Essentials of college algebra

Cover......Page 1
Title......Page 3
Copyright......Page 4
Contents......Page 7
Preface......Page 13
Resources for Success......Page 16
R Review of Basic Concepts......Page 19
  Basic Definitions......Page 20
  Operations on Sets......Page 22
  Sets of Numbers and the Number Line......Page 26
  Exponents......Page 27
  Order of Operations......Page 28
  Properties of Real Numbers......Page 30
  Absolute Value......Page 32
  Rules for Exponents......Page 39
  Polynomials......Page 41
  Addition and Subtraction......Page 42
  Multiplication......Page 43
  Division......Page 45
  Factoring Out the Greatest Common Factor......Page 50
  Factoring Trinomials......Page 51
  Factoring Binomials......Page 53
  Factoring by Substitution......Page 55
  Lowest Terms of a Rational Expression......Page 59
  Multiplication and Division......Page 60
  Addition and Subtraction......Page 62
  Complex Fractions......Page 64
  Negative Exponents and the Quotient Rule......Page 67
  Rational Exponents......Page 70
  Complex Fractions Revisited......Page 73
  Radical Notation......Page 77
  Simplified Radicals......Page 80
  Operations with Radicals......Page 81
  Rationalizing Denominators......Page 82
 Test Prep......Page 88
 Review Exercises......Page 91
 Test......Page 95
1 Equations and Inequalities......Page 97
  Solving Linear Equations......Page 98
  Identities, Conditional Equations, and Contradictions......Page 100
  Solving for a Specified Variable (Literal Equations)......Page 101
  Solving Applied Problems......Page 104
  Motion Problems......Page 105
  Mixture Problems......Page 106
  Modeling with Linear Equations......Page 108
  Basic Concepts of Complex Numbers......Page 115
  Operations on Complex Numbers......Page 116
 1.4 Quadratic Equations......Page 122
  Solving a Quadratic Equation......Page 123
  Completing the Square......Page 124
  The Quadratic Formula......Page 125
  Solving for a Specified Variable......Page 127
  The Discriminant......Page 128
  Chapter 1 Quiz (Sections 1.1–1.4)......Page 131
  Geometry Problems......Page 132
  Using the Pythagorean Theorem......Page 133
  Height of a Projected Object......Page 134
  Modeling with Quadratic Equations......Page 135
  Rational Equations......Page 142
  Work Rate Problems......Page 144
  Equations with Radicals......Page 146
  Equations Quadratic in Form......Page 149
  Summary Exercises on Solving Equations......Page 154
  Linear Inequalities......Page 155
  Three-Part Inequalities......Page 157
  Quadratic Inequalities......Page 158
  Rational Inequalities......Page 160
  Basic Concepts......Page 167
  Absolute Value Equations......Page 168
  Absolute Value Inequalities......Page 169
  Absolute Value Models for Distance and Tolerance......Page 170
 Test Prep......Page 174
 Review Exercises......Page 178
 Test......Page 184
2 Graphs and Functions......Page 187
  The Rectangular Coordinate System......Page 188
  The Distance Formula......Page 189
  The Midpoint Formula......Page 191
  Graphing Equations in Two Variables......Page 193
  Center-Radius Form......Page 198
  General Form......Page 200
  An Application......Page 202
  Relations and Functions......Page 206
  Domain and Range......Page 208
  Determining Whether Relations Are Functions......Page 209
  Function Notation......Page 212
  Increasing, Decreasing, and Constant Functions......Page 215
  Graphing Linear Functions......Page 221
  Standard Form Ax + By = C......Page 223
  Slope......Page 224
  Linear Models......Page 227
  Chapter 2 Quiz (Sections 2.1–2.4)......Page 234
  Point-Slope Form......Page 235
  Slope-Intercept Form......Page 236
  Parallel and Perpendicular Lines......Page 238
  Modeling Data......Page 241
  Solving Linear Equations in One Variable by Graphing......Page 243
  Summary Exercises on Graphs, Circles, Functions, and Equations......Page 248
  Continuity......Page 249
  The Identity, Squaring, and Cubing Functions......Page 250
  The Absolute Value Function......Page 252
  Piecewise-Defined Functions......Page 253
  The Relation x = y2......Page 255
  Stretching and Shrinking......Page 260
  Reflecting......Page 262
  Symmetry......Page 263
  Even and Odd Functions......Page 266
  Translations......Page 267
  Chapter 2 Quiz (Sections 2.5–2.7)......Page 275
  Arithmetic Operations on Functions......Page 276
  The Difference Quotient......Page 279
  Composition of Functions and Domain......Page 280
 Test Prep......Page 289
 Review Exercises......Page 293
 Test......Page 298
3 Polynomial and Rational Functions......Page 301
  Quadratic Functions......Page 302
  Graphing Techniques......Page 303
  Completing the Square......Page 304
  The Vertex Formula......Page 307
  Quadratic Models......Page 308
 3.2 Synthetic Division......Page 318
  Synthetic Division......Page 319
  Evaluating Polynomial Functions Using the Remainder Theorem......Page 321
  Testing Potential Zeros......Page 322
  Factor Theorem......Page 324
  Rational Zeros Theorem......Page 326
  Number of Zeros......Page 327
  Conjugate Zeros Theorem......Page 329
  Finding Zeros of a Polynomial Function......Page 330
  Descartes’ Rule of Signs......Page 331
  Graphs of General Polynomial Functions......Page 336
  Behavior at Zeros......Page 338
  Turning Points and End Behavior......Page 339
  Graphing Techniques......Page 341
  Intermediate Value and Boundedness Theorems......Page 344
  Polynomial Models......Page 346
  Summary Exercises on Polynomial Functions, Zeros, and Graphs......Page 355
 3.5 Rational Functions: Graphs, Applications, and Models......Page 357
  The Reciprocal Function f (x )......Page 358
  The Function f (x )......Page 360
  Asymptotes......Page 361
  Steps for Graphing Rational Functions......Page 364
  Rational Function Models......Page 369
  Summary Exercises on Solving Equations and Inequalities......Page 377
  Direct Variation......Page 379
  Combined and Joint Variation......Page 381
 Test Prep......Page 388
 Review Exercises......Page 393
 Test......Page 398
4 Inverse, Exponential, and Logarithmic Functions......Page 401
  One-to-One Functions......Page 402
  Inverse Functions......Page 404
  Equations of Inverses......Page 406
  An Application of Inverse Functions to Cryptography......Page 410
  Exponents and Properties......Page 415
  Exponential Functions......Page 417
  Exponential Equations......Page 420
  Compound Interest......Page 422
  The Number e and Continuous Compounding......Page 424
  Exponential Models......Page 425
  Logarithms......Page 431
  Logarithmic Equations......Page 432
  Logarithmic Functions......Page 433
  Properties of Logarithms......Page 437
  Summary Exercises on Inverse, Exponential, and Logarithmic Functions......Page 444
  Common Logarithms......Page 445
  Applications and Models with Common Logarithms......Page 446
  Applications and Models with Natural Logarithms......Page 448
  Logarithms with Other Bases......Page 449
  Exponential Equations......Page 457
  Logarithmic Equations......Page 460
  Applications and Models......Page 463
  Growth Function Models......Page 469
  Decay Function Models......Page 471
  Summary Exercises on Functions: Domains and Defining Equations......Page 480
 Test Prep......Page 483
 Review Exercises......Page 486
 Test......Page 490
5 Systems and Matrices......Page 493
  Substitution Method......Page 494
  Elimination Method......Page 495
  Special Systems......Page 496
  Applying Systems of Equations......Page 498
  Solving Linear Systems with Three Unknowns (Variables)......Page 499
  Using Systems of Equations to Model Data......Page 501
  The Gauss-Jordan Method......Page 511
  Special Systems......Page 515
  Determinants......Page 522
  Cofactors......Page 523
  Evaluating n x n Determinants......Page 524
  Determinant Theorems......Page 526
  Cramer’s Rule......Page 527
  Decomposition of Rational Expressions......Page 534
  Distinct Linear Factors......Page 535
  Repeated Linear Factors......Page 536
  Distinct Linear and Quadratic Factors......Page 537
  Repeated Quadratic Factors......Page 538
  Chapter 5 Quiz (Sections 5.1–5.4)......Page 540
  Solving Nonlinear Systems with Real Solutions......Page 541
  Applying Nonlinear Systems......Page 545
  Summary Exercises on Systems of Equations......Page 551
  Solving Linear Inequalities......Page 552
  Solving Systems of Inequalities......Page 554
  Linear Programming......Page 555
  Adding Matrices......Page 563
  Special Matrices......Page 564
  Multiplying Matrices......Page 565
  Applying Matrix Algebra......Page 570
  Identity Matrices......Page 576
  Multiplicative Inverses......Page 577
  Solving Systems Using Inverse Matrices......Page 581
 Test Prep......Page 587
 Review Exercises......Page 593
 Test......Page 598
 C......Page 601
 H......Page 602
 O......Page 603
 R......Page 604
 Z......Page 605
Solutions to Selected Exercises......Page 607
Answers to Selected Exercises......Page 629
Index of Applications......Page 661
 C......Page 665
 F......Page 666
 I......Page 667
 M......Page 668
 P......Page 669
 S......Page 670
 Z......Page 671
Photo Credits......Page 672