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Algebra and Trigonometry

Review Exercises

Algebra and TrigonometryReview Exercises
  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Chapter Review
      1. Key Terms
      2. Key Concepts
    6. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Unit Circle: Sine and Cosine Functions
    1. Introduction to The Unit Circle: Sine and Cosine Functions
    2. 7.1 Angles
    3. 7.2 Right Triangle Trigonometry
    4. 7.3 Unit Circle
    5. 7.4 The Other Trigonometric Functions
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Periodic Functions
    1. Introduction to Periodic Functions
    2. 8.1 Graphs of the Sine and Cosine Functions
    3. 8.2 Graphs of the Other Trigonometric Functions
    4. 8.3 Inverse Trigonometric Functions
    5. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    6. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 9.1 Solving Trigonometric Equations with Identities
    3. 9.2 Sum and Difference Identities
    4. 9.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 9.4 Sum-to-Product and Product-to-Sum Formulas
    6. 9.5 Solving Trigonometric Equations
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 10.1 Non-right Triangles: Law of Sines
    3. 10.2 Non-right Triangles: Law of Cosines
    4. 10.3 Polar Coordinates
    5. 10.4 Polar Coordinates: Graphs
    6. 10.5 Polar Form of Complex Numbers
    7. 10.6 Parametric Equations
    8. 10.7 Parametric Equations: Graphs
    9. 10.8 Vectors
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 11.1 Systems of Linear Equations: Two Variables
    3. 11.2 Systems of Linear Equations: Three Variables
    4. 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 11.4 Partial Fractions
    6. 11.5 Matrices and Matrix Operations
    7. 11.6 Solving Systems with Gaussian Elimination
    8. 11.7 Solving Systems with Inverses
    9. 11.8 Solving Systems with Cramer's Rule
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 12.1 The Ellipse
    3. 12.2 The Hyperbola
    4. 12.3 The Parabola
    5. 12.4 Rotation of Axes
    6. 12.5 Conic Sections in Polar Coordinates
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. 13 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 13.1 Sequences and Their Notations
    3. 13.2 Arithmetic Sequences
    4. 13.3 Geometric Sequences
    5. 13.4 Series and Their Notations
    6. 13.5 Counting Principles
    7. 13.6 Binomial Theorem
    8. 13.7 Probability
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  15. A | Proofs, Identities, and Toolkit Functions
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index

Review Exercises

Real Numbers: Algebra Essentials

For the following exercises, perform the given operations.

1.

( 532 ) 2 6 ( 532 ) 2 6

2.

64÷( 28 )+14÷7 64÷( 28 )+14÷7

3.

2 5 2 +6÷2 2 5 2 +6÷2

For the following exercises, solve the equation.

4.

5x+9=−11 5x+9=−11

5.

2y+ 4 2 =64 2y+ 4 2 =64

For the following exercises, simplify the expression.

6.

9( y+2 )÷32+1 9( y+2 )÷32+1

7.

3m( 4+7 )m 3m( 4+7 )m

For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

8.

11

9.

0

10.

5 6 5 6

11.

11 11

Exponents and Scientific Notation

For the following exercises, simplify the expression.

12.

2 2 2 4 2 2 2 4

13.

4 5 4 3 4 5 4 3

14.

( a 2 b 3 ) 4 ( a 2 b 3 ) 4

15.

6 a 2 a 0 2 a −4 6 a 2 a 0 2 a −4

16.

( xy ) 4 y 3 2 x 5 ( xy ) 4 y 3 2 x 5

17.

4 −2 x 3 y −3 2 x 0 4 −2 x 3 y −3 2 x 0

18.

( 2 x 2 y ) −2 ( 2 x 2 y ) −2

19.

( 16 a 3 b 2 ) ( 4a b −1 ) −2 ( 16 a 3 b 2 ) ( 4a b −1 ) −2

20.

Write the number in standard notation: 2.1314× 10 −6 2.1314× 10 −6

21.

Write the number in scientific notation: 16,340,000

Radicals and Rational Expressions

For the following exercises, find the principal square root.

22.

121 121

23.

196 196

24.

361 361

25.

75 75

26.

162 162

27.

32 25 32 25

28.

80 81 80 81

29.

49 1250 49 1250

30.

2 4+ 2 2 4+ 2

31.

4 3 +6 3 4 3 +6 3

32.

12 5 13 5 12 5 13 5

33.

−243 5 −243 5

34.

250 3 −8 3 250 3 −8 3

Polynomials

For the following exercises, perform the given operations and simplify.

35.

(3 x 3 +2x1)+(4 x 2 2x+7) (3 x 3 +2x1)+(4 x 2 2x+7)

36.

( 2y+1 )( 2 y 2 2y5 ) ( 2y+1 )( 2 y 2 2y5 )

37.

(2 x 2 +3x6)+(3 x 2 4x+9) (2 x 2 +3x6)+(3 x 2 4x+9)

38.

( 6 a 2 +3a+10 )( 6 a 2 −3a+5 ) ( 6 a 2 +3a+10 )( 6 a 2 −3a+5 )

39.

(k+3)(k6) (k+3)(k6)

40.

(2h+1)(3h2) (2h+1)(3h2)

41.

( x+1 )( x 2 +1 ) ( x+1 )( x 2 +1 )

42.

(m2)( m 2 +2m3) (m2)( m 2 +2m3)

43.

( a+2b )( 3ab ) ( a+2b )( 3ab )

44.

( x+y )( xy ) ( x+y )( xy )

Factoring Polynomials

For the following exercises, find the greatest common factor.

45.

81p+9pq27 p 2 q 2 81p+9pq27 p 2 q 2

46.

12 x 2 y+4x y 2 −18xy 12 x 2 y+4x y 2 −18xy

47.

88 a 3 b+4 a 2 b144 a 2 88 a 3 b+4 a 2 b144 a 2

For the following exercises, factor the polynomial.

48.

2 x 2 9x18 2 x 2 9x18

49.

8 a 2 +30a27 8 a 2 +30a27

50.

d 2 5d66 d 2 5d66

51.

x 2 +10x+25 x 2 +10x+25

52.

y 2 6y+9 y 2 6y+9

53.

4 h 2 12hk+9 k 2 4 h 2 12hk+9 k 2

54.

361 x 2 121 361 x 2 121

55.

p 3 +216 p 3 +216

56.

8 x 3 125 8 x 3 125

57.

64 q 3 27 p 3 64 q 3 27 p 3

58.

4x (x1) 1 4 +3 (x1) 3 4 4x (x1) 1 4 +3 (x1) 3 4

59.

3p ( p+3 ) 1 3 −8 ( p+3 ) 4 3 3p ( p+3 ) 1 3 −8 ( p+3 ) 4 3

60.

4r ( 2r1 ) 2 3 5 ( 2r1 ) 1 3 4r ( 2r1 ) 2 3 5 ( 2r1 ) 1 3

Rational Expressions

For the following exercises, simplify the expression.

61.

x 2 x12 x 2 8x+16 x 2 x12 x 2 8x+16

62.

4 y 2 25 4 y 2 20y+25 4 y 2 25 4 y 2 20y+25

63.

2 a 2 a3 2 a 2 6a8 5 a 2 19a4 10 a 2 13a3 2 a 2 a3 2 a 2 6a8 5 a 2 19a4 10 a 2 13a3

64.

d4 d 2 9 d3 d 2 16 d4 d 2 9 d3 d 2 16

65.

m 2 +5m+6 2 m 2 5m3 ÷ 2 m 2 +3m9 4 m 2 4m3 m 2 +5m+6 2 m 2 5m3 ÷ 2 m 2 +3m9 4 m 2 4m3

66.

4 d 2 7d2 6 d 2 17d+10 ÷ 8 d 2 +6d+1 6 d 2 +7d10 4 d 2 7d2 6 d 2 17d+10 ÷ 8 d 2 +6d+1 6 d 2 +7d10

67.

10 x + 6 y 10 x + 6 y

68.

12 a 2 +2a+1 3 a 2 −1 12 a 2 +2a+1 3 a 2 −1

69.

1 d + 2 c 6c+12d dc 1 d + 2 c 6c+12d dc

70.

3 x 7 y 2 x 3 x 7 y 2 x

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