math 1300 fundamentals of mathematicsjac/1300/fa2019/math1300... · inequalities section 2.6:...

MATH 1300

Fundamentals of

Mathematics

University of Houston Department of Mathematics

MATH 1300, Fundamentals of Mathematics

MATH 1300 Fundamentals of Mathematics i

MATH 1300 – Fundamentals of Mathematics

Table of Contents

CHAPTER 1: INTRODUCTORY INFORMATION AND REVIEW ................................... 1

Section 1.1: Numbers ................................................................................................................ 1

Types of Numbers................................................................................................................... 1

Order on a Number Line ....................................................................................................... 16

Exercise Set 1.1 ..................................................................................................................... 23

Section 1.2: Integers ................................................................................................................ 26

Operations with Integers ....................................................................................................... 26

Exercise Set 1.2 ..................................................................................................................... 35

Section 1.3: Fractions .............................................................................................................. 36

Greatest Common Divisor and Least Common Multiple ..................................................... 36

Addition and Subtraction of Fractions.................................................................................. 43

Multiplication and Division of Fractions.............................................................................. 50

Exercise Set 1.3 ..................................................................................................................... 56

Section 1.4: Exponents and Radicals ...................................................................................... 58

Evaluating Exponential Expressions..................................................................................... 58

Square Roots ......................................................................................................................... 66

Exercise Set 1.4 ..................................................................................................................... 72

Section 1.5: Order of Operations............................................................................................. 75

Evaluating Expressions Using the Order of Operations ....................................................... 75

Exercise Set 1.5 ..................................................................................................................... 79

Section 1.6: Solving Linear Equations .................................................................................... 82

Linear Equations ................................................................................................................... 82

Exercise Set 1.6 ..................................................................................................................... 85

Section 1.7: Interval Notation and Linear Inequalities............................................................ 86

Linear Inequalities ................................................................................................................ 86

Exercise Set 1.7 ..................................................................................................................... 94

University of Houston Department of Mathematics ii

Section 1.8: Absolute Value and Equations ............................................................................ 96

Absolute Value...................................................................................................................... 96

Exercise Set 1.8 ................................................................................................................... 103

CHAPTER 2: POINTS, LINES, AND FUNCTIONS .......................................................... 104

Section 2.1: An Introduction to the Coordinate Plane........................................................... 104

Points in the Coordinate Plane............................................................................................ 104

Exercise Set 2.1 ................................................................................................................... 117

Section 2.2: The Distance and Midpoint Formulas ............................................................... 120

The Distance Formula......................................................................................................... 120

The Midpoint Formula........................................................................................................ 129

Exercise Set 2.2 ................................................................................................................... 134

Section 2.3: Slope and Intercepts of Lines ............................................................................ 136

The Slope of a Line............................................................................................................. 136

Intercepts of Lines............................................................................................................... 142

Exercise Set 2.3 ................................................................................................................... 149

Section 2.4: Equations of Lines............................................................................................. 152

Writing Equations of Lines................................................................................................. 152

Exercise Set 2.4 ................................................................................................................... 160

Section 2.5: Parallel and Perpendicular Lines....................................................................... 162

Pairs of Lines - Parallel and Perpendicular Lines............................................................... 162

Exercise Set 2.5 ................................................................................................................... 168

Section 2.6: An Introduction to Functions ............................................................................ 170

Definition of a Function...................................................................................................... 170

Domain of a Function ......................................................................................................... 177

Exercise Set 2.6 ................................................................................................................... 181

Section 2.7: Functions and Graphs........................................................................................ 185

Graphing a Function ........................................................................................................... 185

Exercise Set 2.7 ................................................................................................................... 200

CHAPTER 3: POLYNOMIALS ............................................................................................ 203

Section 3.1: An Introduction to Polynomial Functions......................................................... 203

Polynomials and Polynomial Functions.............................................................................. 203

Exercise Set 3.1 ................................................................................................................... 213

MATH 1300 Fundamentals of Mathematics iii

Section 3.2: Adding, Subtracting, and Multiplying Polynomials.......................................... 216

Operations with Polynomials.............................................................................................. 216

Exercise Set 3.2 ................................................................................................................... 223

Section 3.3: Dividing Polynomials........................................................................................ 225

Polynomial Long Division and Synthetic Division ............................................................ 225

Exercise Set 3.3 ................................................................................................................... 238

Section 3.4: Quadratic Functions .......................................................................................... 240

The Definition and Graph of a Quadratic Function ............................................................ 240

Exercise Set 3.4 ................................................................................................................... 249

CHAPTER 4: FACTORING .................................................................................................. 250

Section 4.1: Greatest Common Factor and Factoring by Grouping.................................... 250

GCF and Grouping.............................................................................................................. 250

Exercise Set 4.1 ................................................................................................................... 258

Section 4.2: Factoring Special Binomials and Trinomials .................................................... 260

Special Factor Patterns........................................................................................................ 260

Exercise Set 4.2 ................................................................................................................... 268

Section 4.3: Factoring Polynomials....................................................................................... 270

Techniques for Factoring Trinomials.................................................................................. 270

Exercise Set 4.3 ................................................................................................................... 281

Section 4.4: Using Factoring to Solve Equations.................................................................. 283

Solving Quadratic Equations by Factoring ......................................................................... 283

Solving Other Polynomial Equations by Factoring ............................................................ 290

Exercise Set 4.4 ................................................................................................................... 295

CHAPTER 5: RATIONAL EXPRESSIONS, EQUATIONS, AND FUNCTIONS ........... 297

Section 5.1: Simplifying Rational Expressions..................................................................... 297

Rational Expressions........................................................................................................... 297

Exercise Set 5.1 ................................................................................................................... 302

Section 5.2: Multiplying and Dividing Rational Expressions................................................ 304

Multiplication and Division ................................................................................................ 304

Exercise Set 5.2 ................................................................................................................... 309

University of Houston Department of Mathematics iv

Section 5.3: Adding and Subtracting Rational Expressions.................................................. 311

Addition and Subtraction .................................................................................................... 311

Exercise Set 5.3 ................................................................................................................... 319

Section 5.4: Complex Fractions ............................................................................................ 321

Simplifying Complex Fractions.......................................................................................... 321

Exercise Set 5.4 ................................................................................................................... 328

Section 5.5: Solving Rational Equations............................................................................... 331

Rational Equations .............................................................................................................. 331

Exercise Set 5.5 ................................................................................................................... 340

Section 5.6: Rational Functions............................................................................................. 342

Working with Rational Functions....................................................................................... 342

Exercise Set 5.6 ................................................................................................................... 354

ODD-NUMBERED ANSWERS TO EXERCISE SETS ....................................................... 357

MATH 1300 Fundamentals of Mathematics v

MATH 1300 – Fundamentals of Mathematics

Online Resources

Math 1300 Online:

All materials found in this textbook can also be found online at:

http://online.math.uh.edu/Math1300/

The Math 1300 online site also contains flash lectures which match each additional example in

the text.

Additional Resources:

Math 1300 is designed to prepare students for Math 1310 (College Algebra). There is some

overlap in course material between these two courses, and the online resources for Math 1310

may prove to be helpful.

The Math 1310 materials can be found online at:

http://online.math.uh.edu/Math1310/

In addition to the textbook material, the online site for Math 1310 contains flash lectures

pertaining to most of the topics in the College Algebra course. These lectures simulate the

classroom experience, with audio of course instructors as they present the material on prepared

lesson notes. The lectures are useful in furthering understanding of the course material and can

only be viewed online.

The next page contains a list of some of the overlapping topics between the courses. This should

help students to locate the flash lectures and other textbook materials in Math 1310 that may be

useful for Math 1300. (Keep in mind that Math 1300 and Math 1310 do not cover identical

topics. It may be necessary to search through the Math 1310 sections to find specific types of

examples covered in Math 1300, and some topics in one course may not be covered at all in the

other.)

University of Houston Department of Mathematics vi

Overlapping Topics between Math 1300 and Math 1310

(to use Math 1310 online resources as a reference for Math 1300)

Math 1300 Math 1310

Section 1.6: Linear Equations Section 2.1: Linear Equations

Section 1.7: Interval Notation and Linear

Inequalities

Section 2.6: Linear Inequalities

Section 1.8: Absolute Value and Equations Section 2.8: Absolute Value

Section 2.1: An Introduction to the Coordinate

Section 1.1: Points, Regions, Distance

and Midpoints

Section 2.2: The Distance and Midpoint Formulas Section 1.1: Points, Regions, Distance

and Midpoints

Section 2.3: Slope and Intercepts of Lines Section 1.2: Lines

Section 2.4: Equations of Lines Section 1.2: Lines

Section 2.5: Parallel and Perpendicular Lines Section 1.2: Lines

Section 2.6: An Introduction to Functions Section 3.1: Basic Ideas

Section 2.7: Functions and Graphs Section 3.2: Functions and Graphs

Section 3.1: An Introduction to Polynomial

Functions

Section 4.1: Polynomial Functions

Section 3.3: Dividing Polynomials Section 4.2: Dividing Polynomials

Section 3.4: Quadratic Functions Section 3.5: Maximum and Minimum

Values

Section 4.1: Greatest Common Factor and

Factoring by Grouping

Section 2.3: Quadratic Equations

Section 2.5: Other Equations

Section 4.2: Factoring Special Binomials and

Trinomials

Section 2.3: Quadratic Equations

Section 4.3: Factoring Polynomials Section 2.3: Quadratic Equations

Section 4.4: Using Factoring to Solve Equations Section 2.3: Quadratic Equations

Section 5.6: Rational Functions Section 4.4: Rational Functions

SECTION 1.1 Numbers

MATH 1300 Fundamentals of Mathematics 1

Chapter 1 Introductory Information and Review

Section 1.1: Numbers

Types of Numbers

Order on a Number Line

Types of Numbers

Natural Numbers:

CHAPTER 1 Introductory Information and Review

University of Houston Department of Mathematics 2

Example:

Solution:

Even/Odd Natural Numbers:

SECTION 1.1 Numbers

Whole Numbers:

Example:

Solution:

Integers:

Example:

Solution:

Even/Odd Integers:

Example:

Solution:

SECTION 1.1 Numbers

Rational Numbers:

Example:

Solution:

Irrational Numbers:

SECTION 1.1 Numbers

Real Numbers:

Example:

Solution:

Note About Division Involving Zero:

Additional Example 1:

Solution:

SECTION 1.1 Numbers

Solution:

Natural Numbers:

Whole Numbers:

Integers:

Prime/Composite Numbers:

Positive/Negative Numbers:

Even/Odd Numbers:

Rational Numbers:

SECTION 1.1 Numbers

Solution:

Natural Numbers:

Whole Numbers:

Integers:

Prime/Composite Numbers:

Positive/Negative Numbers:

Even/Odd Numbers:

Rational Numbers:

SECTION 1.1 Numbers

Solution:

SECTION 1.1 Numbers

Order on a Number Line

The Real Number Line:

Example:

Solution:

SECTION 1.1 Numbers

Inequality Symbols:

The following table describes additional inequality symbols.

Example:

Solution:

Example:

Solution:

Example:

Solution:

SECTION 1.1 Numbers

Solution:

SECTION 1.1 Numbers

Solution:

Exercise Set 1.1: Numbers

State whether each of the following numbers is prime,

composite, or neither. If composite, then list all the

factors of the number.

1. (a) 8 (b) 5 (c) 1

(d) 7 (e) 12

2. (a) 11 (b) 6 (c) 15

(d) 0 (e) 2

Answer the following.

3. In (a)-(e), use long division to change the

following fractions to decimals.

9 (b) 2

9 (c) 3

9 (e) 5

9 Note: 3 1

Notice the pattern above and use it as a

shortcut in (f)-(m) to write the following

fractions as decimals without performing

long division.

9 (g) 7

9 (h) 8

9 (j) 10

9 (k) 14

(l) 25

9 (m) 29

9 Note: 6 2

4. Use the patterns from the problem above to

change each of the following decimals to either a

proper fraction or a mixed number.

(a) 0.4 (b) 0.7 (c) 2.3

(d) 1.2 (e) 4.5 (f) 7.6

State whether each of the following numbers is

rational or irrational. If rational, then write the

number as a ratio of two integers. (If the number is

already written as a ratio of two integers, simply

rewrite the number.)

5. (a) 0.7 (b) 5 (c) 3

(d) 5 (e) 16 (f) 0.3

(g) 12 (h) 2.3

3.5 (i) e

(j) 4 (k) 0.04004000400004...

6. (a) (b) 0.6 (c) 8

(d) 1.3

4.7 (e)

5 (f) 9

(g) 3.1 (h) 10 (i) 0

9 (k) 0.03003000300003…

Circle all of the words that can be used to describe

each of the numbers below.

Even Odd Positive Negative

Prime Composite Natural Whole

Integer Rational Irrational Real

Undefined

8. 0.7

Undefined

11. Which elements of the set

48, 2.1, 0.4, 0, 7, , , 5, 12 belong

to each category listed below?

(a) Even (b) Odd

(c) Positive (d) Negative

(e) Prime (f) Composite

(g) Natural (h) Whole

(i) Integer (j) Real

(k) Rational (l) Irrational

(m) Undefined

12. Which elements of the set

4 56.25, 4 , 3, 5, 1, , 1, 2, 10

belong to each category listed below?

(a) Even (b) Odd

(c) Positive (d) Negative

(e) Prime (f) Composite

(g) Natural (h) Whole

(i) Integer (j) Real

(k) Rational (l) Irrational

(m) Undefined

Fill in each of the following tables. Use “Y” for yes if

the row name applies to the number or “N” for no if it

does not.

10 55 13.3

Undefined

Natural

Integer

Rational

Irrational

Composite

2.36 0

Undefined

Natural

Integer

Rational

Irrational

Composite

Answer the following. If no such number exists, state

“Does not exist.”

15. Find a number that is both prime and even.

16. Find a rational number that is a composite

number.

17. Find a rational number that is not a whole

number.

18. Find a prime number that is negative.

19. Find a real number that is not a rational number.

20. Find a whole number that is not a natural

number.

21. Find a negative integer that is not a rational

number.

22. Find an integer that is not a whole number.

23. Find a prime number that is an irrational number.

24. Find a number that is both irrational and odd.

Answer True or False. If False, justify your answer.j

25. All natural numbers are integers.

26. No negative numbers are odd.

27. No irrational numbers are even.

28. Every even number is a composite number.

29. All whole numbers are natural numbers.

30. Zero is neither even nor odd.

31. All whole numbers are integers.

32. All integers are rational numbers.

33. All nonterminating decimals are irrational

numbers.

34. Every terminating decimal is a rational number.

35. List the prime numbers less than 10.

36. List the prime numbers between 20 and 30.

37. List the composite numbers between 7 and 19.

38. List the composite numbers between 31 and 41.

39. List the even numbers between 13 and 97 .

40. List the odd numbers between 29 and 123 .

Fill in the appropriate symbol from the set , , .

41. 7 ______ 7

42. 3 ______ 3

43. 7 ______ 7

44. 3 ______ 3

45. 81 ______ 9

46. 5 ______ 25

47. 5.32 ______53

100______ 0.07

3 ______

6 ______

3 ______

6 ______

53. 15 ______ 4

54. 7 ______ 49

55. 3 ______ 9

56. 29 ______ 5

57. Find the additive inverse of the following

numbers. If undefined, write “undefined.”

(a) 3 (b) 4 (c) 1

(d) 23

(e) 37

58. Find the multiplicative inverse of the following

(a) 3 (b) 4 (c) 1

(d) 23

(e) 37

59. Find the multiplicative inverse of the following

(a) 2 (b) 59

(d) 35

1 (e) 1

60. Find the additive inverse of the following

(a) 2 (b) 59

(d) 35

1 (e) 1

61. Place the correct number in each of the following

blanks:

(a) The sum of a number and its additive

inverse is _____. (Fill in the correct

number.)

(b) The product of a number and its

multiplicative inverse is _____. (Fill in the

correct number.)

62. Another name for the multiplicative inverse is

the ____________________.

Order the numbers in each set from least to greatest

and plot them on a number line.

(Hint: Use the approximations 2 1.41 and

3 1.73 .)

63. 0 9

1, 2, 0.4, , , 0.495 4

3 ,1 , 0.65 , , 1.5 , 0.643

Section 1.2: Integers

Operations with Integers

Absolute Value:

SECTION 1.2 Integers

Addition of Integers:

Example:

Solution:

Subtraction of Integers:

Example:

Solution:

Multiplication of Integers:

Example:

Solution:

Division of Integers:

Example:

Solution:

Exercise Set 1.2: Integers

Evaluate the following.

1. (a) 3 7 (b) 3 ( 7) (c) 3 7

(d) 3 ( 7) (e) 3 0

2. (a) 8 5 (b) 8 5 (c) 8 ( 5)

(d) 8 ( 5) (e) 0 ( 5)

3. (a) 0 4 (b) 4 0 (c) 0 ( 4)

(d) 4 0

4. (a) 6 0 (b) 0 ( 6) (c) 0 6

(d) 6 0

5. (a) 10 2 (b) 10 ( 2) (c) 10 2

(d) 2 ( 10) (e) 2 ( 10) (f) 2 10

(g) 2 10 (h) 10 ( 2)

6. (a) 7 ( 9) (b) 7 9 (c) 7 9

(d) 9 ( 7) (e) 9 ( 7) (f) 9 7

(g) 7 ( 9) (f) 9 7

7. (a) 1(4) ____ 0 (b) 7( 2) ____ 0

(c) 5( 1)( 2) ____ 0 (d) 3( 1)(0) ____ 0

8. (a) 3( 2) ____ 0 (b) 7( 1) ____ 0

(c) 5(0)( 2) ____ 0 (d) 2( 2)( 2) ___ 0

Evaluate the following. If undefined, write

“Undefined.”

9. (a) 6(0) (b) 6

(d) 6( 1) (e) 6(1) (f) 6( 1)

(g) 6( 1) (h) 6

(k) 6( 1)( 1) (l)

10. (a) 1(7) (b) 7

(c) 7( 1)

(d) 0( 7) (e) 1( 7) (f) 0

(j) 7( 1)( 1) (k) 7(0)( 1) (l) 7

11. (a) 10( 2) (b) 10

(c) 10(2)

(d) 10

12. (a) 6

(b) 6( 3) (c)

(d) 6(3) (e) 6( 3) (f) 6

13. (a) 2( 3)( 4) (b) ( 2)( 3)( 4)

(c) 1( 2)( 3)( 4)

(d) 1(2)( 3)( 4)

14. (a) 3( 2)(5) (b) 3( 2)(5)

(c) 3( 2)( 1)(5)

(d) 3( 2)( 2)( 5)

15. (a) 8 2 (b) 8 ( 2) (c) 8( 2)

(e) 8 ( 2) (f) ( 8)(0)

(g) 8( 1) (h) 8 1 (i) 8

(j) 0 8 (k) 2 ( 8) (l) 0

0 (o) 2 8

16. (a) 12

3 (b) 12( 3) (c) 12 3

(d) 3 12 (e) 0( 3) (f) 0 ( 3)

(g) ( 3)(12) (h) 12

(k) 1 ( 3) (l) 1(12)

3 (n) 3 ( 1) (o) 3(1)

Section 1.3: Fractions

Greatest Common Divisor and Least Common Multiple

Addition and Subtraction of Fractions

Multiplication and Division of Fractions

Greatest Common Divisor and Least Common Multiple

Greatest Common Divisor:

SECTION 1.3 Fractions

A Method for Finding the GCD:

Least Common Multiple:

A Method for Finding the LCM:

Example:

Solution:

The LCM is

Solution:

The LCM is 2 2 2 3 5 120 .

Solution:

The LCM is 2 3 3 5 7 630 .

Solution:

The LCM is 2 2 3 3 2 72 .

Solution:

The LCM is 2 3 3 2 5 180 .

Addition and Subtraction of Fractions

Addition and Subtraction of Fractions with Like Denominators:

a b a b

Example:

Solution:

Addition and Subtraction of Fractions with Unlike

Denominators:

Example:

Solution:

(b) We must rewrite the given fractions so that they have a common denominator.

Find the LCM of the denominators 14 and 21 to find the least common denominator.

Solution:

Multiplication and Division of Fractions

Multiplication of Fractions:

Example:

Solution:

Division of Fractions:

Example:

Solution:

Exercise Set 1.3: Fractions

For each of the following groups of numbers,

(a) Find their GCD (greatest common divisor).

(b) Find their LCM (least common multiple).

1. 6 and 8

2. 4 and 5

3. 7 and 10

4. 12 and 15

5. 14 and 28

6. 6 and 22

7. 8 and 20

8. 9 and 18

9. 18 and 30

10. 60 and 210

11. 16, 20, and 24

12. 15, 21, and 27

Change each of the following improper fractions to a

mixed number.

13. (a) 97

(b) 23

14. (a) 103

(b) 17

15. (a) 274

(b) 32

11 (c)

16. (a) 1513

(b) 43

Change each of the following mixed numbers to an

improper fraction.

17. (a) 16

5 (b) 49

7 (c) 23

18. (a) 12

3 (b) 78

10 (c) 35

19. (a) 57

2 (b) 23

5 (c) 14

20. (a) 19

4 (b) 45

11 (c) 37

Evaluate the following. Write all answers in simplest

form. (If the answer is a mixed number/improper

fraction, then write the answer as a mixed number.)

21. (a) 2 17 7 (b)

11 11 11

22. (a) 3 1

5 5 (b)

23. (a) 4 15 5

8 2 (b) 7 23

24. (a) 3 21

5 5 (b) 6 2

11 117 5

25. (a) 3 14 4

5 2 (b) 3 45 5

26. (a) 5 37 7

9 2 (b) 511

27. (a) 23

7 (b) 3 910 10

28. (a) 7 1112 12

6 2 (b) 516 6

29. (a) 1 1

4 2 (b)

30. (a) 1 1

8 10 (b)

31. (a) 1 1 1

4 5 6 (b)

32. (a) 1 1 1

2 7 5 (b)

33. (a) 1 1

35 10 (b)

Exercise Set 1.3: Fractions

34. (a) 1 1

6 24 (b)

35. (a) 3 17 6

4 5 (b) 7 110 2

36. (a) 5 17 4

10 3 (b) 3112 8

37. (a) 3 45 7

7 8 (b) 4 29 3

38. (a) 514 6

7 3 (b) 7 138 24

39. (a) 7215 12

5 2 (b) 7 516 6

40. (a) 9 510 8

7 6 (b) 5 314 4

fraction, then write the answer as an improper

fraction.)

41. (a) 2 3

9 4 (b)

42. (a) 7 9

16 10 (b)

43. (a) 13

5 (b) 23

44. (a) 25

9 (b) 27

fraction, then write the answer as an improper

fraction.)

45. (a) 1

53 (b)

46. (a) 3

87 (b)

18 (c)

47. (a) 1 25

7 11 (b)

48. (a) 36 1

(b) 8 7

19 3 (c)

49. (a) 1

43 (c)

50. (a) 3

51. (a) 12 18

35 7 (b)

52. (a)

(b) 36 9

5 50 (c)

53. (a) 1045 77

8 (b) 7 98 10

54. (a) 329 4

2 (b) 7 416 5

55. (a) 1 1732 5 (b) 3 3

5 116 2

56. (a) 1 17 4

3 5 (b) 3 115 1225

57. (a) 5 18 4

5 2 (b) 1719 18

58. (a) 545 7

4 1 (b) 5 111 22

Section 1.4: Exponents and Radicals

Evaluating Exponential Expressions

Square Roots

Evaluating Exponential Expressions

Two Rules for Exponential Expressions:

Example:

SECTION 1.4 Exponents and Radicals

Solution:

Example:

Solution:

Additional Properties for Exponential Expressions:

Two Definitions:

Quotient Rule for Exponential Expressions:

Exponential Expressions with Bases of Products:

Exponential Expressions with Bases of Fractions:

Example:

Evaluate each of the following:

(a) 32 (b)

Solution:

Square Roots

Definitions:

Two Rules for Square Roots:

Writing Radical Expressions in Simplest Radical Form:

Example:

Solution:

Example:

Solution:

Exponential Form:

Solution:

Exercise Set 1.4: Exponents and Radicals

Write each of the following products instead as a base

and exponent. (For example, 26 6 6 )

1. (a) 7 7 7 (b) 10 10

(c) 8 8 8 8 8 8 (d) 3 3 3 3 3 3 3

2. (a) 9 9 9 (b) 4 4 4 4 4

(c) 5 5 5 5 (d) 17 17

3. 27 ______ 0

9 ______ 0

8 ______ 0

6. 68 ______ 0

7. 210 ______ 2

8. 310 ______ 3

9. (a) 13 (b)

23 (c) 33

(d) 13 (e)

23 (f) 33

3 (h) 2

3 (i) 3

(j) 03 (k)

03 (l) 0

(m) 43 (n)

43 (o) 4

10. (a) 05 (b)

05 (c)

(d) 15 (e)

15 (f)

(g) 25 (h)

25 (i)

(j) 35 (k)

35 (l)

(m) 45 (n)

45 (o)

11. (a) 2

0.5 (b)

12. (a) 2

0.03 (b)

Write each of the following products instead as a base

and exponent. (Do not evaluate; simply write the base

and exponent.) No answers should contain negative

exponents.

13. (a) 2 65 5 (b)

2 65 5

14. (a) 8 53 3 (b)

8 53 3

15. (a) 9

16. (a) 9

17. (a) 7 3

18. (a) 12

8 8 (b)

19. (a) 6

37 (b) 3

20. (a) 4

23 (b) 4

Rewrite each expression so that it contains positive

exponent(s) rather than negative exponent(s), and then

evaluate the expression.

21. (a) 15

(b) 25

(c) 35

22. (a) 13

(b) 23

(c) 33

23. (a) 32 (b) 52

24. (a) 27

(b) 410

25. (a)

26. (a)

27. (a) 25 (b)

28. (a) 2

(b) 28

29. (a) 3

30. (a) 1

31. (a) 2

032 (b)

32. (a) 2

Simplify the following. No answers should contain

negative exponents.

33. (a) 3

3 4 23x y z (b) 3

3 4 23x y z

34. (a) 2

5 3 46x y z (b) 2

5 3 46x y z

13 4 6

44 3 7

39. 4 3

40. 7 0

Write each of the following expressions in simplest

radical form or as a rational number (if appropriate).

If it is already in simplest radical form, say so.

45. (a) 1236 (b) 7 (c) 18

46. (a) 20 (b) 49 (c) 1232

47. (a) 1250 (b) 14 (c)

48. (a) 1219 (b)

49 (c) 55

49. (a) 28 (b) 72 (c) 1227

50. (a) 1245 (b) 48 (c) 500

51. (a) 54 (b) 1280 (c) 60

52. (a) 120 (b) 180 (c) 1284

53. (a) 1

54. (a) 1

55. (a) 7

10 (c)

56. (a) 1

57. (a) 53 (b) 4 5 7x y z

58. (a) 72 (b) 2 9 5a b c

59. (a) 2

5 (b) 4

6 (c) 6

60. (a) 2

7 (b) 4

3 (c) 6

We can evaluate radicals other than square roots.

With square roots, we know, for example, that

49 7 , since 2

7 49 , and 49 is not a real

number. (There is no real number that when squared

gives a value of 49 , since 27 and

27 give a value

of 49, not 49 . The answer is a complex number,

which will not be addressed in this course.) In a

similar fashion, we can compute the following:

Cube Roots 3 125 5 , since

35 125 .

3 125 5 , since 3

5 125 .

Fourth Roots 4 10,000 10 , since

410 10,000 .

4 10,000 is not a real number.

Fifth Roots 5 32 2 , since

52 32 .

5 32 2 , since 5

2 32 .

Sixth Roots

1 1664 2 , since

is not a real number.

Evaluate the following. If the answer is not a real

number, state “Not a real number.”

61. (a) 64 (b) 64 (c) 64

62. (a) 25 (b) 25 (c) 25

63. (a) 3 8 (b) 3 8 (c) 3 8

64. (a) 4 81 (b) 4 81 (c) 4 81

65. (a) 6 1,000,000 (b) 6 1,000,000

(c) 6 1,000,000

66. (a) 5 32 (b) 5 32 (c) 5 32

67. (a) 1416

(b) 1416

(c) 1416

68. (a) 1327

(b) 1327

(c) 1327

69. (a) 15100,000

(b) 15100,000

(c) 15100,000

70. (a) 6 1 (b) 6 1 (c) 6 1

SECTION 1.5 Order of Operations

Section 1.5: Order of Operations

Evaluating Expressions Using the Order of Operations

Rules for the Order of Operations:

1) Operations that are within parentheses and other grouping symbols are performed

first. These operations are performed in the order established in the following steps.

If grouping symbols are nested, evaluate the expression within the innermost

grouping symbol first and work outward.

2) Exponential expressions and roots are evaluated first.

3) Multiplication and division are performed next, moving left to right and performing

these operations in the order that they occur.

4) Addition and subtraction are performed last, moving left to right and performing

these operations in the order that they occur.

Upon removing all of the grouping symbols, repeat the steps 2 through 4 until the

final result is obtained.

Example:

Solution:

Example:

Solution:

SECTION 1.5 Order of Operations

Solution:

Exercise Set 1.5: Order of Operations

1. In the abbreviation PEMDAS used for order of

operations,

(a) State what each letter stands for:

P: ____________________

E: ____________________

M: ____________________

D: ____________________

A: ____________________

S: ____________________

(b) If choosing between multiplication and

division, which operation should come first?

(Circle the correct answer.)

Multiplication

Division

Whichever appears first

(c) If choosing between addition and

subtraction, which operation should come

first? (Circle the correct answer.)

Addition

Subtraction

Whichever appears first

2. When performing order of operations, which of

the following are to be viewed as if they were

enclosed in parentheses? (Circle all that apply.)

Absolute value bars

Radical symbols

Fraction bars

3. (a) 3 4 5 (b) (3 4) 5

(c) 3 4 5 (d) (3 4) 5

(e) 3 4 5 (f) 3 (4 5)

4. (a) 10 6 7 (b) (10 6) 7

(c) 10 6(7) (d) 10(6 7)

(e) 7 10 6 (f) 7 (10 6)

5. (a) 3 7 (b) 7 3

(c) 3 7 (d) 7 3

6. (a) 2 5 (b) 2 5

(c) 2 5 (d) 2 5

7. (a) 2 7 5 (b) 2 (7 5)

(c) 2 ( 7) 5 (d) 2 7( 5)

(e) 2(7 ( 5)) (f) 2(7) 5 7

8. (a) 6 2 ( 4) (b) 6 2 ( 4)

(c) 6 2( 4) (d) ( 6 2)( 4)

(e) 2 ( 6) 4 (f) 2 4( 6 2)

9. (a) 2 1 1

5 3 4 (b)

(c) 2 1 1 1

5 3 4 4

10. (a) 3 5

(c) 3 5

11. (a) 2

5 4 7 (b) 2

(c) 5 1 4 7 (d) 2

7 4 1 5

(e) 2 25 1 (f)

12. (a) 22 3 (b)

(c) 2 3(1 4) (d) 3

( 2 3) 1 4

(e) 2 22 3 (f)

13. (a) 20 2(10) (b) 20 2 10

(c) 20 10 ( 2) 10 5

14. (a) 24 4( 2) (b) (24 4) 2

(c) 24( 2) 4 2( 2)

15. (a) 210 5 2 (b)

210 5 2

(c) 22 10 2 5 5

16. (a) (3 9) 3 4 (b) 3 (9 3) 4

(c) 33 9 3 4

17. (a) 1

18. (a) 1

19. 1 17 4 5

20. 1 18 3 7

21. 2 47 5 2 3

22. 3 23 2 3 4

23. 1 1 3

24. 3 3 10

5 10 3

25. 25

26. 16

3 2 16

27. 2 3 4 1

28. 2 3 4 1

29. 2 3 4 1

30. 2 3 4 1

2 3 4 1

33. 3 7 7 3

12 2 3 3

3 52 4 1 1

5 12 6 3

35. 281 2 4 3 2

36. 2 364 5 4 2

37. 2 24 121 5 4 3

38. 2 2144 5 2 6 12 3

39. 249 3 2

40. 23 49 2

41. 29 16 1

42. 29 16 1

222 3 5

2 8 2 4

22 3 5

2 8 2 4

2 2 3 2 4

5 3 3 7 2 4 1

4 2 2 1 81 2 3

5 2 25 2 2 2 3 3

81 16 2 1 3 1 1 4 2

Evaluate the following expressions for the given values

of the variables.

for 5, 1, and 7P r k .

48. x y

y z for 4, 3, and 8x y z .

8b b c

for 4b and 2c .

50. 2 4

b b ac

for 1, 3, and 18a b c .

Section 1.6: Solving Linear Equations

Linear Equations

Rules for Solving Equations:

Linear Equations:

Example:

SECTION 1.6 Solving Linear Equations

Solution:

Example:

Solution:

Exercise Set 1.6: Solving Linear Equations

Solve the following equations algebraically.

1. 5 12x

2. 8 9x

3. 4 7x

4. 2 8x

5. 6 30x

6. 4 28x

7. 6 10x

8. 8 26x

9. 1373 x

10. 6115 x

11. 7432 xx

12. 6425 xx

13. 3)8(59)2(3 xx

14. 3)4(25)3(4 xx

15. )37(4)52(3 xx

16. )51(648327 xx

17. 75

18. 103

23. 7152 x

24. 2743 x

25. 1)7(52

26. 3)12(1261

27. xxx

28. 12

Section 1.7: Interval Notation and Linear Inequalities

Linear Inequalities

Rules for Solving Inequalities:

SECTION 1.7 Interval Notation and Linear Inequalities

Interval Notation:

Example:

Solution:

Example:

Solution:

Example:

Solution:

Exercise Set 1.7: Interval Notation and Linear Inequalities

For each of the following inequalities:

(a) Write the inequality algebraically.

(b) Graph the inequality on the real number line.

(c) Write the inequality in interval notation.

1. x is greater than 5.

2. x is less than 4.

3. x is less than or equal to 3.

4. x is greater than or equal to 7.

5. x is not equal to 2.

6. x is not equal to 5 .

7. x is less than 1.

8. x is greater than 6 .

9. x is greater than or equal to 4 .

10. x is less than or equal to 2 .

11. x is not equal to 8 .

12. x is not equal to 3.

13. x is not equal to 2 and x is not equal to 7.

14. x is not equal to 4 and x is not equal to 0.

Write each of the following inequalities in interval

notation.

15. 3x

16. 5x

17. 2x

18. 7x

19. 53 x

20. 27 x

21. 7x

22. 9x

Write each of the following inequalities in interval

notation.

Given the set 31,3,4,2 S , use substitution to

determine which of the elements of S satisfy each of

the following inequalities.

29. 1052 x

30. 1424 x

31. 712 x

32. 013 x

33. 1012 x

For each of the following inequalities:

(a) Solve the inequality.

(b) Graph the solution on the real number line.

(c) Write the solution in interval notation.

35. 102 x

36. 243 x

Exercise Set 1.7: Interval Notation and Linear Inequalities

37. 305 x

38. 404 x

39. 1152 x

40. 1743 x

41. 2038 x

42. 010 x

43. 47114 xx

44. 7395 xx

45. 62710 xx

46. xx 5648

47. 1485 xx

48. 9810 xx

49. )7(2)54(3 xx

50. )20()23(4 xx

51. )5(21

52. xx 1031

53. 82310 x

54. 13329 x

55. 17734 x

56. 34519 x

57. 54

58. 35

Which of the following inequalities can never be true?

59. (a) 95 x

(b) 59 x

(c) 73 x

(d) 35 x

60. (a) 53 x

(b) 18 x

(c) 82 x

(d) 107 x

61. You go on a business trip and rent a car for $75

per week plus 23 cents per mile. Your employer

will pay a maximum of $100 per week for the

rental. (Assume that the car rental company

rounds to the nearest mile when computing the

mileage cost.)

(a) Write an inequality that models this

situation.

(b) What is the maximum number of miles

that you can drive and still be

reimbursed in full?

62. Joseph rents a catering hall to put on a dinner

theatre. He pays $225 to rent the space, and pays

an additional $7 per plate for each dinner served.

He then sells tickets for $15 each.

(a) Joseph wants to make a profit. Write an

inequality that models this situation.

(b) How many tickets must he sell to make

a profit?

63. A phone company has two long distance plans as

follows:

Plan 1: $4.95/month plus 5 cents/minute

Plan 2: $2.75/month plus 7 cents/minute

How many minutes would you need to talk each

month in order for Plan 1 to be more cost-

effective than Plan 2?

64. Craig’s goal in math class is to obtain a “B” for

the semester. His semester average is based on

four equally weighted tests. So far, he has

obtained scores of 84, 89, and 90. What range of

scores could he receive on the fourth exam and

still obtain a “B” for the semester? (Note: The

minimum cutoff for a “B” is 80 percent, and an

average of 90 or above will be considered an

“A”.)

Section 1.8: Absolute Value and Equations

Absolute Value

Equations of the Form |x| = C:

Special Cases for |x| = C:

Example:

SECTION 1.8 Absolute Value and Equations

Solution:

Example:

Solution:

Example:

Solution:

Example:

Solution:

Example:

Solution:

Exercise Set 1.8: Absolute Value and Equations

Solve the following equations.

4. 10x

5. 122 x

6. 303 x

7. 54 x

8. 27 x

9. 4 5x

10. 7 2x

11. 843 x

12. 345 x

13. 3 4 8x

14. 5 4 3x

15. 1732 x

16. 31

17. 10734 x

18. 2825 x

19. 115123 x

20. 46922 x

21. 1131421 x

22. 875 x

23. 1523 xx

24. 674 xx

CHAPTER 2 Points, Lines, and Functions

Chapter 2 Points, Lines, and Functions

Section 2.1: An Introduction to the Coordinate Plane

Points in the Coordinate Plane

The Rectangular Coordinate System:

SECTION 2.1 An Introduction to the Coordinate Plane

Plotting Points in the Coordinate Plane:

Example:

Solution:

Graphing Horizontal and Vertical Lines:

Example:

Solution:

Graphing Other Lines:

Example:

Solution:

(c) Draw a line through the points.

Solution:

Exercise Set 2.1: An Introduction to the Coordinate Plane

Plot the following points in a coordinate plane.

1. A(3, 4)

2. B(2, -5)

3. C(-3, -1)

4. D(-4, -6)

5. E(-5, 0)

6. F(0, -2)

Write the coordinates of each of the points shown in

the figure below. Then identify the quadrant or axis

in which the point is located.

Plot each of the following sets of points in a coordinate

plane. Then identify the quadrant or axis in which

each point is located.

13. (a) A(2, 5)

(b) B(-2, -5)

(c) C(2, -5)

(d) D(-2, 5)

14. (a) A(4, -3)

(b) B(-4, -3)

(c) C(-4, 3)

(d) D(4, 3)

15. (a) A(0, -2)

(b) B(-2, 0)

(c) C(2, 0)

(d) D(0, 2)

16. (a) A(-3, 0)

(b) B(3, 0)

(c) (0, -3)

(d) D(0, 3)

17. If the point (a, b) is in Quadrant I, identify the

quadrant of each of the following points:

(a) (-a, -b) (b) (-a, b) (c) (a, a)

18. If the point (a, b) is in Quadrant I, identify the

(a) (-b, a) (b) (b, b) (c) (-b, -a)

19. If the point (a, b) is in Quadrant II, then 0a <

and 0b > . Identify the quadrant of each of the

following points:

(a) (-a, -b) (b) (b, a) (c) (a, -b)

20. If the point (a, b) is in Quadrant III, then 0a <

and 0b < . Identify the quadrant of each of the

following points:

(a) (-a, b) (b) (b, a) (c) (-a, -b)

21. If the point (a, b) is in Quadrant IV, identify the

(a) (b, -b) (b) (-a, -a) (c) (b, a)

22. If the point (a, b) is in Quadrant II, identify the

(a) (-a, b) (b) (b, b) (c) (a, -a)

23. If the point (a, b) is in Quadrant III, identify the

axis on which each of the following points lies:

(a) (a, 0) (b) (0, b) (c) (-b, 0)

24. If the point (a, b) is in Quadrant IV, identify the

axis on which each of the following points lies:

(a) (0, -b) (b) (-a, 0) (c) (b, 0)

Answer True or False.

25. The point (0, 5) is on the x-axis.

26. The point (-4, 0) is in Quadrant II.

27. The point (1, -3) is in Quadrant IV.

28. The point (-2, -5) is in Quadrant III.

29. The point (0, 0) is in Quadrant I.

30. The point (-6, 1) is in Quadrant IV.

31. If the point (a, b) is in Quadrant IV, then 0b < .

32. If the point (a, b) is in Quadrant II, then 0a > .

33. If the point (a, b) is in Quadrant I, then the point

(b, a) is also in Quadrant I.

−4 −2 2 4 6

34. If the point (a, b) is in Quadrant I, then the point

(a, -b) is in Quadrant II.

35. If the point (a, b) is in Quadrant II, then the point

(-a, -b) is in Quadrant III .

36. If the point (a, b) is in Quadrant IV, then the

point (-b, a) is in Quadrant I.

37. If the point (a, b) is in Quadrant III, then 0b > .

38. If the point (a, b) is on the y-axis, then 0a > .

39. If the point (a, b) is on the y-axis, then 0b > .

40. If the point (a, b) is on the y-axis, then 0a = .

41. If the point (a, b) is on the y-axis, then the point

(b, a) is on the x-axis.

42. If the point (a, b) is on the x-axis, then the point

(a, 3) lies in Quadrant I .

43. Given the following points:

A(3, 5), B(3, 1), C(3, 0), D(3, -2)

(a) Plot the above points on a coordinate plane.

(b) What do the above points have in common?

(c) Draw a line through the above points.

(d) What is the equation of the line drawn in

part (c)?

A(-3, 4), B(0, 4), C(1, 4), D(3, 4)

(b) What do the above points have in common?

(c) Draw a line through the above points.

(d) What is the equation of the line drawn in

part (c)?

45. (a) List four points that are on the x-axis.

(b) Analyze the coordinates of the points you

have listed. What do they have in common?

(c) Give the equation of the x-axis.

46. (a) List four points that are on the y-axis.

(b) Analyze the coordinates of the points you

have listed. What do they have in common?

(c) Give the equation of the y-axis.

47. Graph the line 2x = .

48. Graph the line 5y = − .

49. Graph the line 4y = .

50. Graph the line 3x = − .

51. On the same set of axes, graph the lines 1x = −

and 3y = .

52. On the same set of axes, graph the lines 5x =

and 2y = − .

53. On the same set of axes, graph the lines 72

and 0y = .

54. On the same set of axes, graph the lines 0x =

and 52

y = − .

Graph the following lines by first completing the table

and then plotting the points on a coordinate plane.

55. 3 2y x= +

56. 2 5y x= − +

57. 4 7y x= − +

58. 5 1y x= −

59. Graph the line segment with endpoints (-7, 0)

and (0, 7).

60. Graph the line segment with endpoints (3, 5) and

and (-5, -3).

61. Graph the line segment with endpoints (1, -4)

and (-1, 4)

62. Graph the line segment with endpoints (-2, 6)

and (6, 2).

Section 2.2: The Distance and Midpoint Formulas

The Distance Formula

The Midpoint Formula

The Distance Formula

Finding the Distance Between Two Points:

Example:

SECTION 2.2 The Distance and Midpoint Formulas

Solution:

Use the Pythagorean Theorem to determine c.

The Midpoint Formula

Finding the Midpoint of a Line Segment:

Example:

Solution:

Exercise Set 2.2: The Distance and Midpoint Formulas

Use the Pythagorean Theorem to find the missing side

of each of the following triangles.

Pythagorean Theorem: In a right triangle, if a and b

are the measures of the legs, and c is the measure of the hypotenuse, then a2 + b2 = c2.

(1, 2)A and (4, 7)B

(b) Draw segment AB. This will be the

hypotenuse of triangle ABC.

(c) Find a point C such that triangle ABC is a

right triangle. Draw triangle ABC.

(d) Use the Pythagorean theorem to find the

distance between A and B (the length of the

hypotenuse of the triangle).

( 3, 1)A and (1, 5)B

(b) Draw segment AB. This will be the

hypotenuse of triangle ABC.

(c) Find a point C such that triangle ABC is a

right triangle. Draw triangle ABC.

(d) Use the Pythagorean theorem to find the

distance between A and B (the length of the

hypotenuse of the triangle).

Use the distance formula to find the distance between

the two given points. (You can also use the method from

the previous two problems to double-check your answer.)

7. (3, 6) and (5, 9)

8. (4, 7) and (2, 3)

9. ( 5, 0) and ( 2, 6)

10. (9, 4) and (2, 3)

11. (4, 0) and (0, 7)

12. ( 4, 8) and ( 10, 1)

13. 12

5, and 56

14. 23

, 1 and 34

Find the midpoint of the line segment joining points A

and B.

15. (7, 6)A and (3, 8)B

16. (5, 9)A and (1, 3)B

17. ( 7, 0)A and ( 4, 8)B

18. (7, 5)A and (4, 3)B

19. (3, 0)A and (0, 9)B

20. ( 6, 7)A and ( 10, 6)B

21. 13

, 5A and 35

22. 12

3,A and 56

Exercise Set 2.2: The Distance and Midpoint Formulas

23. (a) Graph the line segment with endpoints

( 2, 6)A and (5, 4)B .

(b) Find the distance from A to B.

(c) Find the midpoint of AB .

24. (a) Graph the line segment with endpoints

(4, 0)A and ( 2, 5)B .

(b) Find the distance from A to B.

(c) Find the midpoint of AB .

25. If (4, 7)M is the midpoint of the line segment

joining points A and B, and A has coordinates

(2, 3) , find the coordinates of B.

joining points A and B, and A has coordinates

(1, 6) , find the coordinates of B.

joining points A and B, and B has coordinates

( 1, 2) ,

(a) Find the coordinates of A.

(b) Find the length of AB .

28. If ( 2, 1)M is the midpoint of the line segment

joining points A and B, and B has coordinates

( 5, 3) ,

(a) Find the coordinates of A.

(b) Find the length of AB .

29. Determine which of the following points is

closer to the origin: (5, 6)A or ( 3, 7)B ?

30. Determine which of the following points is

closer to the point (4, 1) : ( 2, 3)A or

(6, 6)B ?

31. A circle has a diameter with endpoints

( 5, 9)A and (3, 5)B .

(a) Find the coordinates of the center of the

circle.

(b) Find the length of the radius of the circle.

32. A circle has a diameter with endpoints (2, 7)A

and (8, 1)B .

(a) Find the coordinates of the center of the

circle.

(b) Find the length of the radius of the circle.

Section 2.3: Slope and Intercepts of Lines

The Slope of a Line

Intercepts of Lines

The Slope of a Line

Finding the Slope of a Line:

SECTION 2.3 Slope and Intercepts of Lines

Example:

Solution:

Intercepts of Lines

Finding Intercepts of Lines:

Horizontal Lines:

Vertical Lines:

Example:

Solution:

Example:

Solution:

Exercise Set 2.3: Slope and Intercepts of Lines

State whether the slope of each of the following lines is

positive, negative, zero, or undefined.

Find the slope of the line that passes through the

following points. If undefined, state „Undefined.‟

7. )7,3( and )0,0(

8. (8, 0) and (3, 6)

9. )10,4( and )5,2(

10. )9,5( and )3,7(

11. (6, 4) and (2, 4)

12. (5, 1) and (5, 8)

13. )7,6( and )3,2(

14. ( 2, 6) and ( 5, 10)

15. ( 3, 8) and ( 3, 4)

16. )7,1( and )7,8(

17. (2, 8) and (0, 3)

18. (1, 4) and ( 7, 2)

19. 12

, 1 and 2 13 6

20. 34

2, and 515 8

21. 2 47 9

, and 5 16 2

22. 3 75 10

, and 714 8

Find the slope of each of the following lines. If

undefined, state „Undefined.‟

For each of the following:

(a) Complete the given table.

(b) Plot the points on a coordinate plane and

graph the line.

(c) Use two points from the table to find the slope

of the line.

27. 4 1y x

28. 3 2y x

29. 23

30. 35

31. Examine the relationship in numbers 27-30

between each of the equations and the

corresponding slope that you found for each line.

Do you see any pattern? Can you determine the

slope of the line from simply looking at its

equation?

32. Based on the pattern found in the previous

problem, state the slope of the following lines

without graphing the line or performing any

calculations:

(a) 2 9y x

(b) 7 5y x

(c) 45

(d) 37

For each of the following graphs:

(a) State the x-intercept.

(b) State the y-intercept.

(c) State the coordinates of the x-intercept.

(d) State the coordinates of the y-intercept.

(e) Find the slope of the line.

For each of the following equations:

(a) Find the x- and y-intercepts of the line.

(b) State the coordinates of the intercepts.

(c) Plot the x- and y-intercepts on a coordinate

plane.

(d) Graph the line, based on the intercepts.

35. 2 8y x

36. 3 6y x

37. 54 xy

38. 73 xy

39. 2025 yx

40. 2 3 18x y

41. 3 5 30x y

42. 3 24 4x y

43. 2 3 10x y

44. 4 6 9x y

45. 5 3 21 0x y

46. 4 7 8 0x y

47. 2 2 7x y

48. 3 15x

49. 4 12y

50. 4 4 15x y

51. 6 24x

52. 2 14y

For each of the following:

(a) Complete the given table.

(b) Plot the points on a coordinate plane and

graph the line.

(c) Find the x- and y-intercepts of the line.

(d) Find the slope of the line.

53. 2 8y x

54. 3y x

55. Examine the relationship in numbers 53 and 54

between each of the equations and the

corresponding y-intercept that you found for

each line. Do you see any pattern? Can you

determine the y-intercept of the line from simply

looking at its equation?

56. Based on the pattern found in the previous

problem, state the y-intercept of the following

lines without graphing the line or performing any

calculations:

(a) 2 9y x

(b) 7 5y x

(c) 45

(d) 37

Section 2.4: Equations of Lines

Writing Equations of Lines

Different Forms for Equations of Lines:

Example:

Solution:

SECTION 2.4 Equations of Lines

Example:

Solution:

Example:

Solution:

Example:

Solution:

To sketch the graph, begin by using

the y-intercept to plot the point 0,1 .

Solution:

Exercise Set 2.4: Equations of Lines

Write an equation in slope-intercept form for each of

the following lines.

For each of the following equations,

(a) Write the equation in slope-intercept form. (b) Identify the slope and the y-intercept of the

(c) Graph the line.

5. 52 yx

6. 4 0y x

7. 5 1x y

8. 63 yx

9. 04 yx

10. 3 9x y

11. 5 4 12x y

12. 1052 yx

13. 5 2 30 0y x

14. 3 2 8 0x y

15. 5 14 2

16. 2 13 2

Each set of conditions below describes the properties

of a particular line. Using these conditions,

(a) Graph the line.

(b) Write an equation for the line in point-slope

(c) Write an equation for the line in slope-

intercept form. (Do this algebraically, and

then check to see if your result matches your

graph.)

17. Slope 3

2; passes through 6, 4

18. Slope 2

5 ; passes through 4, 3

19. Passes through 8, 2 and 4, 7

Exercise Set 2.4: Equations of Lines

Write an equation in slope-intercept form for the line

that satisfies the given conditions.

21. Slope 4

7 ; y-intercept 3

22. Slope 4 ; y-intercept 5

23. Slope 4

24. Slope 3

25. Slope 9

26. Slope 5

31. x-intercept 7 ; y-intercept 5

32. x-intercept 2 ; y-intercept 6

33. Slope 2

3 ; x-intercept 4

34. Slope 5

1; x-intercept 6

Answer the following, assuming that each situation

can be modeled by a linear equation.

35. If a company can make 21 computers for

$23,000, and can make 40 computers for

$38,200, write an equation that represents the

cost C of x computers.

36. A certain electrician charges a $40 traveling fee,

and then charges $55 per hour of labor. Write an

equation that represents the cost C of a job that

takes x hours.

Section 2.5: Parallel and Perpendicular Lines

Pairs of Lines – Parallel and Perpendicular Lines

Pairs of Lines - Parallel and Perpendicular Lines

Parallel Lines:

Perpendicular Lines:

Two lines with slopes 1m and

2m perpendicular if and only if 1 2 1m m .

SECTION 2.5 Parallel and Perpendicular Lines

Example:

Solution:

Example:

Solution:

Exercise Set 2.5: Parallel and Perpendicular Lines

State whether the following pairs of lines are parallel,

perpendicular, or neither.

1. 3 5y x

3 7y x

5. 2 5y x

2 5y x

6. 5 7y x

7. 2 5 7x y

5 2 6x y

8. 3 4 8x y

3 4 8x y

9. 2 3 5x y

4 6 11x y

10. 5 0x y

11. The line passing through (2, 5) and (7, 9)

The line passing through ( 2, 6) and (2, 1)

12. The line passing through ( 4, 7) and (0, 5)

The line passing through ( 3, 8) and ( 5, 9)

The line passing through (3, 7) and (7, 11)

The line passing through ( 6, 6) and ( 2, 5)

15. 4y

16. 3x

17. 2y

18. 5x

19. The line passing through (4, 5) and ( 1, 5)

The line passing through (2, 3) and (0, 3)

20. The line passing through (2, 6) and (2, 8)

The line passing through ( 3, 4) and (5, 4)

Each set of conditions below describes a particular

line. Using these conditions, write an equation for each

line in the following two forms:

(a) Point-slope form

(b) Slope-intercept form

21. Passes through (4, 7) ; parallel to the line

2 5y x

22. Passes through (4, 7) ; perpendicular to the line

2 5y x

23. Passes through ( 12, 5) ; perpendicular to the

line 6 1y x

24. Passes through ( 12, 5) ; parallel to the line

6 1y x

26. Passes through (3, 7) ; perpendicular to the

line 54

27. Passes through ( 1, 6) ; perpendicular to the line

2 3 7x y

Exercise Set 2.5: Parallel and Perpendicular Lines

Write an equation for the line that satisfies the given

conditions. With the exception of vertical lines, write

all equations in slope-intercept form.

29. Passes through (1, 4) ; parallel to the x-axis

30. Passes through (1, 4) ; parallel to the y-axis

33. Passes through ( 2, 3) ; and is

(a) parallel to the line 23

(b) perpendicular to the line 23

34. Passes through (20, 2) ; and is

(a) parallel to the line 35 xy

(b) perpendicular to the line 35 xy

625 yx

834 yx

37. Passes through (2, 3) ; perpendicular to the line

625 yx

38. Passes through ( 1, 5) ; perpendicular to the

line 834 yx

containing (3, 5) and (2, 1)

containing ( 2, 3) and ( 4, 6)

41. Perpendicular to the line containing ( 3, 5) and

(7, 1) ; passes through the midpoint of the line

segment connecting these points

42. Perpendicular to the line containing (4, 2) and

(10, 4) ; passes through the midpoint of the line

segment connecting these points

Section 2.6: An Introduction to Functions

Definition of a Function

Domain of a Function

Definition of a Function

Definition:

SECTION 2.6 An Introduction to Functions

Defining a Function by an Equation in the Variables x and y:

The Function Notation:

Example:

Solution:

Example:

Solution:

Domain of a Function

Finding the Domain of a Function:

Example:

Solution:

Example:

Solution:

Exercise Set 2.6: An Introduction to Functions

For each of the examples below, determine whether

the mapping makes sense within the context of the

given situation, and then state whether or not the

mapping represents a function.

1. Erik conducts a science experiment and maps the

temperature outside his kitchen window at

various times during the morning.

2. Dr. Kim counts the number of people in

attendance at various times during his lecture this

afternoon.

State whether or not each of the following mappings

represents a function.

Express each of the following rules in function

notation. (For example, “Subtract 3, then square”

would be written as 2( ) ( 3)f x x .)

7. (a) Divide by 7, then add 4

(b) Add 4, then divide by 7

8. (a) Multiply by 2, then square

(b) Square, then multiply by 2

9. (a) Take the square root, then subtract 6 squared

(b) Take the square root, subtract 6, then square

10. (a) Add 4, square, then subtract 2

(b) Subtract 2, square, then add 4

Complete the table for each of the following functions.

11. 3( ) 5f x x

12. 2( ) ( 4) 1g x x

Find the domain of each of the following functions.

Write the domain first as an inequality, and then

express it in interval notation.

( )f xx

x ( )f x

7 9 -3

8 4 -7

Time Temp. (oF)

# of People

( )2 5

( )3 4

21. 4 1

( )4 9

22. 5 7

( )3 7

25. 242)( 2 xxxf

26. xxf 27)(

27. ( ) 3 5g x x

28. 2( ) 16h x x

29. ttf )(

30. 3)( xxh

31. 5)( xxf

32. 7)( xxg

33. 3( ) 5f x x

34. 3( ) 7g x x

35. ( ) 2 9h x x

36. ( ) 3 2h t t

37. ( ) 1 5g x x

38. ( ) 4f x x

39. ( ) 8 5 2f x x

40. ( ) 2 7 4f x x

43. 3( ) 1f t t

44. 3( ) 2 9g x x

45. 31

46. 32 9

( )4 7

47. 5( )h x x

48. 4( )h x x

49. 6( ) 3 5g x x

50. 5( ) 2 7g x x

51. ( )f x x

52. ( ) 2g x x

53. ( ) 2 6H x x

54. ( ) 3 5f x x

( )f xx

59. If 45)( xxf ,

(a) Find (3)f

(b) Find x when ( ) 3f x

(c) Find 12

(d) Find x when 12

( )f x

(e) Find 0f

(f) Find x when ( ) 0f x

60. If ( ) 3 1f x x ,

(a) Find ( 5)f

(b) Find x when ( ) 5f x

(c) Find 34

(d) Find x when 34

( )f x

(e) Find 0f

(f) Find x when ( ) 0f x

61. If ( ) 3h x x ,

(a) Find (1)h

(b) Find x when ( ) 1h x

(c) Find 2h

(d) Find x when ( ) 2h x

(e) Find 7h

(f) Find x when ( ) 7h x

62. If ( ) 7g x x ,

(a) Find (0)g

(b) Find x when ( ) 0g x

(c) Find 2g

(d) Find x when ( ) 2g x

(e) Find 3g

(f) Find x when ( ) 3g x

63. If ( ) 2h x x , find

(a) (7)h

(b) (25)h

(c) 14

64. If ( ) 2h x x , find

(a) (7)h

(b) (25)h

(c) 14

65. If ( ) 3f x x , find

(a) (16)f

(b) (12)f

(c) 9f

66. If ( ) 3f x x , find

(a) (16)f

(b) (12)f

(c) 9f

67. If 2( ) 5 6g x x x ,

(a) Find (3)g

(b) Find 4g

(c) Find 12

(d) Find 0g

68. If 2( ) 2 15h t t t ,

(a) Find (0)h

(b) Find (6)h

(c) Find 5h

(d) Find 23

69. If 3

(a) Find ( 7)f

(b) Find (0)f

(c) Find 5f

(d) Find 3f

(e) Find 2f

70. If 5 2

(a) Find (2)g

(b) Find ( 4)g

(c) Find 52

(d) Find 3g

(e) Find (0)g

SECTION 2.7 Functions and Graphs

Section 2.7: Functions and Graphs

Graphing a Function

The Graph of a Function:

The Vertical Line Test:

Example:

Solution:

Example:

Solution:

Example:

Solution:

The graph of y f x is shown below.

(a) Find the domain of f.

(b) Find the range of f.

(c) Find the following function values: 3 ; 1 ; 0 ; 1f f f f .

(d) For what value(s) of x is 2f x ?

Solution:

Part (a):

Part (b):

Part (c):

Part (d):

Solution:

Exercise Set 2.7: Functions and Graphs

yDetermine whether or not each of the following graphs

represents a function.

For each set of points,

(a) Graph the set of points.

(b) Determine whether or not the set of points

represents a function. Justify your answer.

11. (1, 5), (2, 4), ( 3, 4), (2, 1), (3, 6)

12. ( 3, 2), (1, 2), (0, 3), (2,1), ( 2,1)

13. (2, 0), (4, 1), (6, 0), (3, 1), (5, 2)

14. ( 1, 4), ( 2, 3), (4,1), (4, 2), ( 2, 3)

15. Analyze the coordinates in each of the sets

above. Describe a method of determining

whether or not the set of points represents a

function without graphing the points.

16. Determine whether or not each set of points

represents a function without graphing the

points. Justify each answer.

(a) ( 7, 3), (3, 7), (1, 5), (5,1), ( 2,1)

(b) (6, 3), ( 4, 3), (2, 3), ( 3, 3), (5, 3)

(c) (3, 6), (3, 4), (3, 2), (3, 3), (3, 5)

(d) ( 2, 5), ( 5, 2), (2, 5), (5, 2), (5, 2)

17. The graph of )(xfy is shown below.

(a) Find the domain of the function. Write your

answer in interval notation.

(b) Find the range of the function. Write your

(c) Find the following function values: )6();4();0();2( ffff

(d) For what value(s) of x is ( ) 9f x ?

18. The graph of )(xgy is shown below.

(c) Find the following function values:

( 2); (0); (1); (3); (6)g g g g g

(d) For what value(s) of x is ( ) 2g x ?

19. The graph of )(xgy is shown below.

(c) Find the following function values:

( 2); (0); (2); (4); (6)g g g g g

(d) Which is greater, ( 2)g or (3)g ?

20. The graph of )(xfy is shown below.

(c) Find the following function values: )4();1();1();2();3( fffff

(d) Which is smaller, )0(f or )3(f ?

For each of the following functions:

(a) State the domain of the function. Write your

(b) Choose x-values corresponding to the domain

of the function, calculate the corresponding y-

values, plot the points, and draw the graph of

the function.

21. 6)(23 xxf

22. 4)(32 xxf

23. 31,53)( xxxh

24. 23,2)( xxxh

25. 3)( xxg

26. 4)( xxg

27. 3)( xxf

28. xxf 5)(

29. xxxF 4)( 2

30. 1)3()( 2 xxG

For each of the following equations,

(a) Solve for y.

(b) Determine whether the equation defines y as a

function of x. (Do not graph.)

31. 3 5 8y x

32. 2692 yx

33. 22 3 7y x

34. 2 1 5y x

35. 23 yx

36. 32 yx

37. xy 2

38. 43 yx

39. 2 5 7 0y x

40. 3 4 8 0x y

SECTION 3.1 An Introduction to Polynomial Functions

Chapter 3 Polynomials

Section 3.1: An Introduction to Polynomial Functions

Polynomials and Polynomial Functions

Polynomials:

A polynomial in a single variable x is the sum of a finite number of terms of the form nax , where a is a constant and the exponent n is a whole number. Recall that the set

of whole numbers is 0,1, 2, ...

Examples of polynomials in x are 33 , 5 8 ,x x x and 24 7 1x x . They can be

classified according to the number of terms:

CHAPTER 3 Polynomials

Degree of a Polynomial:

Example:

Solution:

Polynomial Functions:

Evaluating Polynomial Functions:

Example:

Solution:

Graphs of Polynomial Functions:

Example:

Solution:

Leading term:

Degree:

Leading Coefficient:

Constant term:

Solution:

Exercise Set 3.1: An Introduction to Polynomial Functions

(a) State whether or not each of the following

expressions is a polynomial. (Yes or No.)

(b) If the answer to part (a) is yes, then state the

degree of the polynomial.

(c) If the answer to part (a) is yes, then classify

the polynomial as a monomial, binomial,

trinomial, or none of these. (Polynomials of

four or more terms are not generally given

specific names.)

1. 34 3x

2. 5 3 86 3x x

3. 3 5x

4. 3 22 4 7 4x x x

5. 3 2

7. 3 25 8x x x

8. 27 52 3

9. 3 2

7 5 32

10. 1 4 13 7 2x x

11. 11 1

34 29 2 4x x x

12. 2 3 1x x

13. 632

14. 3 26 8x x

15. 43 7x

16. 2 110 3 5x x

17. 10

18. 7 4x

19. 1 5 1 3 1 25 2 3 4x x x

20. 2 4 93 5 6 3x x x

21. 3 4 2 23 2a b a b

22. 5 2 4 94 3x y x y

23. 5 3

24. 2 9 3 4 22 15 4

3x y z xy x y z

25. 3 7 4 3 2325 7

4xyz y x y z

26. 7 3 5 6 2 42 3a a b b a b

27. (a) 37 2x x is a trinomial.

(b) 37 2x x is a third degree polynomial.

(c) 37 2x x is a binomial.

(d) 37 2x x is a first degree polynomial

28. (a) 25 3 2x x is a trinomial.

(b) 25 3 2x x is a third degree polynomial.

(c) 25 3 2x x is a binomial.

(d) 25 3 2x x is a second degree polynomial.

29. (a) 36x is a monomial.

(b) 36x is a third degree polynomial.

(c) 36x is a first degree polynomial.

(d) 36x is a trinomial.

30. (a) 2 34 7x x x is a second degree

polynomial.

(b) 2 34 7x x x is a binomial.

(c) 2 34 7x x x is a third degree polynomial.

(d) 2 34 7x x x is a trinomial.

31. (a) 7 4 6 83 2 3x x y y is a tenth degree

polynomial.

(b) 7 4 6 83 2 3x x y y is a binomial.

(c) 7 4 6 83 2 3x x y y is an eighth degree

polynomial.

(d) 7 4 6 83 2 3x x y y is a trinomial.

32. (a) 4 53a b is a fifth degree polynomial.

(b) 4 53a b is a trinomial.

(c) 4 53a b is a ninth degree polynomial.

(d) 4 53a b is a monomial.

Each of the graphs below represents a polynomial

function. Use the graph to determine the x- and y-

intercept(s). The equation of each graph is given for

informational purposes, but the intercepts can be

determined entirely from the graph.

33. 3 2( ) 5 2 8f x x x x

34. 2( ) 8 12f x x x

35. 5 4 3( ) 2 16 36 54f x x x x x

36. 4 3 27 5 3113 3 3 3

( ) 10f x x x x x

For each of the following polynomial functions,

(a) Classify the function as linear, quadratic, or

cubic.

(b) Find the x- and y-intercept(s) of the function.

(Do this algebraically without drawing the

graph.)

(c) Find ( 4), ( 1)f f and (6)f .

37. 2( ) 64f x x

38. ( ) 3 8f x x

39. 3( ) 32 4f x x

40. 2( ) 50 2f x x

41. ( ) 12 5f x x

42. 3( ) 2 54f x x

Follow the directions above for numbers 43 and 44,

but in part (b), find the y-intercept only. (Do not

find the x-intercept(s).)

43. 2( ) 3 28f x x x

44. 2( ) 18 9 2f x x x

Section 3.2: Adding, Subtracting, and Multiplying

Polynomials

Operations with Polynomials

Like Terms:

Addition of Polynomials:

Example:

SECTION 3.2 Adding, Subtracting, and Multiplying Polynomials

Solution:

Subtraction of Polynomials:

Example:

Solution:

Multiplication of Polynomials:

Example:

Solution:

Special Case - Multiplying Two Binomials:

Example:

Solution:

Special Products:

Example:

Solution:

(a) 26 2 7x x

(b) 25 3 6x x x

Solution:

Exercise Set 3.2: Adding, Subtracting, and Multiplying Polynomials

Multiply. Write your final answer with the terms in

descending order, from highest to lowest degree.

1. 5 8x x

2. 2 10x x

3. 1 3x x

4. 9 2x x

5. 3 12x x

6. 15 10x x

7. 4 20x x

8. 7 8x x

9. 4 4x x

10. 9 9x x

11. 7 7x x

12. 6 6x x

13. 2 1 3x x

14. 4 3 2x x

15. 5 1 4 7x x

16. 3 1 2 5x x

17. 3 2 4x x

18. 6 1 3 4x x

19. 2 25 7x x

20. 2 23 4 2 1x x

21. 4 3 2 37 2 5x x x x

22. 5 2 42 5 3x x x

Perform the indicated operations. Write your final

answer with the terms in descending order, from

highest to lowest degree.

23. (a) 5 2x x

(b) 5 2x x

(c) 5 2x x

24. (a) 3 37 4x x

(b) 3 37 4x x

(c) 3 37 4x x

25. (a) 3 4 32 5 6x x x

(b) 3 4 32 5 6x x x

(c) 3 4 32 5 6x x x

26. (a) 7 4 45 3x x x

(b) 7 4 45 3x x x

(c) 7 4 45 3x x x

27. (a) 2 33 2 9x x x

(b) 2 33 2 9x x x

(c) 2 33 2 9x x x

28. (a) 37 5x x x

(b) 37 5x x x

(c) 37 5x x x

29. (a) 2 210 5 7x x x

(b) 2 210 5 7x x x

(c) 2 210 5 7x x x

30. (a) 38 2 5x x

(b) 38 2 5x x

(c) 38 2 5x x

31. (a) 3 + 7x x

(b) 3 7x x

(c) 3 7x x

Exercise Set 3.2: Adding, Subtracting, and Multiplying Polynomials

32. (a) 2 8x x

(b) 2 8x x

(c) 2 8x x

33. (a) 2 3 2 7x x x

(b) 2 3 2 7x x x

(c) 2 3 2 7x x x

34. (a) 2 3 3 26 2 7x x x x

(b) 2 3 3 26 2 7x x x x

(c) 2 3 3 26 2 7x x x x

35. (a) 25 5 4x x x

(b) 25 5 4x x x

(c) 25 5 4x x x

36. (a) 2 27 2 4 9x x x

(b) 2 27 2 4 9x x x

(c) 2 27 2 4 9x x x

37. (a) 22 3 4 12x x x

(b) 22 3 4 12x x x

(c) 22 3 4 12x x x

38. (a) 23 1 4 2 6x x x

(b) 23 1 4 2 6x x x

(c) 23 1 4 2 6x x x

39. (a) 3 2 32 5 8 2x x x x x

(b) 3 2 32 5 8 2x x x x x

(c) 3 22 5 8 2x x x x

40. (a) 4 3 25 3 4 2 3x x x x

(b) 4 3 25 3 4 2 3x x x x

(c) 4 3 25 3 4 2 3x x x x

41. (a) 2 23 2 1 2 5 3x x x x

(b) 2 23 2 1 2 5 3x x x x

(c) 2 23 2 1 2 5 3x x x x

42. (a) 2 25 7 2 4 3x x x x

(b) 2 25 7 2 4 3x x x x

(c) 2 25 7 2 4 3x x x x

43. (a) 4 3 5 32 5 3 2x x x x x x

(b) 4 3 5 32 5 3 2x x x x x x

(c) 4 3 5 32 5 3 2x x x x x x

44. (a) 7 4 5 44 2 6 3 5x x x x x x

(b) 7 4 5 44 2 6 3 5x x x x x x

(c) 7 4 5 44 2 6 3 5x x x x x x

45. (a) 3 2 34 2 3 5 4x x x x

(b) 3 2 34 2 3 5 4x x x x

(c) 3 2 34 2 3 5 4x x x x

46. (a) 2 4 35 2 8 2 4x x x x x

(b) 2 4 35 2 8 2 4x x x x x

(c) 2 4 35 2 8 2 4x x x x x

47. (a) 2

1 7x x

48. (a) 2

2 3x x

SECTION 3.3 Dividing Polynomials

Section 3.3: Dividing Polynomials

Polynomial Long Division and Synthetic Division

Long Division of Polynomials:

Example:

Solution:

Synthetic Division of Polynomials:

Example:

Solution:

A Comparison of Long Division and Synthetic Division

Let us now analyze the previous two examples, both of which solved the same problem using

long division and then synthetic division.

Long Division Synthetic Division

4 3 22 05 8 3 5x x x x x

Constant: 5

Change the sign of the constant term when

performing synthetic division.

4 3 22 05 8 3 5x x x x x

Notice the coefficients of the dividend: 2, 0, 8, 3, 5

Write the coefficients of the dividend (without

changing any signs). Do not forget the

‘placeholder’ for 30x .

Notice that the coefficients in each column of

the subtraction problems under the division

sign (at the left) are similar to the numbers in

each column of the synthetic division problem

(above). Remember that at the left, the signs

are changed when the expressions are

subtracted.

2 10 42 213

5 2 0 8 3 5

x x x x x

10 50 210 1

5 | 2 0 8 3 5

2 10 42 213 1060

2 0 8| 55 3

| 2 0 3 55 8

Notice that the numbers in the answer line of

the synthetic division problem are the same as

the coefficients of the quotient plus the final

remainder in the long division problem.

In the long division problem, there is one

column for each power of x, and the

arithmetic in each column is done with the

coefficients.

Synthetic division is a shortcut for doing the

arithmetic with the coefficients without having

to write down all the variables. Remember that

this synthetic division procedure ONLY works

when the divisor is of the form D x x c .

The Remainder Theorem:

5 2 0 8 3 5

2 10 42 213

213 1065

x x x x x

5 | 2 0 8 3 5

10 50 210 1

2 10 42 213 1 6

Solution:

Exercise Set 3.3: Dividing Polynomials

Use long division to find the quotient and the

remainder.

1. 2 6 11

2. 2 5 12

3. 2 7 2

4. 2 6 5

5. 3 22 19 12

33222 23

12656 23

8. 3 212 13 22 14

9. 3 2

2 13 28 21

10. 4 3 2

7 4 42 12

x x x x

11. 64

144433222

12. 362

4282201024

14. xx

Use synthetic division to find the quotient and the

remainder.

15. 2 8 4

286133 23

312 23

8732 45

101827113 234

1251832 234

25. 21

26. 31

234 9106

Evaluate P(c) using the following two methods:

(a) Substitute c into the function.

(b) Use synthetic division along with the

Remainder Theorem.

27. 2;254)( 23 cxxxxP

28. 1;3875)( 23 cxxxxP

Exercise Set 3.3: Dividing Polynomials

29. 1;12487)( 23 cxxxxP

30. 3;14672)( 34 cxxxxP

Evaluate P(c) using synthetic division along with the

Remainder Theorem. (Notice that substitution without

a calculator would be quite tedious in these examples,

so synthetic division is particularly useful.)

31. 5;321703883)( 23567 cxxxxxxP

32. 2;11235103)( 2456 cxxxxxxP

33. 43234 ;12254)( cxxxxP

34. 273456 ;135932196)( cxxxxxxP

When the remainder is zero, the dividend can be

written as a product of two factors (the divisor and the

quotient), as shown below.

65 , so 30 5 6 .

, so 2

6 3 2x x x x

In the following examples, use either long division or

synthetic division to find the quotient, and then write

the dividend as a product of two factors.

35. 2 11 24

36. 2 3 40

37. 2 7 18

38. 2 10 21

39. 24 25 21

40. 23 22 24

41. 22 7 5

42. 25 4 12

Section 3.4: Quadratic Functions

The Definition and Graph of a Quadratic Function

Definition:

Graph:

SECTION 3.4 Quadratic Functions

Example:

Solution:

(b) Since 1 0a , the parabola opens upward.

Solution:

Exercise Set 3.4: Quadratic Functions

Each of the quadratic functions below is written in the

form 2

( )f x ax bx c . The graph of a quadratic

function is a parabola with vertex (h, k), where

h and 2ba

(a) Give the coordinates of the vertex of the

parabola.

(b) Does the parabola open upward (with the

vertex being the lowest point on the graph) or

downward (with the vertex being the highest

point on the graph)?

(c) Find the y-intercept of the parabola.

(d) Find the axis of symmetry. (Be sure to write

your answer as an equation of a line.)

(e) Draw a sketch of the parabola which includes

the features from parts (a) through (c). (Do

not worry about the accuracy of the x-

intercepts on your graph; we will learn about

these in a later section.)

1. 2( ) 6 7f x x x

2. 2( ) 8 21f x x x

3. 2( ) 2f x x x

4. 2( ) 6 9f x x x

5. 2( ) 2 8 11f x x x

6. 2( ) 10f x x x

7. 2( ) 14 49f x x x

8. 2( ) 16f x x

9. 2( ) 8 9f x x x

10. 2( ) 3 18 15f x x x

11. 2( ) 2 5f x x

12. 2( ) 4 7f x x x

13. 2( ) 4 40 115f x x x

14. 2( ) 5 10 8f x x x

15. 2( ) 2 8 14f x x x

16. 2( ) 4 24 27f x x x

17. 2( ) 5 3f x x x

18. 2( ) 7 1f x x x

19. 2( ) 2 3 4f x x x

20. 2( ) 7 3f x x x

For each of the following quadratic functions,

(a) Multiply the factors to obtain a function of the

form 2

( )f x ax bx c .

(b) Find the coordinates of the vertex (h, k) of the

parabola, using the formulas 2ba

(c) Match the function to its appropriate graph

from the choices shown below:

I. II.

III. IV.

21. ( ) 3 5f x x x

22. ( ) 2 8f x x x

23. ( ) 2 1 5f x x x

24. ( ) 3 2 4f x x x

CHAPTER 4 Factoring

Chapter 4 Factoring

Section 4.1: Greatest Common Factor and Factoring by

Grouping

GCF and Grouping

Finding the Greatest Common Factor:

Example:

Solution:

SECTION 4.1 Greatest Common Factor and Factoring by Grouping

Factoring Out the Greatest Common Factor:

CHAPTER 4 Factoring

Example:

Solution:

Factoring by Grouping:

Solution:

CHAPTER 4 Factoring

Solution:

The GCF is the product of the factors that are shared by all three monomials.

Solution:

CHAPTER 4 Factoring

Solution:

Exercise Set 4.1: Greatest Common Factor and Factoring by Grouping

Find the GCF (Greatest Common Factor) of the

following monomials.

1. 3 2 2 5 418 , 24 , 12x y x y xy

2. 3 5 7 3 4 920 , 32 , 8x y x y x y

3. 7 4 4 5 7 8, 7 , 14a b a b a b

4. 6 10 4 7 412 , 15 , 21c d c d c d

5. 3 12 10 5 6 7 516 , 32 , 100a b c a bc a c

6. 5 2 7 3 7 330 , 90 , 45a b a c b c

7. 6 9 8 3 5 7 5 6 410 , 18 , 7x y z x y z x y z

8. 7 5 3 4 8 6 5 29 , 50 , 20x y z x y x y z

Find the GCF of the terms of the polynomial and

factor it out. If the first term that appears in the

polynomial has a negative coefficient, then factor out

the negative of the GCF.

9. 5 10a

10. 4 12x

11. 3 15b

12. 4 24y

13. 9 24x y

14. 10 25a c

15. 6 8x xy

16. 8 12ab bc

17. 3 26 2a b ab

18. 5 73 6ac a c

19. 2 215 20r t rt

20. 4 3 3 630 2u v u w

21. 3 24 2 8x x x

22. 5 3 218 36 45x x x

23. 3 2 4 5 8 35 3 7x y x y x y

24. 3 6 4 5 220 8 12a b ab a b

25. 7 4 9 2 5 3 9 635 28 21a b c a b c a b c

26. 3 7 8 2 5 4 2 6 736 12 48x y z x y z x y z

27. 3 2 5 4 7 3 810 21 49a b c a c b c

28. 7 4 6 4 2 6 34 35 9x y z y z x y z

Factor the following expressions.

29. (a) 5xy y

(b) 4 5 4x x x

30. (a) 3xy y

(b) 6 3 6x x x

31. (a) 3b ab

(b) 3 5 5c a c

32. (a) ap cp

(b) 2 2a b c b

33. 3 ( 5) 4 ( 5)a a b a

34. 4 ( 7) 3 ( 7)x x y x

35. 2 ( 8) ( 8)x x x

36. 3 ( 2) ( 2)b b b

37. ( 3)( 5) ( 2)( 5)x x x x

38. ( 4)( 1) ( 4)( 6)x x x x

39. ( 2)(4 3) ( 8)( 2)a a a a

40. (3 1)(2 6) (3 1)( 2)a a a a

Factor by grouping.

41. 2 2b c ab ac

42. 3 3x y xz yz

43. 5 5y z xy xz

44. 4 4a b ca cb

45. 2 3 3x x xy y

46. 3 5 15xy x y

47. ac ad bc bd

48. 2p pr tp tr

49. 4 4xy x y

50. 2 2b ab a

Exercise Set 4.1: Greatest Common Factor and Factoring by Grouping

51. 2y xy y x

52. 2px p x p

53. 12 8 3 2b ab a

54. 18 24 15 20xy x y

55. 26 2 3t tx t x

56. 215 5 6 2x xy x y

57. 12 3 8 2ac ad bc bd

58. 24 15 8 5xy xz y z

Factor by grouping. (Hint: Use groups of three.)

59. ad ae af bd be bf

60. 4 3 3 12xy xz x y z

61. 23 12 15 2 8 10x xz x xy yz y

62. 212 8 20 3 2 5ab ac ad b bc bd

Each of the following expressions contains like terms.

Do not combine the like terms; instead, simply factor

by grouping. (This method will be helpful in the next

section when factoring trinomials.)

63. 2 3 2 6x x x

64. 2 5 7 35x x x

65. 2 4 3 12x x x

66. 2 3 6 18x x x

67. 26 10 9 15x x x

68. 221 3 14 2x x x

69. 29 21 6 14x x x

70. 225 5 20 4x x x

71. 24 14 14 49x x x

72. 29 15 15 25x x x

CHAPTER 4 Factoring

Section 4.2: Factoring Special Binomials and Trinomials

Special Factor Patterns

Factoring the Difference of Two Squares:

Example:

Solution:

SECTION 4.2 Factoring Special Binomials and Trinomials

Factoring the Difference of Two Cubes:

Example:

CHAPTER 4 Factoring

Solution:

Factoring the Sum of Two Cubes:

Example:

Solution:

Factoring Perfect Square Trinomials:

Example:

CHAPTER 4 Factoring

Solution:

CHAPTER 4 Factoring

(c) The monomials 310yz and 10y share a common factor of 10 .y The first step in

factoring the given binomial to factor out the GCF of 10y .

Solution:

Exercise Set 4.2: Factoring Special Binomials and Trinomials

Multiply the following.

1. (a) 4 4x x

2. (a) 9 9x x

3. 22 49 7x x

4. 2 64 8 8x x x

5. 2 26 12 36x x x

6. 2 24 16x x

7. 210 10 100x x x

8. 22 24 144 12x x x

9. 22 81 9x x

10. 2 25 25x x

Factor the following polynomials. If the polynomial

can not be factored any further within the real

number system, then write the original polynomial as

your answer.

11. (a) 2 9x

(b) 2 9x

(c) 2 6 9x x

(d) 2 6 9x x

12. (a) 2 25x

(b) 2 25x

(c) 2 10 25x x

(d) 2 10 25x x

13. 2 49x

14. 2 36x

15. 2 144x

16. 2 81a

17. 2 1p

18. 21 p

19. 2 100x

20. 2 4x

21. 225 c

22. 2144 d

23. 24 9b

24. 225 49x

25. 216 1x

26. 236 1x

27. 2 249 100x y

28. 2 264 25a b

29. 2 225 16c d

30. 2 24 9z w

33. 2 2

34. 2 2

35. 2 216

36. 2 2

37. 2 20 100x x

38. 2 8 16x x

39. 2 2 1x x

40. 2 14 49x x

41. 2 18 81x x

42. 2 24 144x x

43. 24 12 9x x

Exercise Set 4.2: Factoring Special Binomials and Trinomials

44. 29 30 25x x

45. 225 40 16x x

46. 236 12 1x x

47. 2 22x bx b

48. 2 22x cx c

49. 2 2 24 20 25b c bcd d

50. 2 2 29 24 16x xyz y z

When the remainder is zero, the dividend can be

written as a product of two factors (the divisor and the

quotient), as shown below.

65 , so 30 5 6 .

, so 2

6 3 2x x x x

In the following examples, use either long division or

synthetic division to find the quotient, and then write

the dividend as a product of two factors.

51. 3 8

52. 3 27

Factor the following polynomials.

53. 3 64x

54. 3 1m

55. 3 27p

56. 3 125x

57. 3 3x y

58. 3 3c d

59. 3 3125 8a b

60. 3 364 27x y

CHAPTER 4 Factoring

Section 4.3: Factoring Polynomials

Techniques for Factoring Trinomials

Factorability Test for Trinomials:

Example:

Solution:

SECTION 4.3 Factoring Polynomials

Factoring Trinomials with Leading Coefficient 1:

CHAPTER 4 Factoring

Example:

Solution:

Factoring Trinomials with Leading Coefficient Different from 1:

CHAPTER 4 Factoring

Example:

Solution:

(a) 22 3 8x x

(b) 242 25 3x x

Solution:

CHAPTER 4 Factoring

Solution:

CHAPTER 4 Factoring

Solution:

Exercise Set 4.3: Factoring Polynomials

At times, it can be difficult to tell whether or not a

quadratic of the form 2

ax bx c can be factored

into the form dx e fx g , where a, b, c, d, e, f,

and g are integers. If 2

4b ac is a perfect square, then

the quadratic can be factored in the above manner.

For each of the following problems,

(a) Compute 2

4b ac .

(b) Use the information from part (a) to

determine whether or not the quadratic can

be written as factors with integer coefficients.

(Do not factor; simply answer Yes or No.)

1. 2 5 3x x

2. 2 7 10x x

3. 2 6 16x x

4. 2 6 4x x

5. 29 x

6. 27x x

7. 22 7 4x x

8. 26 1x x

9. 22 2 5x x

10. 25 4 1x x

Factor the following polynomials. If the polynomial

can not be rewritten as factors with integer

coefficients, then write the original polynomial as your

answer.

11. 2 4 5x x

12. 2 9 14x x

13. 2 5 6x x

14. 2 6x x

15. 2 7 12x x

16. 2 8 15x x

17. 2 12 20x x

18. 2 7 18x x

19. 2 5 24x x

20. 2 9 36x x

21. 2 16 64x x

22. 2 6 9x x

23. 2 15 56x x

24. 2 6 27x x

25. 2 11 60x x

26. 2 19 48x x

27. 2 17 42x x

28. 2 12 64x x

29. 2 49x

30. 2 36x

31. 2 3x

32. 2 8x

33. 29 25x

34. 216 81x

35. 22 5 3x x

36. 23 16 15x x

37. 28 2 3x x

38. 24 16 15x x

39. 29 9 4x x

40. 25 17 6x x

41. 24 3 10x x

42. 29 21 10x x

43. 212 17 6x x

44. 28 26 7x x

Factor the following. Remember to first factor out the

Greatest Common Factor (GCF) of the terms of the

polynomial, and to factor out a negative if the leading

coefficient is negative.

45. 2 9x x

46. 2 16x x

47. 25 20x x

48. 24 28x x

Exercise Set 4.3: Factoring Polynomials

49. 22 18x

50. 28 8x

51. 4 25 20x x

52. 33 75x x

53. 22 10 8x x

54. 23 12 63x x

55. 210 10 420x x

56. 24 40 100x x

57. 3 29 22x x x

58. 3 27 6x x x

59. 3 24 4x x x

60. 5 4 310 21x x x

61. 4 3 26 6x x x

62. 3 22 80x x x

63. 5 39 100x x

64. 12 1049 64x x

65. 250 55 15x x

66. 230 24 72x x

Factor the following polynomials. (Hint: Factor first

by grouping, and then continue to factor if possible.)

67. 3 22 25 50x x x

68. 3 23 4 12x x x

69. 3 25 4 20x x x

70. 3 29 18 25 50x x x

71. 3 24 36 9x x x

72. 3 29 27 4 12x x x

SECTION 4.4 Using Factoring to Solve Equations

Section 4.4: Using Factoring to Solve Equations

Solving Quadratic Equations by Factoring

Solving Other Polynomials Equations by Factoring

Solving Quadratic Equations by Factoring

Zero-Product Property:

Example:

Solution:

CHAPTER 4 Factoring

Example:

Solution:

The x-Intercepts of the Graph of a Quadratic Function:

Example:

Solution:

CHAPTER 4 Factoring

Solution:

CHAPTER 4 Factoring

Solution:

CHAPTER 4 Factoring

(c) Since 1 0a , the parabola opens upward.

Solving Other Polynomial Equations by Factoring

Solving Polynomial Equations by Factoring:

Example:

Solution:

Example:

Solution:

The x-Intercepts of the Graph of a Polynomial Function:

CHAPTER 4 Factoring

Example:

Solution:

CHAPTER 4 Factoring

Exercise Set 4.4: Using Factoring to Solve Equations

Solve the following equations by factoring.

1. 021102 xx

2. 040132 xx

3. 01282 xx

4. 04062 xx

5. 35)2( xx

6. 20)8( xx

7. 72142 xx

8. xx 11602

9. 01572 2 xx

10. 0473 2 xx

11. 12176 2 xx

12. 6710 2 xx

13. 253 2 xx

14. 0568 2 xx

15. 0252 x

16. 0492 x

17. 094 2 x

18. 02536 2 x

Solve the following equations by factoring. To simplify

the process, remember to first factor out the Greatest

Common Factor (GCF) and to factor out a negative if

the leading coefficient is negative.

19. 082 xx

20. 0102 xx

21. 2 5 36 0x x

22. 2 14 48 0x x

23. 0213 2 xx

24. 0305 2 xx

25. 0123 2 x

26. 077 2 x

27. 090155 2 xx

28. 024204 2 xx

29. 03023080 2 xx

30. 0187512 2 xx

31. 3 25 6 0x x x

32. 3 27 18 0x x x

Each of the quadratic functions below is written in the

form 2

( )f x ax bx c . The graph of a quadratic

function is a parabola with vertex (h, k), where

h and 2ba

(a) Find the x-intercept(s) of the parabola by

setting ( ) 0f x and solving for x.

(b) Write the coordinates of the x-intercept(s)

found in part (a).

(c) Find the y-intercept of the parabola and write

its coordinates.

(d) Give the coordinates of the vertex (h, k) of the

parabola, using the formulas 2ba

(e) Does the parabola open upward (with the

vertex being the lowest point on the graph) or

downward (with the vertex being the highest

point on the graph)?

(f) Find the axis of symmetry. (Be sure to write

your answer as an equation of a line.)

(g) Draw a graph of the parabola that includes

the features from parts (b) through (e).

33. 2( ) 6 8f x x x

34. 2( ) 2 15f x x x

35. 2( ) 8 16f x x x

36. 2( ) 10 16f x x x

37. 2( ) 4 21f x x x

38. 2( ) 10 25f x x x

39. 2( ) 3 12 36f x x x

40. 2( ) 4 8 5f x x x

41. 2( ) 16f x x

Exercise Set 4.4: Using Factoring to Solve Equations

42. 2( ) 25f x x

43. 2( ) 9 4f x x

44. 2( ) 9 100f x x

Find the x-intercept(s) of the following.

45. 3 2( ) 7 10f x x x x

46. 3 2( ) 2 99f x x x x

47. 3( ) 25f x x x

48. 3( ) 4f x x x

49. 3 2( ) 2 9 18f x x x x

50. 3 2( ) 4 4f x x x x

For each of the following problems:

(a) Model the situation by writing appropriate

equation(s).

(b) Solve the equation(s) and then answer the

question posed in the problem.

51. The length of a rectangular frame is 5 cm longer

than its width. If the area of the frame is 36 cm2,

find the length and width of the frame.

52. The width of a rectangular garden is 8 m shorter

than its length. If the area of the field is 180 m2,

find the length and the width of the garden

53. The height of a triangle is 3 cm shorter than its

base. If the area of the triangle is 90 cm2, find the

base and height of the triangle.

54. Find x if the area of the figure below is 26cm2.

(Note that the figure may not be drawn to scale.)

SECTION 5.1 Simplifying Rational Expressions

Chapter 5 Rational Expressions, Equations, and Functions

Section 5.1: Simplifying Rational Expressions

Rational Expressions

Definition:

Simplifying:

CHAPTER 5 Rational Expressions, Equations, and Functions

Example:

Solution:

Exercise Set 5.1: Simplifying Rational Expressions

Simplify the following rational expressions. If the

expression cannot be simplified any further, then

simply rewrite the original expression.

5. 2 5

6. 4 9

9. x y

10. c d

a b c d

x y w z

z w x y

13. 4 8

17. 2 2a b

18. 2 16

30. 27 14

Exercise Set 5.1: Simplifying Rational Expressions

6 24 18

4 8 60

5 10 40

10 30 20

4 17 15

5 13 6

4 8 21

8 24 14

10 9 2

15 4 4

5 22 8

8 30 7

6 13 5

43. 3 2

1m m m

m m m n n

44. 2 2

ax ay bx by

ax ay x y

45. 3 2 6

3 5 15

xy x y

yz z y

46. 2 2

5 2 10

4 5 20

ab a b

a b b a

47. 3 8

Section 5.2: Multiplying and Dividing Rational Expressions

Multiplication and Division

Multiplication of Rational Expressions:

To multiply two fractions, place the product of the numerators over the product

of the denominators.

Example:

Solution:

SECTION 5.2 Multiplying and Dividing Rational Expressions

Division of Rational Expressions:

Example:

Solution:

SECTION 5.2 Multiplying and Dividing Rational Expressions

Solution:

Exercise Set 5.2: Multiplying and Dividing Rational Expressions

Multiply the following rational expressions and

simplify. No answers should contain negative

exponents.

1. 6 14

2. 8 45

5. 4 7 3

5 8 6 9

ab c d

c d a b

6. 5 6 3

3 8 10 9

x y wz

w z x y

7. 8 2 4 6 6 2

3 5 5 3 7

m n n t p t

p t m m n

8. 3 4 8 2 2 3 4

2 7 7 5

x y a b x y a

ab x y b y

10. 4 56

11. 5 3

12. 6 5

17. 3 2

18. 2 1

( 4)5 20

(4 28)7

21. 3 4

(2 8)3 12

22. 2 2

(3 3)4 4

23. 6 12 4 12

24. 7 6 24

2 8 5 35

25. 6 10 3

5 15 9

27. 2 2

3 4 2 15

x x x x

28. 2 2

8 15 9 14

x x x x

29. 2 2

3 10 2 4

x x x x

30. 2 2

6 30 4 21

6 40 8

x x x x

Exercise Set 5.2: Multiplying and Dividing Rational Expressions

32. 2 225 12 36

33. 2 2

2 9 10 7 12

5 6 2 3 5

x x x x

34. 2 2

2 8 3 14 5

3 16 5 20

x x x x

35. 7 2 14

7 3 21 2 2

ax bx ay by ax x a

ax x a ax bx a b

36. 2 22 2

ac ad bc bd c d

ac ad bc bd ac ad bc bd

Divide the following rational expressions and simplify.

No answers should contain negative exponents.

37. 5 15

38. 6 4

39. 25

40. 12

43. 4 3

44. 3 7 5 9

a c b c

45. 5 6

a ba d

46. 3 2

x yx z

47. 3 5

48. 4 3

49. 27 7 1

51. 2 1 1

6 3 18

416 xx

4 9 2 3

510 25

57. 2 2

3 10 6

3 28 12

x x x x

58. 2 2

4 4 8 20

6 16 9 8

x x x x

59. 2 2

6 1 3 2 1

6 5 1 3 4 1

x x x x

60. 2 2

10 17 6 6 5 4

5 4 12 3 2 8

x x x x

61. 3 3

am an bm bn am an bm bn

2 2 5 5

5 53 3

cx dx cy dy cx cy dx dy

cx dx c dx x xy y

SECTION 5.3 Adding and Subtracting Rational Expressions

Section 5.3: Adding and Subtracting Rational Expressions

Addition and Subtraction

Addition and Subtraction of Rational Expressions with Like

Denominators:

Example:

Perform the following operations. All results should be in simplified form.

Solution:

Addition and Subtraction of Rational Expressions with Unlike

Denominators:

Example:

Solution:

Perform the following operations. All results should be in simplified form.

Solution:

Perform the addition. Give the result in simplified form.

Solution:

Perform the subtraction. Give the result in simplified form.

Solution:

Perform the subtraction. Give the result in simplified form.

Solution:

Perform the following operations. Give all results in simplified form.

Solution:

Exercise Set 5.3: Adding and Subtracting Rational Expressions

Perform the indicated operations and simplify.

(Whenever possible, write both the numerator and

denominator of the answer in factored form.)

1. 2 3

2. 4 2

3. 3 2

4 9a b

4. 7 5

2 3c c

5. 2 2 5

x y xy

6. 4 7 5 4

a b a b

7. 8 7

8. 3 4 6

9. 3 2 2 6

5 20 5 20

10. 2 3 10 9

4 3 4 3

11. 2 3

1 5x x

12. 5 6

4 7x x

13. 3 8

14. 5 2

15. 3 4

1 2x x

16. 1 2

2 2x x

17. 6 2

3 7x x

18. 7 4

9 2x x

19. 5 4

25. 4 2

26. 3 1

27. 2 3

29. 1 1

30. 2 3 6

31. 3 1

Exercise Set 5.3: Adding and Subtracting Rational Expressions

32. 1 2

33. 5 2

34. 4 2

35. 7 5

8 12 6 6x x

36. 5 2

12 6 10 40x x

37. 3 8 6

1 2x x x

38. 2 4 3

3 2x x x

41. 2 2 2

2 8 2 4

x x x x x x

42. 2 2 2

3 18 6 3

x x x x x x

43. 2 2 2

10 24 12 32 14 48

x x x x x x

44. 2 2 2

7 12 4 3 5 4

x x x x x x

SECTION 5.4 Complex Fractions

Section 5.4: Complex Fractions

Simplifying Complex Fractions

Definition:

Simplifying:

Example:

Solution:

Method 1:

Method 2:

Solution:

Exercise Set 5.4: Complex Fractions

Simplify the following. No answers should contain

negative exponents.

3 41 2

6 23 2

21. 3 2

22. 4 5

5 4a b

3 43 2

1 23 2

For each of the following expressions,

(a) Rewrite the expression so that it contains

positive exponents rather than negative

exponents.

(b) Simplify the expression.

37. 1 1

38. 1 1

39. 2 2

40. 1 1

41. 1 1

42. 3 3

43. 3 3

44. 2 2

SECTION 5.5 Solving Rational Equations

Section 5.5: Solving Rational Equations

Rational Equations

Definition of a Rational Equation:

Solving a Rational Equation:

Example:

Solution:

Example:

Solution:

Extraneous Solutions:

Example:

Solution:

Exercise Set 5.5: Solving Rational Equations

Solve the following. Remember to identify any

extraneous solutions.

2. 3 2

3. 3 2

5. 5 3

6. 7 3

7. 4 7 3

8. 3 4 8

9. 5 2 6

10. 3 4 5 7

11. 34

12. 24

17. 3 1

23. 5 9

24. 3 12

25. 7 8

26. 2 1

27. 7 2

28. 2 2

29. 1 3 13

1 4 12

30. 4 1 9

9 2 14

31. 5 3 7

32. 3 3

Exercise Set 5.5: Solving Rational Equations

33. 153

35. 3 2 1

4 8 3 6 36a a

36. 5 1 7

3 15 2 10 12c c

5 3 2 15x x x x

1 2 2x x x x

3 1 2 3x x x x

7 2 10

4 5 9 20x x x x

2 2 4x x x

3 6 24

4 4 16x x x

47. 6 1

48. 7 4

49. 4 1

14 1x x

50. 5 2

14 2x x

51. 7 8

15 8x x

52. 5 6

17 9x x

53. 4 2

54. 2 1

55. 1 4

2 5 3 2 5

56. 2 1 6

3 1 3 1

57. 4 3 3

3 2 1 1

58. 5 2

2 3 2 3

Section 5.6: Rational Functions

Working with Rational Functions

Definition of a Rational Function:

Domain of a Rational Function:

Example:

SECTION 5.6 Rational Functions

Solution:

Graph of a Rational Function:

Example:

Solution:

The graph of the function is shown below, labeled with the information from parts (b)-(d).

Vertical Asymptotes:

Finding Vertical Asymptotes

Example:

Solution:

Horizontal Asymptotes:

Solution:

Exercise Set 5.6: Rational Functions

Find the indicated function values. If undefined, state

“Undefined.”

1. If ( )3

, find

(a) (0)f (b) ( 1)f (c) 13

2. If 5

, find

(a) (0)f (b) ( 5)f (c) 15

3. If 3 2

, find

(a) (0)f (b) ( 3)f (c) 45

4. If 2 7

, find

(a) (0)f (b) (4)f (c) 34

5. If 2

, find

(a) ( 2)f (b) (0)f (c) (5)f

6. If 2

, find

(a) ( 4)f (b) (0)f (c) (1)f

7. If 2

( )121

, find

(a) ( 3)f (b) (0)f (c) (12)f

8. If 2

5 14x x , find

(a) (0)f (b) ( 1)f (c) (7)f

9. If 2

, find

(a) (3)f (b) ( 4)f (c) (0)f

10. If 2

, find

(a) (0)f (b) ( 2)f (c) ( 5)f

The graph of each of the following functions has a

horizontal asymptote at 1y . (You will learn how to

find horizontal asymptotes in a later mathematics

course.) For each function,

(a) Find the domain of the function and express it

as an inequality.

(b) Write the equation of the vertical

asymptote(s) of the function.

(c) Find the x- and y-intercept(s) of the function,

if they exist. If an intercept does not exist,

state “None.”

(d) Find (1)f and ( 1)f .

(e) Based on the features from (a)-(d), match the

function with its corresponding graph, using

the choices (Graphs I-IV) below.

Graph IV:

Graph I: Graph II:

Graph III:

The graph of each of the following functions has a

horizontal asymptote at 0y . (You will learn how to

find horizontal asymptotes in a later mathematics

course.) For each function,

as an inequality.

(c) Find the x- and y-intercept(s) of the function,

if they exist. If an intercept does not exist,

state “None.”

(d) Find (1)f and ( 1)f .

(e) Based on the features from (a)-(d), match the

function with its corresponding graph, using

the choices (Graphs I-IV) below.

( )f xx

For each of the following functions,

as an inequality. Then write the domain of the

function in interval notation.

(c) Find the x- and y- intercept(s) of the function.

If an intercept does not exist, state “None."

19. If 10

20. 12

22. ( )8

8 12f x

Graph IV:

Graph I: Graph II:

Graph III:

33. 2 10 25

x xf x

34. 2 7 18

x xf x

37. 2 5 14

( )5 7

x xf x

38. 29 1

( )3 2

25 36( )

7 6( )

x xf x

Odd-Numbered Answers to Exercise Set 1.1: Numbers

1. (a) Composite; 1, 2, 4, 8

(b) Prime

(c) Neither

(d) Neither

(e) Composite; 1, 2, 3, 4, 6, 12

3. (a) 0.1

(b) 0.2

(c) 0.3

(d) 0.4

(e) 0.5

(f) 0.6

(g) 0.7

(h) 0.8

(i) 0.9 1=

(j) 1.1 (since 10 1

9 91= )

(k) 1.5 (since 514

9 91= )

(l) 2.7 (since 25 7

9 92= )

(m) 3.2 (since 29 2

9 93= )

5. (a) Rational; 7

(b) Irrational

(c) Rational; 3

(d) Rational; 5

(e) Rational; 4

(f) Rational; 1

(g) Rational; 12

(h) Rational; 23

(i) Irrational

(j) Rational; 2

(k) Irrational

7. Odd, Negative, Integer, Rational, Real

9. Positive, Irrational, Real

11. (a) 8, 0,12−

(c) 15

47, , , 5, 12π

(d) 8, 2.1, 0.4− − −

(f) 12

(g) 5,12

(h) 0, 5, 12

(i) 8, 0, 5, 12−

(j) All numbers in the set:

48, 2.1, 0.4, 0, 7, , , 5, 12π− − −

(k) 15

48, 2.1, 0.4, 0, , 5,12− − −

(l) 7, π

(m) None

15. 2 is the only number that is both prime and even.

17. Answers vary. Some possible answers are:

3 7, , 0.6, 0.37, 0.2, 8− − −

(Note: Any repeating decimal is a rational number.

There are methods for changing repeating decimals to

fractions which will not be learned in this course.)

19. Answers vary. Some possible answers are:

2, 3, 5, 6, 10, , , 0.080080008...eπ− −

21. Does not exist

23. Does not exist

25. True

27. True

29. False. The number 0 is a whole number but not a

natural number.

31. True

33. False. A repeating decimal such as 4

90.4 = is a

nonterminating decimal, but is a rational number.

35. 2, 3, 5, 7

37. 8, 9. 10, 12, 14, 15, 16, 18

39. 4, 6, 8

10 −55 13.3

Undefined Y N N N N

Natural N Y N N N

Whole N Y N N N

Integer N Y N Y N

Rational N Y Y Y Y

Irrational N N N N N

Prime N N N N N

Composite N N N N N

Real N Y Y Y Y

Odd-Numbered Answers to Exercise Set 1.1: Numbers

−3 −2 −1 0 1

57. (a) 3−

(c) 1−

59. (a) 1

(c) Undefined

8 (Note: 3 8

5 51 = )

(e) 1−

61. (a) 0

63. Numbers ordered from least to greatest:

, 2, 1, , 0.4, 0.494 5

− − −

The above numbers plotted on a number line:

Odd-Numbered Answers to Exercise Set 1.2: Integers

1. (a) 10

(b) 10−

(d) 4−

(e) 3−

3. (a) 4−

(d) 4−

5. (a) 12−

(b) 8−

(e) 12

(f) 8−

(g) 12−

(h) 12

7. (a) <

9. (a) 0

(b) Undefined

(d) 6−

(f) 6−

(i) 6−

(j) Undefined

(k) 6−

11. (a) 20

(c) 20−

(d) 5−

(e) 5−

13. (a) 24

(b) 24−

(c) 24

(d) 24−

15. (a) 6

(b) 10−

(c) 16

(d) 4−

(e) 6−

(h) 9−

(i) 8−

(j) 8−

(k) 10

4, or 0.25

(n) Undefined

Odd-Numbered Answers to Exercise Set 1.3: Fractions

1. (a) GCD: 2 (b) LCM: 24

3. (a) GCD: 1 (b) LCM: 70

5. (a) GCD: 14 (b) LCM: 28

7. (a) GCD: 4 (b) LCM: 40

9. (a) GCD: 6 (b) LCM: 90

11. (a) GCD: 4 (b) LCM: 240

13. (a) 27

1 (b) 35

4 (c) 13

15. (a) 34

6− (b) 1011

2− (c) 3

107−

17. (a) 316

(b) 679

(c) 263

19. (a) 197

− (b) 173

− (c) 494

21. (a) 37

23. (a) 35

6 (b) 13

25. (a) 2 14 2

3 3= (b) 25

27. (a) 13

6 (b) 4 2

10 53 3=

29. (a) 34

(b) 421

31. (a) 1760

(b) 3135

33. (a) 5 1

70 14− −= (b)

35. (a) 2542

9 (b) 2 1

10 52 2=

37. (a) 6

3516 (b)

39. (a) 33 1160 20

2 2= (b) 1348

41. (a) 3536

(b) 2845

43. (a) 163

(b) 193

45. (a) 53

(b) 352

(c) 20−

47. (a) 2577

(b) 1528

(c) 425

49. (a) 100 (b) 23

(c) 507

51. (a) 2

15 (b)

(c) 92

53. (a) 17

1 (b) 1116

55. (a) 12 (b) 15

57. (a) 12

2 (b) 57

Odd-Numbered Answers to Exercise Set 1.4: Exponents and Radicals

1. (a) 37 (b)

(c) 68 (d)

9. (a) 3 (b) 9 (c) 27

(d) 3− (e) 9− (f) 27−

(g) 3− (h) 9 (i) 27−

(j) 1 (k) 1− (l) 1

(m) 81 (n) 81− (o) 81

11. (a) 0.25, or 1

25 (c)

13. (a) 85 (b)

15. (a) 76 (b)

17. (a) 24 (b)

19. (a) 187 (b)

21. (a) 1

55= (b)

255= (c)

23. (a) 3

82= (b)

25. (a)

15= (b)

27. (a) 2

255− = − (b)

29. (a) 1

32− (b) 16

31. (a) 1 (b) 1

33. (a)

9 1227

35. 20x

39. 4 68a b−

43. 12 8

9a b−

45. (a) 6 (b) 7 (c) 9 2 3 2=

47. (a) 25 2 5 2= (b) 14 (c) 9

49. (a) 4 7 2 7= (b) 36 2 6 2=

(c) 9 3 3 3=

51. (a) 9 6 3 6= (b) 16 5 4 5=

(c) 4 15 2 15=

53. (a) 5

55. (a) 7

10 (c)

57. (a) 9 3 (b) 2 2 3x y z yz

59. (a) 5 (b) 36 (c) 8

61. (a) 8 (b) Not a real number (c) 8−

63. (a) 2 (b) 2− (c) 2−

65. (a) 10 (b) Not a real number (c) 10−

67. (a) 1

2 (b) Not a real number (c)

69. (a) 1

10 (b)

10− (c)

Odd-Numbered Answers to Exercise Set 1.5: Order of Operations

1. (a) P: Parentheses

E: Exponents

M: Multiplication

D: Division

A: Addition

S: Subtraction

(b) Whichever appears first

(c) Whichever appears first

3. (a) 23 (b) 35

(c) 17− (d) 5−

(e) 4 (f) 6−

5. (a) 10 (b) 4

(c) 4− (d) 10

7. (a) 4− (b) 14−

(c) 10 (d) 33

(e) 24− (f) 16

9. (a) 1960

(b) 1160

30− (d) 1

11. (a) 45 (b) 49−

(c) 8 (d) 57

(e) 26− (f) 36

13. (a) 100 (b) 1 (c) 5,000

15. (a) 40 (b) 16 (c) 33

17. (a) 619

(b) 2 (c) 18

19. 1920

21. 281

23. 18

25. 514

27. 7−

29. 2 3 5−

31. 25−

35. 19

37. 18−

39. 13

41. 57

45. 115

47. 67

Odd-Numbered Answers to Exercise Set 1.6: Solving Linear Equations

1. 7x =

3. 3x = −

5. 5x =

9. 2−=x

11. 5=x

11822 ==x

15. 136=x

17. 35−=x

19. 6x =

21. 185

23. 20=x

25. 10=x

27. 5227=x

Odd-Numbered Answers to Exercise Set 1.7:

Interval Notation and Linear Inequalities

1 2 3 4 5 6 7 8 9

1. (a) 5x >

(c) ( )5, ∞

3. (a) 3x ≤

(c) ( ], 3−∞

5. (a) 2x ≠

(c) ( ) ( ), 2 2,−∞ ∞∪

7. (a) 1x < −

(c) ( ), 1−∞ −

9. (a) 4x ≥ −

(c) [ )4,− ∞

11. (a) 8x ≠ −

(c) ( ) ( ), 8 8,−∞ − − ∞∪

13. (a) 2x ≠ and 7x ≠

(c) ( ) ( ) ( ), 2 2, 7 7,−∞ ∞∪ ∪

15. ( )∞,3

17. ( ]2, −∞−

19. ( ]5,3

21. ( ) ( ), 7 7,−∞ − − ∞∪

23. ( )3,1−

25. ( ]4,∞−

27. ( ) ( ), 0 0,−∞ ∞∪

29. { }31,3,2 −

31. { }31,3,2 −

33. { }31,2

35. (a) 5<x

(c) ( )5,∞−

37. (a) 6−≤x

(c) ( ]6, −∞−

39. (a) 3−≥x

(c) [ )∞− ,3

41. (a) 4−<x

(c) ( )4, −∞−

43. (a) 5−>x

(c) ( )∞− ,5

45. (a) 8

13≥x

(c) [ )∞,8

47. (a) 31≤x

(c) ( ]31,∞−

2 3 4 5 6 7 8

−9 −8 −7 −6 −5 −4 −3

−5 −4 −3 −2 −1 0 1

−7 −6 −5 −4 −3 −2 −1

−11 −10 −9 −8 −7 −6 −5 −4

−6 −5 −4 −3 −2 −1 0

−4 −3 −2 −1 0 1 2

−1 0 1 2 3 4 5 6

0 1 2 3 4 5 6

3 4 5 6 7 8 9 10

Interval Notation and Linear Inequalities

−5 −4 −3 −2 −1 0 1 2 3 4

−3 −2 −1 0 1 2 3

49. (a) 172>x

(c) ( )∞,172

51. (a) 2−≥x

(c) [ )∞− ,2

53. (a) 24 <≤− x

(c) [ )2,4−

55. (a) 12 i.e.,21 ≤≤−−≥≥ xx

(c) [ ]1,2−

57. (a) 3

20 << x

(c) ( )3

59. b (since 9 is not less than 5), and

d (since -5 is not greater than -3)

61. (a) 10023.075 ≤+ x , where x represents the # of

miles driven.

(b) 7.108≤x , so you can drive a maximum of 108

miles and still be reimbursed in full.

63. You would need to talk for more than 110 minutes in

order for Plan 1 to be more cost-effective than Plan 2.

Odd-Numbered Answers to Exercise Set 1.8: Absolute Value and Equations

1. 7±=x

3. No solution (since the absolute value of any quantity

is always 0≥ , and therefore cannot be negative)

5. 6±=x

7. 9,1 −== xx

9. 1x = ±

11. 34,4 −== xx

13. 4x = ±

15. 9,12 == xx

17. 37

31 , == xx

19. 23

21 , −== xx

21. No solution (since the absolute value of any quantity

is always 0≥ , and therefore cannot be negative)

23. 81

23 , −== xx

An Introduction to the Coordinate Plane

1, 3, 5:

7. (-4, 2); Quadrant II

9. (0, 3); y-axis

11. (-3, -4); Quadrant III

13. Graph:

(a) Quadrant I

(b) Quadrant III

(c) Quadrant IV

(d) Quadrant II

15. Graph:

(a) y-axis

(b) x-axis

(c) x-axis

(d) y-axis

17. (a) III (b) II (c) I

19. (a) IV (b) IV (c) III

21. (a) II (b) III (c) II

23. (a) x-axis (b) y-axis (c) x-axis

25. False

27. True

29. False

31. True

33. True

35. False

37. False

39. False

41. True

−6 −4 −2 2 4 6

−4 −2 2 4

43. (a), (c): See graph below.

(b) They all have an x-value of 3

(d) 3x =

45. (a) Answers vary for part (a). One possible set of

points is ( 3, 0), (0, 0), (2, 0), (6, 0)− .

(b) All points on the x-axis have a y-value of zero.

(c) 0y =

−2 2 4

−4 −2 2 4

−2 2 4

−2 2 4 6

55. 3 2y x= +

Graph:

57. 4 7y x= − +

Graph:

41.25==== 2

− 13

−6 −4 −2 2 4 6

−8 −6 −4 −2 2 4 6 8 10

−8 −6 −4 −2 2 4

−4 −2 2 4

The Distance and Midpoint Formulas

1. 13c =

3. 32 16 2 4 2b = = =

5. (a) – (c): See graph.

Note: Point C could also be placed at (1, 7).

(d) 222

)(53 AB=+

9. 535945 ==

11. 65

13. 373

15. ( )7 ,5

17. ( )112

, 4−

19. ( )3 92 2

21. ( )215

, 1−

23. (a)

(b) 149

(c) ( )32

25. (6,11)

27. (a) (7, 8)−

29. Point B is closer to the origin, since 6158 <

31. (a) The center of the circle is ( 1, 2)− − .

(b) The length of the radius of the circle is 65 .

−2 2 4 6 8

−4 −2 2 4 6 8

Slope and Intercepts of Lines

1. Positive

3. Zero

5. Negative

13. 54

15. Undefined

17. 52

19. 5−

23. 34

25. Undefined

27. (a) 4 1y x= − +

(c) Slope: 4−

29. (a) 23

4y x= −

(c) Slope: 23

31. Summary of slopes from numbers 27-30:

4 1y x= − + Slope: 4−

3 2y x= + Slope: 3

4y x= − Slope: 23

6y x= − + Slope: 35

The slope is the coefficient of the x-term. The

equation of a line is often written in the form

y mx b= + , and m represents the slope of the line.

33. (a) x-intercept: 4

(b) y-intercept: 2

(c) Coordinates of x-intercept: ( )4, 0

(d) Coordinates of y-intercept: ( )0, 2

(e) Slope: 12

2− 9

1 3−

0 4−

3−−−−

6−−−− 8−

3−−−−

−6 −4 −2 2 4 6 8

−6 −4 −2 2 4 6 8 10

y-intercept: 8

(b) Coordinates of x-intercept: ( )4, 0

Coordinates of y-intercept: ( )0, 8

(c), (d): See graph below.

37. (a) x-intercept: 5 14 4

y-intercept: 5−

(b) Coordinates of x-intercept: ( )14

Coordinates of y-intercept: ( )0, 5−

y-intercept: 10

y-intercept: 6−

−4 −2 2 4 6 8

2 4 6 8 10

−4 −2 2 4 6

43. (a) x-intercept: 5−

y-intercept: 10 13 3

(b) Coordinates of x-intercept: ( )5, 0−

Coordinates of y-intercept: ( )13

45. (a) x-intercept: 21 15 5

4− = −

y-intercept: 7

4 , 0−

47. (a) x-intercept: 7 12 2

y-intercept: 7 12 2

Coordinates of y-intercept: ( )12

49. (a) x-intercept: None

y-intercept: 3

(b) Coordinates of x-intercept: N/A

−6 −4 −2 2

−8 −6 −4 −2 2 4

−2 2 4 6

−4 −2 2 4

51. (a) x-intercept: 4−

y-intercept: None

(b) Coordinates of x-intercept: ( )4, 0−

Coordinates of y-intercept: N/A

53. (a) 2 8y x= −

(c) x-intercept: 4

y-intercept: 8−

(d) Slope: 2

55. Summary of y-intercepts from numbers 53 and 54:

2 8y x= − y-intercept: 8−

3y x= − + y-intercept: 3

The y-intercept is the constant term. The equation of

a line is often written in the form y mx b= + , and b

represents the y-intercept of the line.

0 8−−−−

2 4−−−−

0.5− 9−−−−

−6 −4 −2 2

−6 −4 −2 2 4 6 8 10 12

Equations of Lines

2y x= −

3. 3−−= xy

5. (a) 52 +−= xy

(b) Slope: 2− ; y-intercept: 5

7. (a) 5 1y x= −

(b) Slope: 5; y-intercept: 1−

9. (a)4xy = −

(b) Slope: 14

− ; y-intercept: 0

11. (a) 54

3y x= −

(b) Slope: 54

; y-intercept: 3−

13. (a) 25

6y x= − +

(b) Slope: 25

− ; y-intercept: 6

15. (a) 52

2y x= −

(b) Slope: 52

; y-intercept: 2−

−2 2 4 6

−6 −4 −2 2 4 6

−4 −2 2 4 6

−2 2 4 6 8 10 12 14 16

−6 −4 −2 2 4 6

Equations of Lines

17. (a)

(b) ( ) ( )23

4 6y x− = +

(c) 23

8y x= +

19. (a)

(b) ( ) ( )34

2 8y x− = − + or ( ) ( )34

7 4y x+ = − −

(c) 34

4y x= − −

21. 47

3y x= − +

23. 45

7y x= −

25. 2 49 3

y x= − +

27. 35

4y x= +

29. 7 35 5

y x= − −

31. 57

5y x= −

33. 32

6y x= − +

35. 6200800 += xC

−12 −10 −8 −6 −4 −2 2

−10 −8 −6 −4 −2 2 4 6

Parallel and Perpendicular Lines

1. Parallel

3. Perpendicular

5. Neither

7. Perpendicular

9. Parallel

11. Neither

13. Perpendicular

15. Parallel

17. Perpendicular

19. Parallel

21. (a) ( )7 2 4y x− = −

(b) 2 1y x= −

23. (a) ( )16

5 12y x− = +

(b) 16

7y x= +

25. (a) ( )54

7 3y x+ = − −

(b) 5 134 4

y x= − −

27. (a) ( )32

6 1y x− = − +

(b) 3 92 2

y x= − +

29. 4=y

31. 2x =

33. (a) 132

3 3y x= +

(b) 32

y x= −

35. 52

2y x= −

37. 192

5 5y x= − +

39. 186 +−= xy

41. 5 43 3

y x= −

An Introduction to Functions

1. This mapping does not make sense, since Erik could

not record two different temperatures at 9AM. The

mapping does not represent a function.

3. Yes, the mapping represents a function.

5. No, the mapping does not represent a function.

7. (a) ( ) 47

xf x = +

9. (a) 2( ) 6 36f x x x= − = −

(b) ( )2

( ) 6f x x= −

13. 0x ≠

Interval notation: ( ) ( ), 0 0,−∞ ∞∪

15. 3x ≠

17. 4x ≠ −

Interval notation: ( ) ( ), 4 4,−∞ − − ∞∪

19. 52

t ≠ −

Interval notation: ( ) ( )5 52 2

, ,−∞ − − ∞∪

21. 94

Interval notation: ( ) ( )9 94 4

, ,−∞ ∞∪

23. 3x ≠ − and 3x ≠

Interval notation: ( ) ( ) ( ), 3 3, 3 3,−∞ − − ∞∪ ∪

25. All real numbers

Interval Notation: ( ),−∞ ∞

29. 0≥t

Interval Notation: [ )∞,0

31. 5≥x

Interval Notation: [ )∞,5

35. 92

x ≥ −

Interval Notation: )92

,− ∞

37. 15

Interval Notation: ( 15

, −∞

39. 52

x ≤ −

Interval Notation: ( 52

, −∞ −

41. 2x ≥ and 6x ≠

Interval notation: [ ) ( )2, 6 6, ∞∪

45. 5t ≠ −

49. 53

Interval Notation: )53

x 3( ) 5f x x= −= −= −= −

2− 13−−−−

1− 6−−−−

0 5−−−−

1 4−−−−

An Introduction to Functions

55. 7x ≠

57. 4x ≠ −

59. (a) (3) 11f =

(b) 75

(c) ( ) 1312 2

f − = −

(e) ( )0 4f = −

(f) 45

61. (a) (1) 2h =

(b) 4, 2x x= =

(c) ( )2 5h − =

(d) No such value of x exists (since 3x − cannot

be negative).

(e) ( )7 4h =

(f) 10, 4x x= = −

63. (a) (7) 3h =

(b) (25) 27 9 3 3 3h = = =

(c) ( ) 39124 4

65. (a) (16) 1f =

(b) (12) 12 3 4 3 3 2 3 3f = − = − = −

(c) ( )9 0f =

67. (a) (3) 0g =

(b) ( )4 42g − =

(c) ( ) 3512 4

g − =

(d) ( )0 6g =

69. (a) 12

( 7)f − =

(b) 23

(0)f = −

(c) ( ) 72

(d) ( )3f is undefined.

(e) ( )2 0f − =

Functions and Graphs

1. No, the graph does not represent a function.

3. Yes, the graph represents a function.

7. No, the graph does not represent a function.

11. (a)

(b) No, the set of points does not represent a

function. The graph does not pass the vertical

line test at 2x = .

13. (a)

(b) Yes, the set of points represents a function. The

graph passes the vertical line test.

15. If each x value is paired with only one y value, then

the set of points represents a function. If an x value is

paired with more than one y value (i.e. two or more

coordinates have the same x value but different y

values), then the set of points does not represent a

function.

17. (a) Domain: [ ]4, 6−

(b) Range: [ ]3, 9−

(c) ( 2) 3f − =

(0) 3f = −

(4) 9f =

(6) 3f =

(d) 4x = − , 4x =

19. (a) Domain: ( ), 6−∞

(b) Range: ( ], 5−∞

(c) ( 2) 1g − =

(0) 5g =

(2) 1g = −

(4) 1g =

(6)g is undefined

(d) ( 2)g − is greater than (3)g , since 1 0> .

21. (a) Domain: ( ),−∞ ∞

2( ) 6f x x= − += − += − += − +

2− 9

1− 15

27.5====

24.5====

−4 −2 2 4

−2 2 4 6

−6 −4 −2 2 4 6

23. (a) Domain: [ )1, 3−

25. (a) Domain: ( ),−∞ ∞

27. (a) Domain: [ )3, ∞

29. (a) Domain: ( ),−∞ ∞

x ( ) 3 5h x x= −= −= −= −

1− 8−−−−

0 5−−−−

1 2−−−−

3 4 (open circle)

x ( ) 3g x x= −= −= −= −

x ( ) 3f x x= −= −= −= −

5 2 1.4≈≈≈≈

x 2( ) 4F x x x= −= −= −= −

1− 5

1 3−−−−

2 4−−−−

3 3−−−−

−6 −4 −2 2 4 6 8

−2 2 4 6 8 10 12

−4 −2 2 4 6 8

−2 2 4 6 8

31. (a) 5 8 5 8

3 3 3xy x+= = +

(b) Yes, the equation defines y as a function of x.

33. (a) 2 23 7 3 72 2 2

xy x− += = − +

35. (a) 3y x= ± +

(b) No, the equation does not define y as a function

37. (a) ( )2y x= ± +

(b) No, the equation does not define y as a function

39. (a) 5 7 5 7

+= = +

An Introduction to Polynomial Functions

1. (a) Yes

(b) Degree: 3

(c) Binomial

3. (a) Yes

(b) Degree: 1

(c) Binomial

5. (a) No

(b) N/A

(c) N/A

7. (a) No

(b) N/A

(c) N/A

9. (a) No

(b) N/A

(c) N/A

11. (a) No

(b) N/A

(c) N/A

13. (a) Yes

(b) Degree: 6

(c) Monomial

15. (a) No

(b) N/A

(c) N/A

17. (a) Yes

(b) Degree: 0

(c) Monomial

19. (a) Yes

(b) Degree: 5

(c) None of these

21. (a) Yes

(b) Degree: 7

(c) Binomial

23. (a) No

(b) N/A

(c) N/A

25. (a) Yes

(b) Degree: 9

(c) Trinomial

27. (a) False

(b) True

(c) True

(d) False

29. (a) True

(b) True

(c) False

(d) False

31. (a) True

(b) False

(c) False

(d) True

33. x-intercepts: 1, 2, 4−

y-intercept: 8

35. x-intercepts: 1, 0, 3−

y-intercept: 0

37. (a) Quadratic

(b) x-intercepts: 8, 8−

y-intercept: 64−

(c) ( 4) 48f − = −

( 1) 63f − = −

(6) 28f = −

39. (a) Cubic

(b) x-intercept: 2

y-intercept: 32

(c) ( 4) 288f − =

( 1) 36f − =

(6) 832f = −

41. (a) Linear

(b) x-intercept: 125

y-intercept: 12

(c) ( 4) 32f − =

( 1) 17f − =

(6) 18f = −

43. (a) Quadratic

(b) y-intercept: 28−

(c) ( 4) 0f − =

( 1) 24f − = −

(6) 10f = −

Adding, Subtracting, and Multiplying Polynomials

1. 2 3 40x x+ −

3. 2 4 3x x− +

5. 2 15 36x x+ +

7. 2 16 80x x− −

9. 2 16x −

11. 2 14 49x x− +

13. 22 5 3x x+ −

15. 220 39 7x x+ +

17. 23 14 8x x− +

19. 4 22 35x x+ −

21. 7 6 57 37 10x x x− + −

23. (a) 3x−

(b) 7x−

(c) 210x−

25. (a) 4 35 4x x− −

(b) 4 35 28x x− −

(c) 1060x−

27. (a) 5 327 6x x−

(b) 3 29 3 2x x x− −

(c) 315x−

29. (a) 4 370 35x x− −

(b) 23 5x x+

(c) 5350x−

31. (a) 2 10x +

(b) 4−

(c) 2 10 21x x+ +

33. (a) 2 7x x+ −

(b) 2 7x x+ +

(c) 3 22 21x x x− −

35. (a) 2 10 4x x− + −

(b) 2 20x− −

(c) 3 25 25 20x x x− +

37. (a) 2 2 9x x− −

(b) 2 6 15x x− + +

(c) 3 22 5 36 36x x x− − −

39. (a) 3 24 13x x x− −

(b) 4 3 216 6 40x x x− −

(c) 5 4 3 22 16 40 10x x x x x+ − − −

41. (a) 25 3 2x x− +

(b) 2 7 4x x+ −

(c) 4 3 26 11 3 11 3x x x x− − + −

43. (a) 5 4 32 7 4x x x x− + −

(b) 5 4 32 3 2x x x x− − + −

(c) 9 8 7 6 5 4 22 5 4 7 2 11 3x x x x x x x− + − + + − +

45. (a) 3 22 4 2 9x x x− − + +

(b) 3 24 4 2 1x x x− + + −

(c) 6 5 4 3 23 12 2 19 16 10 20x x x x x x− + + − − + +

47. (a) 2 15 48x x+ +

(b) 2 13 50x x− − −

(c) 3 213 35 49x x x+ + −

Odd-Numbered Answers to Exercise Set 3.3

Dividing Polynomials

1. Quotient: 4x − ;

Remainder: 3

3. Quotient: 6x + ;

Remainder: 8−

5. Quotient: 452

−− xx ;

Remainder: 0

7. Quotient: 5432

++ xx ;

Remainder: 7−

9. Quotient: 72 +x ;

Remainder: 145 +x

11. Quotient: 1823

21 −+ xx ;

Remainder: 8−

13. Quotient: 1532

−− xx ;

Remainder: 605 +x

15. Quotient: 2+x ;

Remainder: 24

17. Quotient: 4232

+− xx ;

Remainder: 8

19. Quotient: 4423

−+− xxx ;

Remainder: 0

21. Quotient: 1774323

−−+ xxx ;

Remainder: 75−

23. Quotient: 422

+− xx ;

Remainder: 0

25. Quotient: 6242

−+ xx ;

Remainder: 2

27. (a) Using substitution, 4)2( =P

(b) The remainder is 4 , so 4)2( =P .

29. (a) 7)1( −=−P

(b) The remainder is 7− , so 7)1( −=−P .

31. 97)5( −=P

33. ( ) 1043 −=−P

35. ( )( )2 11 24 8 3x x x x− + = − −

37. ( )( )2 7 18 2 9x x x x− − = + −

39. ( )( )24 25 21 7 4 3x x x x− − = − +

41. ( )( )22 7 5 1 2 5x x x x+ + = + +

Quadratic Functions

1. (a) Vertex: ( )3, 2− −

(b) The parabola opens upward.

(c) y-intercept: 7

(d) Axis of symmetry: 3x = −

3. (a) Vertex: ( )1, 1−

(c) y-intercept: 0

(d) Axis of symmetry: 1x =

5. (a) Vertex: ( )2, 3

(c) y-intercept: 11

7. (a) Vertex: ( )7, 0

(c) y-intercept: 49

−10 −8 −6 −4 −2 2 4

−2 2 4

−6 −4 −2 2 4 6 8 10

−7 7 14 21

Quadratic Functions

9. (a) Vertex: ( )4, 7−

(b) The parabola opens downward.

(c) y-intercept: 9−

11. (a) Vertex: ( )0, 5−

13. (a) Vertex: ( )5,15

(c) y-intercept: 115

15. (a) Vertex: ( )2, 6− −

−14 −12 −10 −8 −6 −4 −2 2 4 6

−8 −6 −4 −2 2 4 6 8

−4 −2 2 4 6 8 10 12

−8 −6 −4 −2 2 4 6

Quadratic Functions

17. (a) Vertex: ( )5 132 4

(c) y-intercept: 3

(d) Axis of symmetry: 52

19. (a) Vertex: ( )3 418 16

(c) y-intercept: 2

(d) Axis of symmetry: 38

x = −

21. (a) 2( ) 2 15f x x x= + −

(b) ( )1, 16− −

(c) Graph III

23. (a) 2( ) 2 12 10f x x x= − + −

(b) ( )3, 8

(c) Graph II

−6 −4 −2 2 4 6 8 10

−2 2

Greatest Common Factor and Factoring by Grouping

1. 26xy

3. 4 4a b

5. 3 54a c

7. 3 5 4x y z

9. ( )5 2a +

11. ( )3 5b− −

13. ( )3 3 8x y−

15. ( )2 3 4x y−

17. ( )22 3 1ab a b +

19. ( )5 3 4rt r t−

21. ( )22 2 4x x x+ −

23. ( )3 2 3 55 3 7x y xy x y− − +

25. ( )2 4 5 8 5 57 5 4 3a b c a c b ab c− +

27. ( )5 3 2 4 2 3 310 21 49c a b a c b c− − +

29. (a) ( )5y x −

(b) ( )( )4 5x x− −

31. (a) ( )3b a+

(b) ( ) ( )5 3c a+ +

33. ( )( )5 3 4a a b+ +

35. ( )( )8 2 1x x+ +

37. ( ) ( )5 2 1x x+ −

39. ( ) ( )2 3 11a a− −

41. ( )( )2b c a+ +

43. ( )( )5y z x+ −

45. ( )( )3x x y− +

47. ( )( )c d a b− −

49. ( ) ( )4 1y x− +

51. ( )( )1y x y+ −

53. ( )( )3 2 4b a+ +

55. ( ) ( )3 2 1t x t− +

57. ( )( )4 3 2c d a b− −

59. ( )( )d e f a b− − +

61. ( )( )4 5 3 2x z x y+ − −

63. ( )( )3 2x x− +

65. ( )( )4 3x x− −

67. ( )( )3 5 2 3x x+ +

69. ( )( )3 7 3 2x x+ −

71. ( ) ( )2 7 2 7x x+ + , or ( )2

2 7x +

Factoring Special Binomials and Trinomials

1. (a) 2 16x −

(b) 2 8 16x x+ +

(c) 2 8 16x x− +

3. False

5. True

7. True

9. False

11. (a) ( )( )3 3x x+ −

(b) 2 9x +

(c) ( )2

(d) ( )2

3x −

13. ( )( )7 7x x+ −

15. 2 144x +

17. ( ) ( )1 1p p+ −

19. ( )( )10 10x x+ −

21. ( )( )5 5c c+ −

23. ( ) ( )2 3 2 3b b+ −

25. ( )( )4 1 4 1x x+ −

27. ( )( )7 10 7 10x y x y+ −

29. 2 225 16c d+

31. 2 23 3

x x + −

33. x a x a

y b y b

35. 4 4

5 3 5 3

xy xy + −

37. ( )2

10x −

39. ( )2

41. ( )2

43. ( )2

2 3x −

45. ( )2

5 4x +

47. ( )2

x b−

49. ( )2

2 5bc d−

51. ( )( )3 28 2 2 4x x x x− = − + +

53. ( ) ( )24 4 16x x x+ − +

55. ( )( )23 3 9p p p− + +

57. ( )( )2 2x y x xy y− + +

59. ( )( )2 25 2 25 10 4a b a ab b− + +

Factoring Polynomials

1. (a) 13

(b) No

3. (a) 100

(b) Yes

5. (a) 36

(b) Yes

7. (a) 81

(b) Yes

9. (a) 36−

(b) No

11. ( )( )5 1x x+ −

13. ( )( )3 2x x− −

15. 2 7 12x x− −

17. ( )( )10 2x x+ +

19. ( )( )3 8x x+ −

21. ( )2

23. ( )( )7 8x x− −

25. ( )( )4 15x x+ −

27. ( ) ( )3 14x x+ +

29. ( )( )7 7x x+ −

31. 2 3x −

33. 29 25x +

35. ( )( )2 1 3x x+ −

37. ( )( )2 1 4 3x x+ −

39. ( )( )3 4 3 1x x+ −

41. ( )( )4 5 2x x+ −

43. ( )( )4 3 3 2x x− −

45. ( )9x x +

47. ( )5 4x x− −

49. ( )( )2 3 3x x+ −

51. ( )( )25 2 2x x x− + −

53. ( )( )2 4 1x x+ +

55. ( )( )10 7 6x x− − +

57. ( )( )11 2x x x+ −

59. ( )2

2x x− +

61. ( )2 2 6 6x x x+ +

63. ( )( )3 3 10 3 10x x x+ −

65. ( )( )5 2 1 5 3x x+ +

67. ( )( )( )2 5 5x x x+ + −

69. ( )( )25 4x x− +

71. ( )( )( )9 2 1 2 1x x x+ + −

Using Factoring to Solve Equations

1. 3,7 == xx

3. 6,2 −=−= xx

5. 7,5 =−= xx

7. 4,18 =−= xx

, 5x x= − =

11. 3 42 3

,x x= − = −

13. 23

1,x x= =

15. 5,5 =−= xx

17. 3 32 2

,x x= − =

19. 8,0 == xx

21. 4, 9x x= = −

23. 7,0 == xx

25. 2,2 −== xx

27. 3,6 =−= xx

29. 18

, 3x x= = −

31. 0, 3, 2x x x= = − = −

33. (a) x-intercepts: 4, 2

(b) ( )4, 0 , ( )2, 0

(c) y-intercept: 8

(d) Vertex: ( )3, 1−

(e) The parabola opens upward.

(f) Axis of symmetry: 3x =

(b) ( )4, 0

(c) y-intercept: 16

(d) Vertex: ( )4, 0

37. (a) x-intercepts: 7, 3−

(b) ( )7, 0− , ( )3, 0

(c) y-intercept: 21

(d) Vertex: ( )2, 25−

(e) The parabola opens downward.

(f) Axis of symmetry: 2x = −

−4 −2 2 4 6 8 10

−6 −4 −2 2 4 6 8 10 12 14

−12 −10 −8 −6 −4 −2 2 4 6 8 10

(b) ( )6, 0− , ( )2, 0

(d) Vertex: ( )2, 48− −

(f) Axis of symmetry: 2x = −

(b) ( )4, 0− , ( )4, 0

(d) Vertex: ( )0, 16−

43. (a) x-intercepts: 3 32 2

(b) ( )32

, 0− , ( )32

(c) y-intercept: 9

(d) Vertex: ( )0, 9

(e) The parabola opens downward.

45. 0, 2, 5− −

47. 0, 5, 5−

49. 2, 3, 3−

Note: In the following problems, there are other correct

ways of modeling the situation in part (a). The final

answer in part (b), however, is unique, regardless of the

method used for solving the problem.

51. (a) Let the width of the rectanglew =

5 the length of the rectanglew + =

Equation: 36)5( =+ww

(b) 4=w (Note that 9w = − does not make sense as

the width of the rectangle.)

Answer: The length of the rectangle is 9 cm,

and the width of the rectangle is 4

−12 −10 −8 −6 −4 −2 2 4 6 8 10

−12 −10 −8 −6 −4 −2 2 4 6 8 10 12

−8 −6 −4 −2 2 4 6 8

53. (a) Let the base of the trianglex =

3 the height of the trianglex − =

))(( trianglea of Area21 heightbase=

Equation: 90)3)((21 =−xx

(b) 15=x (Note that 12x = − does not make sense

as the base of the triangle.)

Answer: The base of the triangle is 15 cm, and

the height of the triangle is 12 cm.

Simplifying Rational Expressions

7. ( )

9. 1−

11. ( )

, or 3 3

c d d c− −−

5x −

2 2a b

19. 7 7 7

, or , or 2 2 2

+ − − +−

− − −

31. 2( 3)

33. ( 6)

35. 3( 1)

37. 4 5

39. 3 4

41. 2 7

43. 1m

47. 2 2 4x x+ +

49. 3x +

Multiplying and Dividing Rational Expressions

3 3p t

13. 5x −

15. 5x

17. ( )3 2 , or 3 2x x− + − −

21. ( )2 3 4

29. ( )

31. ( )( )

+ +−

1x −

51. ( )3 1x +

53. ( )

2 4 x+

55. ( )( )

( )( )

61. a b

Adding and Subtracting Rational Expressions

1. 14 15

3. 27 8

7 2y x

7. 2 15

11. ( )( )

− −

13. ( )

15. ( )( )

17. ( )

( ) ( )

19. 1 1

, or 3 3

− −

21. 3 13

23. 2 3

25. ( )( )

( )( )

27. ( )( )

29. ( )( )

x x+ +

31. ( )

( )( )

22 2 7

33. ( )( )

4 3x x

35. ( )( )

12 2 3 1

37. ( )( )

− −

39. ( )( )

210 37

− −

41. ( ) ( )

( )( )

43. ( )( )

Complex Fractions

15. ( )2 2

17. ( )2

ab a b

19. ( )

21. 5−

23. ( )2 1x −

25. 2 3

27. ( )

( )( )

29. 25

31. ( )( )

( )( )

− −

35. (a)

37. (a)

(b) y x

39. (a)

(b) y x

41. (a)

1 1c d

d cd c+ +

43. (a) 3 3

1 1a b

( )( )

2 23 3

2 2 2 2

b a b ab bb a

ab b a ab b a

+ + ++=

Complex Fractions

45. (a) 1

(b) 2 1

47. (a) 5

(b) 15 4

Solving Rational Equations

1. 30x =

3. 20c = −

5. 3017

x = −

7. 103

x = −

9. No solution.

( 1x = − is an extraneous solution.)

11. 4360

x = −

13. 54

15. No solution.

17. 9x = −

19. 4, 4x x= − =

21. 1, 7x x= =

23. 4t =

25. 8x =

27. 352

x = −

29. 2w =

31. 1x = −

33. 25x =

35. 5a =

37. 114

39. No solution.

( 1x = − is an extraneous solution.)

41. 22x = −

43. 2, 3x x= − =

45. 12

4,x x= − =

47. 1, 4x x= − =

49. 2, 2x x= − =

51. 12, 2x x= − = −

53. 192

0,x x= = −

55. 43

5,x x= − =

57. 29

x = −

( 1x = is an extraneous solution.)

Rational Functions

1. (a) (0) 0f =

(b) 14

( 1)f − =

(c) ( ) 1183

f = −

3. (a) 27

(0)f =

(b) 1110

( 3)f − =

(c) ( ) 24315

f = −

5. (a) ( 2)f − is undefined.

(b) 13

(0)f = −

(c) 17

(5)f =

7. (a) 3

112( 3)f − = −

(b) (0) 0f =

(c) 1223

(12)f = −

9. (a) (3) 0f =

(b) ( 4)f − is undefined.

28(0)f = −

11. (a) Domain: 3x ≠

(b) Vertical asymptote at 3x =

(c) x-intercept: 4

y-intercept: 43

(d) 32

(1)f = ; 54

( 1)f − =

(e) Graph IV

(c) x-intercept: 6−

y-intercept: 2−

(d) 72

(1)f = − ; 54

( 1)f − = −

(e) Graph II

(c) x-intercept: None

y-intercept: 2

(d) (1) 4f = ; 43

( 1)f − =

(e) Graph III

y-intercept: None

(d) (1) 4f = ; ( 1) 4f − = −

(e) Graph II

19. (a) Domain: 5x ≠ −

( ) ( ), 5 5,−∞ − − ∞∪

(b) Vertical asymptote: 5x = −

y-intercept: 2

21. (a) Domain: 2x ≠ −

( ) ( ), 2 2,−∞ − − ∞∪

(b) Vertical asymptote: 2x = −

(c) x-intercept: 6

y-intercept: 3−

( ) ( ), 0 0,−∞ ∞∪

(b) Vertical asymptote: 0x =

y-intercept: None

25. (a) Domain: 3, 3x x≠ − ≠

( ) ( ) ( ), 3 3, 3 3,−∞ − − ∞∪ ∪

(b) Vertical asymptotes: 3, 3x x= − =

(c) x-intercept: 9

y-intercept: 1−

Rational Functions

27. (a) Domain: 2, 6x x≠ ≠

( ) ( ) ( ), 2 2, 6 6,−∞ ∞∪ ∪

(b) Vertical asymptotes: 2, 6x x= =

y-intercept: 2−

( ) ( ), 1 1,−∞ ∞∪

y-intercept: 5

31. (a) Domain: 0, 8x x≠ ≠ −

( ) ( ) ( ), 8 8, 0 0,−∞ − − ∞∪ ∪

(b) Vertical asymptotes: 0, 8x x= = −

y-intercept: None

( ) ( ), 5 5,−∞ ∞∪

y-intercept: 5

35. (a) Domain: 5, 5x x≠ − ≠

( ) ( ) ( ), 5 5, 5 5,−∞ − − ∞∪ ∪

(b) Vertical asymptotes: 5, 5x x= − =

(c) x-intercept: 0

y-intercept: 0

37. (a) Domain: 75

x ≠ −

( ) ( )7 75 5

, ,−∞ − − ∞∪

(b) Vertical asymptote: 75

x = −

(c) x-intercepts: 7, 2−

y-intercept: 2−

39. (a) Domain: 1, 4x x≠ ≠

( ) ( ) ( ), 1 1, 4 4,−∞ ∞∪ ∪

(b) Vertical asymptotes: 1, 4x x= =

(c) x-intercepts: 6 65 5

y-intercept: 9−

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