 90. Round to the nearest cent.

87.Suppose that $2125 is invested at 6.2%, compounded semiannually. How much is in the account at the end of 5 yr? $2883.67

88.Suppose that $9550 is invested at 5.4%, compounded semiannually. How much is in the account at the end of 7 yr? $13,867.23

89.Suppose that $6700 is invested at 4.5%, compounded quarterly. How much is in the account at the end of 6 yr? $8763.54

90.Suppose that $4875 is invested at 5.8%, compounded quarterly. How much is in the account at the end of 9 yr? $8185.56

Collaborative Discussion and Writing

91.Are the parentheses necessary in the expression ? Why or why not?

92.Is for any negative value(s) ofx? Why or why not?

Synthesis

Savings Plan. The formula

gives the amount S accumulated in a savings plan when a deposit of P dollars is made each month for t years in an account with interest rate r, compounded monthly. Use this formula for Exercises 93 – 96.

93.Marisol deposits $250 in a retirement account each month beginning at age 40. If the investment earns 5% interest, compounded monthly, how much will have accumulated in the account when she retires 27 yr later? $170,797.30

94.Gordon deposits $100 in a retirement account each month beginning at age 25. If the investment earns 4% interest, compounded monthly, how much will have accumulated in the account when Gordon retires at age 65? $118,196.13

95.Gina wants to establish a college fund for her newborn daughter that will have accumulated $120,000 at the end of 18 yr. If she can count on an interest rate of 6%, compounded monthly, how much should she deposit each month to accomplish this? $309.79

96.Liam wants to have $200,000 accumulated in a retirement account by age 70. If he starts making monthly deposits to the plan at age 30 and can count on an interest rate of 4.5%, compounded monthly, how much should he deposit each month in order to accomplish this? $149.13

S苷_P

冋

冉

1⫹ r 12

冊

12⭈t ⫺₁

r

12

册

x⫺₂

⬍_x⫺1 4⭈₂₅⫼共₁₀⫺₅兲

关4共8⫺6兲2⫹4兴 共3⫺2⭈8兲 22共23⫹5兲

4共8⫺₆兲2⫺₄⭈₃⫹₂⭈₈ 31⫹₁₉0

Simplify. Assume that all exponents are integers, all denominators are nonzero, and zero is not raised to a nonpositive power.

97. 98. 1

99. 100.

101. 102.

, or x6r

y18t x6ry⫺18t

9x2ay2b

冋冉

xr yt

冊

2

冉

x2r

y4t

冊

⫺2

册

⫺3

冋

共3xayb兲3

共⫺_3xa_yb兲2

册

2

mx2nx2 共mx⫺_b

⭈_nx⫹b兲x共_mb_n⫺b兲x t8x

共ta⫹_x

⭈_tx⫺a兲4

共xy⭈_x⫺y兲3 x8t

R

.3

Addition,

Subtraction, and

Multiplication of

Polynomials

• Identify the terms, coefficients, and degree of a polynomial. • Add, subtract, and multiply polynomials.

Polynomials

Polynomials are a type of algebraic expression that you will often encounter in your study of algebra. Some examples of polynomials are

, , , and .

All but the first are polynomials in one variable.

Polynomials in One Variable

A polynomial in one variableis any expression of the type

,

where nis a nonnegative integer and are real numbers, called coefficients. The parts of a polynomial separated by plus signs are called terms. The leading coefficientis , and the constant termis . If the degreeof the polynomial is n. The polynomial is said to be written in descending order, because the exponents decrease from left to right.

EXAMPLE 1 _{Identify the terms of the polynomial}

.

Solution Writing plus signs between the terms, we have

,

so the terms are

, , x, and ⫺_12.

A polynomial, like 23, consisting of only a nonzero constant term has degree 0. It is agreed that the polynomial consisting only of 0 has nodegree.

⫺_7.5x3 2x4

2x4⫺_7.5x3⫹_x⫺₁₂苷_2x4⫹共⫺_7.5x3兲⫹_x⫹共⫺₁₂兲 2x4⫺_7.5x3⫹_x⫺₁₂

an苷0, a0

an a0

an, . . . , anxn⫹an⫺₁xn

⫺₁

⫹ ⭈ ⭈ ⭈ ⫹_a2x2⫹_a1x⫹_a0

z6⫺兹₅ ⫺_2.3a4

5y3⫺7

3y

2_⫹3y⫺2

3x⫺_4y

EXAMPLE 2 _{Find the degree of each polynomial.}

a) b) c) 7

Solution

POLYNOMIAL DEGREE

a) 3

b) 4

c) 0

Algebraic expressions like and

are polynomials in several variables. The degree of a termis the sum of the exponents of the variables in that term. The degree of a polynomialis the degree of the term of highest degree.

EXAMPLE 3 _{Find the degree of the polynomial}

.

Solution The degrees of the terms of are 4, 6, and 0,

respectively, so the degree of the polynomial is 6.

A polynomial with just one term, like , is a monomial. If a poly-nomial has two terms, like , it is a binomial. A polynomial with three terms, like , is a trinomial.

Expressions like

, , and

are not polynomials, because they cannot be written in the form , where the exponents are all nonnegative in-tegers and the coefficients are all real numbers.

Addition and Subtraction

If two terms of an expression have the same variables raised to the same powers, they are called like terms, or similar terms. We can combine, or collect,like termsusing the distributive property. For example, and

are like terms and

.

We add or subtract polynomials by combining like terms.

EXAMPLE 4 _{Add or subtract each of the following.}

a)

b) 共6x2y3⫺_9xy兲⫺共_5x2_y3⫺_4xy兲

共⫺_5x3⫹_3x2⫺_x兲⫹共_12x3⫺_7x2⫹₃兲 苷_8y2

3y2⫹_5y2苷共₃⫹₅兲_y2 5y2

3y2 an⫺1xn⫺1⫹ ⭈ ⭈ ⭈ ⫹a1x⫹a0

anxn⫹ x⫹₁

x4⫹₅ 9⫺兹_x

2x2⫺_5x⫹ 3 x

4x2⫺_4xy⫹₁ x2⫹₄

⫺_9y6

7ab3⫺_11a2_b4⫹₈ 7ab3⫺_11a2_b4⫹₈

5x4y2⫺_3x3_y8⫹_7xy2⫹₆ 3ab3⫺₈

7苷_7x0 y2⫺3

2⫹5y

4_苷5y4_⫹y2_⫺3 2

2x3⫺₉

y2⫺3

2⫹5y

4

Solution

a)

Rearranging using the commutative and associative properties Using the distribu-tive property

b) We can subtract by adding an opposite:

Adding the opposite of

. Combining like terms

Multiplication

Multiplication of polynomials is based on the distributive property—for example,

Using the distributive property Using the distributive property two more times

. Combining like terms

In general, to multiply two polynomials, we multiply each term of one by each term of the other and add the products.

EXAMPLE 5 _{Multiply: .}

Solution We have

Using the distributive property

Using the distributive property three more times . Combining like terms We can also use columns to organize our work, aligning like terms under each other in the products.

Multiplying by Multiplying by 2y

Adding ⫺_12x6_y2⫹_29x4_y2⫺_23x2_y2⫹_6y2

8x4y2⫺_14x2_y2⫹_6y2

ⴚ3x2y ⫺_12x6_y2⫹_21x4_y2⫺ _9x2_y2

2y⫺_3x2_y 4x4y⫺_7x2_y⫹_3y 苷_29x4_y2⫺_12x6_y2⫺_23x2_y2⫹_6y2

苷_8x4_y2⫺_12x6_y2⫺_14x2_y2⫹_21x4_y2⫹_6y2⫺_9x2_y2 苷_4x4_y共_2y⫺_3x2_y兲⫺_7x2_y共_2y⫺_3x2_y兲⫹_3y共_2y⫺_3x2_y兲 共4x4y⫺_7x2_y⫹_3y兲 共_2y⫺_3x2_y兲

共4x4y⫺_7x2_y⫹_3y兲 共_2y⫺_3x2_y兲 苷_x2⫹_7x⫹₁₂

苷_x2⫹_3x⫹_4x⫹₁₂ 共x⫹₄兲 共_x⫹₃兲苷_x共_x⫹₃兲⫹₄共_x⫹₃兲

苷_x2_y3⫺_5xy

苷_6x2_y3⫺_9xy⫺_5x2_y3⫹_4xy

5x2y3ⴚ4xy 苷共_6x2_y3⫺_9xy兲⫹共⫺_5x2_y3⫹_4xy兲

共6x2y3⫺_9xy兲⫺共_5x2_y3⫺_4xy兲 苷_7x3⫺_4x2⫺_x⫹₃

We can find the product of two binomials by multiplying the First terms, then the Outer terms, then the Inner terms, then the Last terms. Then we combine like terms, if possible. This procedure is sometimes called FOIL.

EXAMPLE 6 _{Multiply: .}

Solution We have

F L

F O I L

I O

We can use FOIL to find some special products.

Special Products of Binomials

Square of a sum

Square of a difference

Product of a sum and a difference

EXAMPLE 7 _{Multiply each of the following.}

a) b) c)

Solution

a) b)

c) 共x2⫹_3y兲 共_x2⫺_3y兲苷共_x2兲2⫺共_3y兲2苷_x4⫺_9y2

共3y2⫺₂兲2苷共_3y2兲2⫺₂⭈_3y2⭈₂⫹₂2苷_9y4⫺_12y2⫹₄ 共4x⫹₁兲2苷共_4x兲2⫹₂⭈_4x⭈₁⫹₁2苷_16x2⫹_8x⫹₁

共x2⫹_3y兲 共_x2⫺_3y兲 共3y2⫺₂兲2

共4x⫹₁兲2

共A⫹B兲 共A⫺B兲苷A2⫺B2 共A⫺B兲2苷A2⫺2AB⫹B2 共A⫹B兲2苷A2⫹2AB⫹B2

苷_6x2⫺_13x⫺₂₈ 共2x⫺₇兲 共_3x⫹₄兲苷_6x2⫹_8x⫺_21x⫺₂₈

R

.3

Exercise Set

Determine the terms and the degree of the polynomial.

1. , , ,

, ; 4

2. , ,⫺_{4m, 11; 3}

3. , , 5ab,

; 6

4. , , ,

5; 6

Perform the operations indicated. 5.

6.

7.

3x⫹_2y⫺_2z⫺₃ 共⫺_3x⫹_y⫺_2z⫺₄兲

共2x⫹_3y⫹_z⫺₇兲⫹共_4x⫺2₂x_y⫺_z⫹₈兲⫹

2

y⫺_7xy2⫹_8xy⫹₅ 共⫺_4x2_y⫺_4xy2⫹_3xy⫹₈兲

共6x2y⫺_3xy2⫹_5xy⫺₃兲⫹

3x2y⫺_5xy2⫹_7xy⫹₂ 共⫺_2x2_y⫺_3xy2⫹_4xy⫹₇兲

共5x2y⫺_2xy2⫹_3xy⫺₅兲⫹

⫺_3pq2 ⫺_p2_q4

6p3q2 6p3q2⫺_p2_q4⫺_3pq2⫹₅ ⫺2

⫺_7a3_b3

3a4b 3a4b⫺_7a3_b3⫹_5ab⫺₂

⫺_m2

2m3

2m3⫺_m2⫺_4m⫹₁₁ ⫺₄

⫺_y

7y2

3y3 ⫺_5y4 ⫺_5y4⫹_3y3⫹_7y2⫺_y⫺₄

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42.

Collaborative Discussion and Writing

43.Is the sum of two polynomials of degree nalways a polynomial of degree n? Why or why not?

44.Explain how you would convince a classmate that .

Synthesis

Multiply. Assume that all exponents are natural numbers. 45. 46. 47. 48. 49. 50. 51. 52.

53.共_a⫹_b⫹_c兲2 a2⫹_b2⫹_c2⫹_2ab⫹_2ac⫹_2bc t2m2⫹2n2

共tm⫹n兲m⫹n⭈共_tm⫺n兲m⫺n xa2⫺b2 共xa⫺b兲a⫹b

16x4⫺_32x3⫹_16x2 关共2x⫺₁兲2⫺₁兴2

x6⫺₁ 共x⫺₁兲 共_x2⫹_x⫹₁兲 共_x3⫹₁兲

x6m⫺_2x3m_t5n⫹_t10n 共x3m⫺_t5n兲2

a2n⫹_2an_bn⫹_b2n 共an⫹_bn兲2

t2a⫺_3ta⫺₂₈ 共ta⫹₄兲 共_ta⫺₇兲

a2n⫺_b2n 共an⫹_bn兲 共_an⫺_bn兲

共A⫹B兲2苷A2⫹B2

y4⫺₁₆ 共y⫺₂兲 共_y⫹₂兲 共_y2⫹₄兲

x4⫺₁ 共x⫹₁兲 共_x⫺₁兲 共_x2⫹₁兲

25x2⫹_20xy⫹_4y2⫺₉ 共5x⫹_2y⫹₃兲 共_5x⫹_2y⫺₃兲

4x2⫹_12xy⫹_9y2⫺₁₆ 共2x⫹_3y⫹₄兲 共_2x⫹_3y⫺₄兲

9x2⫺_25y2 共3x⫹_5y兲 共_3x⫺_5y兲

9x2⫺_4y2 共3x⫺_2y兲 共_3x⫹_2y兲

16y2⫺₁ 共4y⫺₁兲 共_4y⫹₁兲

4x2⫺₂₅ 共2x⫺₅兲 共_2x⫹₅兲

b2⫺₁₆ 共b⫹₄兲 共_b⫺₄兲

a2⫺₉ 共a⫹₃兲 共_a⫺₃兲

16x4⫺_40x2_y⫹_25y2 共4x2⫺_5y兲2

4x4⫺_12x2_y⫹_9y2 共2x2⫺_3y兲2

25x2⫹_20xy⫹_4y2 共5x⫹_2y兲2

4x2⫹_12xy⫹_9y2 共2x⫹_3y兲2

9x2⫺_12x⫹₄ 共3x⫺₂兲2

25x2⫺_30x⫹₉ 共5x⫺₃兲2

a2⫺_12a⫹₃₆ 共a⫺₆兲2

x2⫺_8x⫹₁₆ 共x⫺₄兲2

y2⫹_14y⫹₄₉ 共y⫹₇兲2

y2⫹_10y⫹₂₅ 共y⫹₅兲2

4a2⫺_8ab⫹_3b2 共2a⫺_3b兲 共_2a⫺_b兲

4x2⫹_8xy⫹_3y2 共2x⫹_3y兲 共_2x⫹_y兲

3b2⫺_5b⫺₂ 共3b⫹₁兲 共_b⫺₂兲

2a2⫹_13a⫹₁₅ 共2a⫹₃兲 共_a⫹₅兲

n2⫺_13n⫹₄₀ 共n⫺₅兲 共_n⫺₈兲

x2⫹_10x⫹₂₄ 共x⫹₆兲 共_x⫹₄兲

y2⫺_3y⫺₄ 共y⫺₄兲 共_y⫹₁兲

x2⫹_2x⫺₁₅ 共x⫹₅兲 共_x⫺₃兲

n3⫺_5n2⫺_10n⫺₄ 共n⫹₁兲 共_n22⫺a_6n⫺₄兲

4_⫺

2a3b⫺_a2_b⫹_4ab2⫺_3b3 共a⫺_b兲 共_2a3⫺_ab⫹_3b2兲

2x4⫺_5x3⫺_5x2⫹_10x⫺₅

共2x4⫺_3x2⫹_7x兲⫺共_5x3⫹_2x2⫺_3x⫹₅兲 x4⫺_3x3⫺_4x2⫹_9x⫺₃

共x4⫺_3x2⫹_4x兲⫺共_3x3⫹_x2⫺_5x⫹₃兲 ⫺_4x2⫹_8xy⫺_5y2⫹₃

共5x2⫹_4xy⫺_3y2⫹₂兲⫺共_9x2⫺_4xy⫹_2y2⫺₁兲 ⫺_2x2⫹_6x⫺₂

共3x2⫺_2x⫺_x3⫹₂兲⫺共_5x2⫺_8x⫺_x3⫹₄兲

7x2⫹_12xy⫺_2x⫺_y⫺₉ 共⫺_x2⫺_y⫺₂兲

Factor polynomials by removing a common factor. Factor polynomials by grouping.

Factor trinomials of the type .

Factor trinomials of the type , , using the FOIL method and the grouping method.

Factor special products of polynomials.

To factor a polynomial, we do the reverse of multiplying; that is, we find an equivalent expression that is written as a product.

Terms with Common Factors

When a polynomial is to be factor