College Algebra and Trigonometry
UV
– 4th Edition
SOLUTIONS
IA
_A
MANUAL
PP
RO
J S. Ratti
VE
Complete Solutions Manual for Instructors and
D?
Students
??
© J S. Ratti
??
All rights reserved. Reproduction or distribution without permission is prohibited.
©Medexcellence ✅��
, CONTENTS
Chapter P Basic Concepts of Algebra
P.1 The Real Numbers and Their Properties ...................................................... 1
P.2 Integer Exponents and Scientific Notation ................................................... 7
UV
P.3 Polynomials ................................................................................................ 13
P.4 Factoring Polynomials................................................................................ 20
P.5 Rational Expressions .................................................................................. 27
P.6 Rational Exponents and Radicals ............................................................... 37
Chapter P Review Exercises.................................................................................... 46
Chapter P Practice Test ........................................................................................... 51
IA
Chapter 1 Equations and Inequalities
1.1 Linear Equations in One Variable .............................................................. 53
1.2 Applications of Linear Equations: Modeling ............................................. 63
1.3 Quadratic Equations ................................................................................... 74
_A
1.4 Complex Numbers: Quadratic Equations with Complex Solutions ........... 89
1.5 Solving Other Types of Equations ............................................................. 96
1.6 Inequalities ............................................................................................... 116
1.7 Equations and Inequalities Involving Absolute Value ............................. 131
Chapter 1 Review Exercises .................................................................................. 148
Chapter 1 Practice Test A ...................................................................................... 159
PP
Chapter 1 Practice Test B ...................................................................................... 160
Chapter 2 Graphs and Functions
2.1 The Coordinate Plane ............................................................................... 163
2.2 Graphs of Equations ................................................................................. 172
RO
2.3 Lines ......................................................................................................... 185
2.4 Functions .................................................................................................. 200
2.5 Properties of Functions............................................................................. 210
2.6 A Library of Functions ............................................................................. 219
2.7 Transformations of Functions .................................................................. 230
2.8 Combining Functions; Composite Functions ........................................... 248
2.9 Inverse Functions ..................................................................................... 263
VE
Chapter 2 Review Exercises .................................................................................. 275
Chapter 2 Practice Test A ...................................................................................... 285
Chapter 2 Practice Test B ...................................................................................... 286
Cumulative Review Exercises (Chapters P−2)...................................................... 287
Chapter 3 Polynomial and Rational Functions
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3.1 Quadratic Functions ................................................................................. 291
3.2 Polynomial Functions ............................................................................... 310
3.3 Dividing Polynomials ............................................................................... 323
3.4 The Real Zeros of a Polynomial Function ................................................ 334
3.5 The Complex Zeros of a Polynomial Function ........................................ 353
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3.6 Rational Functions.................................................................................... 363
3.7 Variation ................................................................................................... 382
Chapter 3 Review Exercises .................................................................................. 387
Chapter 3 Practice Test A ...................................................................................... 403
Chapter 3 Practice Test B ...................................................................................... 404
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Cumulative Review Exercises (Chapters P−3)...................................................... 405
, Chapter 4 Exponential and Logarithmic Functions
4.1 Exponential Functions .............................................................................. 408
4.2 Logarithmic Functions ............................................................................. 420
4.3 Rules of Logarithms ................................................................................. 433
4.4 Exponential and Logarithmic Equations and Inequalities ........................ 446
UV
4.5 Logarithmic Scales; Modeling ................................................................. 459
Chapter 4 Review Exercises .................................................................................. 470
Chapter 4 Practice Test A ...................................................................................... 478
Chapter 4 Practice Test B ...................................................................................... 479
Cumulative Review Exercises (Chapters P−4)...................................................... 480
Chapter 5 Trigonometric Functions
IA
5.1 Angles and Their Measure ....................................................................... 483
5.2 Right-Triangle Trigonometry ................................................................... 490
5.3 Trigonometric Functions of Any Angle; The Unit Circle ........................ 502
5.4 Graphs of the Sine and Cosine Functions................................................. 514
_A
5.5 Graphs of the Other Trigonometric Functions ......................................... 526
5.6 Inverse Trigonometric Functions ............................................................. 538
Chapter 5 Review Exercises .................................................................................. 546
Chapter 5 Practice Test A ...................................................................................... 551
Chapter 5 Practice Test B ...................................................................................... 552
Cumulative Review Exercises (Chapters P−5)...................................................... 552
PP
Chapter 6 Trigonometric Identities and Equations
6.1 Verifying Identities .................................................................................. 555
6.2 Sum and Difference Formulas .................................................................. 568
6.3 Double-Angle and Half-Angle Formulas ................................................. 582
6.4 Product-to-Sum and Sum-to-Product Formulas ....................................... 598
RO
6.5 Trigonometric Equations I........................................................................ 606
6.6 Trigonometric Equations II ...................................................................... 616
Chapter 6 Review Exercises .................................................................................. 637
Chapter 6 Practice Test A ...................................................................................... 644
Chapter 6 Practice Test B ...................................................................................... 645
Cumulative Review Exercises (Chapters 1−6) ...................................................... 647
VE
Chapter 7 Applications of Trigonometric Functions
7.1 The Law of Sines...................................................................................... 650
7.2 The Law of Cosines.................................................................................. 664
7.3 Areas of Polygons Using Trigonometry ................................................... 678
7.4 Vectors ..................................................................................................... 693
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7.5 The Dot Product ....................................................................................... 703
7.6 Polar Coordinates ..................................................................................... 713
7.7 Polar Form of Complex Numbers; DeMoivre’s Theorem ....................... 729
Chapter 7 Review Exercises .................................................................................. 744
Chapter 7 Practice Test A ...................................................................................... 752
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Chapter 7 Practice Test B ...................................................................................... 754
Cumulative Review Exercises (Chapters 1–7) ...................................................... 756
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, Chapter 8 Systems of Equations and Inequalities
8.1 Systems of Linear Equations in Two Variables ....................................... 759
8.2 Systems of Linear Equations in Three Variables ..................................... 777
8.3 Partial-Fraction Decomposition ............................................................... 798
8.4 Systems of Nonlinear Equations .............................................................. 821
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8.5 Systems of Inequalities............................................................................. 835
8.6 Linear Programming................................................................................. 848
Chapter 8 Review Exercises .................................................................................. 863
Chapter 8 Practice Test A ...................................................................................... 876
Chapter 8 Practice Test B ...................................................................................... 879
Cumulative Review Exercises (Chapters P−8)...................................................... 881
IA
Chapter 9 Matrices and Determinants
9.1 Matrices and Systems of Equations.......................................................... 884
9.2 Matrix Algebra ......................................................................................... 905
9.3 The Matrix Inverse ................................................................................... 925
_A
9.4 Determinants and Cramer’s Rule ............................................................. 946
Chapter 9 Review Exercises .................................................................................. 959
Chapter 9 Practice Test A ...................................................................................... 970
Chapter 9 Practice Test B ...................................................................................... 971
Cumulative Review Exercises (Chapters P−9)...................................................... 972
PP
Chapter 10 The Conic Sections
10.2 The Parabola ............................................................................................. 975
10.3 The Ellipse................................................................................................ 990
10.4 The Hyperbola ........................................................................................ 1008
Chapter 10 Review Exercises .............................................................................. 1032
RO
Chapter 10 Practice Test A .................................................................................. 1041
Chapter 10 Practice Test B .................................................................................. 1043
Cumulative Review Exercises (Chapters P−10).................................................. 1044
Chapter 11 Further Topics in Algebra
11.1 Sequences and Series ............................................................................. 1047
11.2 Arithmetic Sequences; Partial Sums ...................................................... 1058
VE
11.3 Geometric Sequences and Series ............................................................ 1066
11.4 Mathematical Induction .......................................................................... 1075
11.5 The Binomial Theorem .......................................................................... 1086
11.6 Counting Principles ................................................................................ 1094
11.7 Probability .............................................................................................. 1101
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Chapter 11 Review Exercises .............................................................................. 1105
Chapter 11 Practice Test A .................................................................................. 1109
Chapter 11 Practice Test B .................................................................................. 1109
Cumulative Review Exercises (Chapters P−11).................................................. 1110
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??
UV
– 4th Edition
SOLUTIONS
IA
_A
MANUAL
PP
RO
J S. Ratti
VE
Complete Solutions Manual for Instructors and
D?
Students
??
© J S. Ratti
??
All rights reserved. Reproduction or distribution without permission is prohibited.
©Medexcellence ✅��
, CONTENTS
Chapter P Basic Concepts of Algebra
P.1 The Real Numbers and Their Properties ...................................................... 1
P.2 Integer Exponents and Scientific Notation ................................................... 7
UV
P.3 Polynomials ................................................................................................ 13
P.4 Factoring Polynomials................................................................................ 20
P.5 Rational Expressions .................................................................................. 27
P.6 Rational Exponents and Radicals ............................................................... 37
Chapter P Review Exercises.................................................................................... 46
Chapter P Practice Test ........................................................................................... 51
IA
Chapter 1 Equations and Inequalities
1.1 Linear Equations in One Variable .............................................................. 53
1.2 Applications of Linear Equations: Modeling ............................................. 63
1.3 Quadratic Equations ................................................................................... 74
_A
1.4 Complex Numbers: Quadratic Equations with Complex Solutions ........... 89
1.5 Solving Other Types of Equations ............................................................. 96
1.6 Inequalities ............................................................................................... 116
1.7 Equations and Inequalities Involving Absolute Value ............................. 131
Chapter 1 Review Exercises .................................................................................. 148
Chapter 1 Practice Test A ...................................................................................... 159
PP
Chapter 1 Practice Test B ...................................................................................... 160
Chapter 2 Graphs and Functions
2.1 The Coordinate Plane ............................................................................... 163
2.2 Graphs of Equations ................................................................................. 172
RO
2.3 Lines ......................................................................................................... 185
2.4 Functions .................................................................................................. 200
2.5 Properties of Functions............................................................................. 210
2.6 A Library of Functions ............................................................................. 219
2.7 Transformations of Functions .................................................................. 230
2.8 Combining Functions; Composite Functions ........................................... 248
2.9 Inverse Functions ..................................................................................... 263
VE
Chapter 2 Review Exercises .................................................................................. 275
Chapter 2 Practice Test A ...................................................................................... 285
Chapter 2 Practice Test B ...................................................................................... 286
Cumulative Review Exercises (Chapters P−2)...................................................... 287
Chapter 3 Polynomial and Rational Functions
D?
3.1 Quadratic Functions ................................................................................. 291
3.2 Polynomial Functions ............................................................................... 310
3.3 Dividing Polynomials ............................................................................... 323
3.4 The Real Zeros of a Polynomial Function ................................................ 334
3.5 The Complex Zeros of a Polynomial Function ........................................ 353
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3.6 Rational Functions.................................................................................... 363
3.7 Variation ................................................................................................... 382
Chapter 3 Review Exercises .................................................................................. 387
Chapter 3 Practice Test A ...................................................................................... 403
Chapter 3 Practice Test B ...................................................................................... 404
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Cumulative Review Exercises (Chapters P−3)...................................................... 405
, Chapter 4 Exponential and Logarithmic Functions
4.1 Exponential Functions .............................................................................. 408
4.2 Logarithmic Functions ............................................................................. 420
4.3 Rules of Logarithms ................................................................................. 433
4.4 Exponential and Logarithmic Equations and Inequalities ........................ 446
UV
4.5 Logarithmic Scales; Modeling ................................................................. 459
Chapter 4 Review Exercises .................................................................................. 470
Chapter 4 Practice Test A ...................................................................................... 478
Chapter 4 Practice Test B ...................................................................................... 479
Cumulative Review Exercises (Chapters P−4)...................................................... 480
Chapter 5 Trigonometric Functions
IA
5.1 Angles and Their Measure ....................................................................... 483
5.2 Right-Triangle Trigonometry ................................................................... 490
5.3 Trigonometric Functions of Any Angle; The Unit Circle ........................ 502
5.4 Graphs of the Sine and Cosine Functions................................................. 514
_A
5.5 Graphs of the Other Trigonometric Functions ......................................... 526
5.6 Inverse Trigonometric Functions ............................................................. 538
Chapter 5 Review Exercises .................................................................................. 546
Chapter 5 Practice Test A ...................................................................................... 551
Chapter 5 Practice Test B ...................................................................................... 552
Cumulative Review Exercises (Chapters P−5)...................................................... 552
PP
Chapter 6 Trigonometric Identities and Equations
6.1 Verifying Identities .................................................................................. 555
6.2 Sum and Difference Formulas .................................................................. 568
6.3 Double-Angle and Half-Angle Formulas ................................................. 582
6.4 Product-to-Sum and Sum-to-Product Formulas ....................................... 598
RO
6.5 Trigonometric Equations I........................................................................ 606
6.6 Trigonometric Equations II ...................................................................... 616
Chapter 6 Review Exercises .................................................................................. 637
Chapter 6 Practice Test A ...................................................................................... 644
Chapter 6 Practice Test B ...................................................................................... 645
Cumulative Review Exercises (Chapters 1−6) ...................................................... 647
VE
Chapter 7 Applications of Trigonometric Functions
7.1 The Law of Sines...................................................................................... 650
7.2 The Law of Cosines.................................................................................. 664
7.3 Areas of Polygons Using Trigonometry ................................................... 678
7.4 Vectors ..................................................................................................... 693
D?
7.5 The Dot Product ....................................................................................... 703
7.6 Polar Coordinates ..................................................................................... 713
7.7 Polar Form of Complex Numbers; DeMoivre’s Theorem ....................... 729
Chapter 7 Review Exercises .................................................................................. 744
Chapter 7 Practice Test A ...................................................................................... 752
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Chapter 7 Practice Test B ...................................................................................... 754
Cumulative Review Exercises (Chapters 1–7) ...................................................... 756
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, Chapter 8 Systems of Equations and Inequalities
8.1 Systems of Linear Equations in Two Variables ....................................... 759
8.2 Systems of Linear Equations in Three Variables ..................................... 777
8.3 Partial-Fraction Decomposition ............................................................... 798
8.4 Systems of Nonlinear Equations .............................................................. 821
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8.5 Systems of Inequalities............................................................................. 835
8.6 Linear Programming................................................................................. 848
Chapter 8 Review Exercises .................................................................................. 863
Chapter 8 Practice Test A ...................................................................................... 876
Chapter 8 Practice Test B ...................................................................................... 879
Cumulative Review Exercises (Chapters P−8)...................................................... 881
IA
Chapter 9 Matrices and Determinants
9.1 Matrices and Systems of Equations.......................................................... 884
9.2 Matrix Algebra ......................................................................................... 905
9.3 The Matrix Inverse ................................................................................... 925
_A
9.4 Determinants and Cramer’s Rule ............................................................. 946
Chapter 9 Review Exercises .................................................................................. 959
Chapter 9 Practice Test A ...................................................................................... 970
Chapter 9 Practice Test B ...................................................................................... 971
Cumulative Review Exercises (Chapters P−9)...................................................... 972
PP
Chapter 10 The Conic Sections
10.2 The Parabola ............................................................................................. 975
10.3 The Ellipse................................................................................................ 990
10.4 The Hyperbola ........................................................................................ 1008
Chapter 10 Review Exercises .............................................................................. 1032
RO
Chapter 10 Practice Test A .................................................................................. 1041
Chapter 10 Practice Test B .................................................................................. 1043
Cumulative Review Exercises (Chapters P−10).................................................. 1044
Chapter 11 Further Topics in Algebra
11.1 Sequences and Series ............................................................................. 1047
11.2 Arithmetic Sequences; Partial Sums ...................................................... 1058
VE
11.3 Geometric Sequences and Series ............................................................ 1066
11.4 Mathematical Induction .......................................................................... 1075
11.5 The Binomial Theorem .......................................................................... 1086
11.6 Counting Principles ................................................................................ 1094
11.7 Probability .............................................................................................. 1101
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Chapter 11 Review Exercises .............................................................................. 1105
Chapter 11 Practice Test A .................................................................................. 1109
Chapter 11 Practice Test B .................................................................................. 1109
Cumulative Review Exercises (Chapters P−11).................................................. 1110
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