Foreword v
Preface xiii
Resources xvi
Acknowledgments xix
Dedication xx
Chapter P Basic Concepts of Algebra 1
P.1 The Real Numbers and Their Properties 2
Classifying Numbers
Equality of Numbers
Classifying Sets of Numbers
Rational Numbers
Irrational Numbers
Integer Exponents
The Real Number Line
Inequalities
Sets
Definition of Union and Intersection
Intervals
Absolute Value
Distance Between Two Points on a Real Number Line
Order of Operations
Properties of the Real Numbers
Subtraction and Division of Real Numbers
Algebraic Expressions
P.2 Integer Exponents and Scientific Notation 20
Rules of Exponents
Simplifying Exponential Expressions
Scientific Notation
P.3 Polynomials 30
Polynomial Vocabulary
Adding and Subtracting Polynomials
Multiplying Polynomials
Special Products
Squaring a Binomial Sum or Difference
The Product of the Sum and Difference of Terms
P.4 Factoring Polynomials 41
The Greatest Common Monomial Factor
Factoring Out a Monomial
Factoring Trinomials of the Form x2+Bx+Cx2+Bx+C
Factoring Formulas
Perfect-Square Trinomials
Difference of Squares
Difference and Sum of Cubes
Factoring by Grouping
Factoring Trinomials of the Form Ax2+Bx+CAx2+Bx+C
P.5 Rational Expressions 50
Rational Expressions
Lowest Terms for a Rational Expression
Multiplication and Division of Rational Expressions
Addition and Subtraction of Rational Expressions
Complex Fractions
P.6 Rational Exponents and Radicals 61
Square Roots
Simplifying Square Roots
Other Roots
Like Radicals
Radicals with Different Indexes
Rationalizing Radical Expressions
Conjugates
Rational Exponents
Chapter P Review and Tests
Review Exercises
Practice Test
Chapter 1 Equations and Inequalities 80
1.1 Linear Equations in One Variable 81
Definitions
Identities, Conditional Equations, and Inconsistent Equations
Equivalent Equations
Solving Linear Equations in One Variable
Formulas
1.2 Applications of Linear Equations: Modeling 92
Solving Applied Problems
Geometry
Finance
Uniform Motion
Work Rate
Mixtures
1.3 Quadratic Equations 105
Factoring Method
The Square Root Method
Completing the Square
The Quadratic Formula
The Discriminant
Applications
Golden Rectangle
1.4 Complex Numbers: Quadratic Equations with Complex Solutions 119
Complex Numbers
Addition and Subtraction
Multiplying Complex Numbers
Complex Conjugates and Division
Quadratic Equations with Complex Solutions
1.5 Solving Other Types of Equations 130
Solving Equations by Factoring
Rational Equations
Equations Involving Radicals
Equations with Rational Exponents
Equations That Are Quadratic in Form
1.6 Inequalities 144
Linear Inequalities
Combining Two Inequalities
Using Test Points to Solve Inequalities
1.7 Equations and Inequalities Involving Absolute Value 158
Equations Involving Absolute Value
Inequalities Involving Absolute Value
Chapter 1 Review and Tests
Practice Test A
Practice Test B
Chapter 2 Graphs and Functions 174
2.1 The Coordinate Plane 175
The Coordinate Plane
Scales on a Graphing Utility
Distance Formula
Midpoint Formula
2.2 Graphs of Equations 185
Graph of an Equation
Intercepts
Symmetry
Circles
Semicircles
2.3 Lines 198
Slope of a Line
Point–Slope Form
Slope–Intercept Form
Equations of Horizontal and Vertical Lines
General Form of the Equation of a Line
Parallel and Perpendicular Lines
Modeling Data Using Linear Regression
2.4 Functions 215
Functions
Function Notation
Representations of Functions
The Domain of a Function
The Range of a Function
Graphs of Functions
Function Information from Its Graph
Building Functions
Functions in Economics
2.5 Properties of Functions 235
Increasing and Decreasing Functions
Relative Maximum and Minimum Values
Even–Odd Functions and Symmetry
Average Rate of Change
2.6 A Library of Functions 249
Linear Functions
Square Root and Cube Root Functions
Piecewise Functions
Graphing Piecewise Functions
Basic Functions
2.7 Transformations of Functions 263
Transformations
Vertical and Horizontal Shifts
Reflections
Stretching or Compressing
Multiple Transformations in Sequence
2.8 Combining Functions; Composite Functions 281
Combining Functions
Composition of Functions
Domain of Composite Functions
Decomposition of a Function
Applications of Composite Functions
2.9 Inverse Functions 294
Inverses
Finding the Inverse Function
Finding the Range of a One-to-One Function
Chapter 2 Review and Tests
Cumulative Review Exercises Chapters P–2
Chapter 3 Polynomial and Rational Functions 317
3.1 Quadratic Functions 318
Quadratic Functions
Standard Form of a Quadratic Function
Graphing a Quadratic Function f(x)=ax2+bx+cf(x)=ax2+bx+c
3.2 Polynomial Functions 332
Polynomial Functions
Power Functions
End Behavior of Polynomial Functions
Zeros of a Function
Zeros and Turning Points
Graphing a Polynomial Function
3.3 Dividing Polynomials 352
The Division Algorithm
Synthetic Division
The Remainder and Factor Theorems
3.4 The Real Zeros of a Polynomial Function 364
Real Zeros of a Polynomial Function
Rational Zeros Theorem
Descartes’s Rule of Signs
Bounds on the Real Zeros
Find the Real Zeros of a Polynomial Function
3.5 The Complex Zeros of a Polynomial Function 376
Conjugate Pairs Theorem
3.6 Rational Functions 384
Rational Functions
Vertical and Horizontal Asymptotes
Translations of f(x)=1xf(x)=1x
Graphing Rational Functions
Oblique Asymptotes
Graph of a Revenue Curve
3.7 Variation 403
Direct Variation
Inverse Variation
Joint and Combined Variation
Chapter 3 Review and Tests
Cumulative Review Exercises Chapters P–3
Chapter 4 Exponential and Logarithmic Functions 422
4.1 Exponential Functions 423
Exponential Functions
Evaluate Exponential Functions
Graphing Exponential Functions
Transformations on Exponential Functions
Simple Interest
Compound Interest
Continuous Compound Interest Formula
The Natural Exponential Function
Exponential Growth and Decay
4.2 Logarithmic Functions 444
Logarithmic Functions
Evaluating Logarithms
Basic Properties of Logarithms
Domains of Logarithmic Functions
Graphs of Logarithmic Functions
Common Logarithm
Natural Logarithm
Investments
Newton’s Law of Cooling
4.3 Rules of Logarithms 462
Rules of Logarithms
Number of Digits
Change of Base
Growth and Decay
Half-Life
Radiocarbon Dating
4.4 Exponential and Logarithmic Equations and Inequalities 475
Solving Exponential Equations
Applications of Exponential Equations
Solving Logarithmic Equations
Logarithmic and Exponential Inequalities
4.5 Logarithmic Scales; Modeling 488
pH Scale
Earthquake Intensity
Loudness of Sound
Musical Pitch
Star Brightness
Modeling
Chapter 4 Review and Tests
Cumulative Review Exercises Chapters P–4
Chapter 5 Systems of Equations and Inequalities 513
5.1 Systems of Linear Equations in Two Variables 514
System of Equations
Graphical Method
Substitution Method
Elimination Method
5.2 Systems of Linear Equations in Three Variables 527
Systems of Linear Equations
Number of Solutions of a Linear System
Nonsquare Systems
Geometric Interpretation
An Application to CAT Scans
5.3 Partial-Fraction Decomposition 539
Partial Fractions
Q(x) Has Only Distinct Linear Factors
Q(x) Has Repeated Linear Factors
Q(x) Has Distinct Irreducible Quadratic Factors
Q(x) Has Repeated Irreducible Quadratic Factors
5.4 Systems of Nonlinear Equations 551
Systems of Nonlinear Equations
Solving Systems of Nonlinear Equations by Substitution
Solving Systems of Nonlinear Equations by Elimination
5.5 Systems of Inequalities 558
Graph of a Linear Inequality in Two Variables
Systems of Linear Inequalities in Two Variables
Nonlinear Inequality
Nonlinear Systems
5.6 Linear Programming 570
Linear Programming
Solving Linear Programming Problems
Chapter 5 Review and Tests
Cumulative Review Exercises Chapters P–5
Chapter 6 Matrices and Determinants 588
6.1 Matrices and Systems of Equations 589
Definition of a Matrix
Using Matrices to Solve Linear Systems
Gaussian Elimination
Gauss–Jordan Elimination
6.2 Matrix Algebra 606
Equality of Matrices
Matrix Addition and Scalar Multiplication
Matrix Multiplication
Computer Graphics
6.3 The Matrix Inverse 621
The Multiplicative Inverse of a Matrix
Finding the Inverse of a Matrix
A Rule for Finding the Inverse of a 2×22×2 Matrix
Solving Systems of Linear Equations by Using Matrix Inverses
Applications of Matrix Inverses
Cryptography
6.4 Determinants and Cramer’s Rule 636
The Determinant of a 2×22×2 Matrix
Minors and Cofactors
The Determinant of an n×nn×n Matrix
Cramer’s Rule
Chapter 6 Review and Tests
Cumulative Review Exercises Chapters P–6
Chapter 7 Conic Sections 658
7.1 Conic Sections: Overview 659
7.2 The Parabola 661
Geometric Definition of a Parabola
Equation of a Parabola
Translations of Parabolas
Reflecting Property of Parabolas
7.3 The Ellipse 676
Definition of Ellipse
Equation of an Ellipse
Translations of Ellipses
7.4 The Hyperbola 690
Definition of Hyperbola
The Asymptotes of a Hyperbola
Graphing a Hyperbola with Center (0, 0)
Translations of Hyperbolas
Chapter 7 Review and Tests
Cumulative Review Exercises Chapters P–7
Chapter 8 Further Topics in Algebra 715
8.1 Sequences and Series 716
Sequences
Recursive Formulas
Factorial Notation
Summation Notation
Series
8.2 Arithmetic Sequences; Partial Sums 729
Arithmetic Sequence
Sum of a Finite Arithmetic Sequence
8.3 Geometric Sequences and Series 738
Geometric Sequence
Finding the Sum of a Finite Geometric Sequence
Annuities
Infinite Geometric Series
8.4 Mathematical Induction 752
Mathematical Induction
Determining the Statement Pk+1Pk+1 from the Statement PkPk
8.5 The Binomial Theorem 759
Pascal’s Triangle
The Binomial Theorem
Binomial Coefficients
8.6 Counting Principles 767
Fundamental Counting Principle
Permutations
Combinations
Distinguishable Permutations
Deciding Whether to Use Permutations, Combinations, or the Fundamental Counting Principle
8.7 Probability 778
The Probability of an Event
The Additive Rule
Mutually Exclusive Events
The Complement of an Event
Experimental Probabilities
Chapter 8 Review and Tests
Cumulative Review Exercises Chapters P–8
Answers to Selected Exercises A-1
Credits C
Index I-1