College Algebra

1. Prerequisites

• 1.1 Introduction
  • 1.1.1 Introduction to Algebra
  • 1.1.2 The Top Ten List of Mistakes

• 1.2 Inequalities
  • 1.2.1 Concepts of Inequality
  • 1.2.2 Inequalities and Interval Notation

• 1.3 Absolute Value
  • 1.3.1 Properties of Absolute Value
  • 1.3.2 Evaluating Absolute Value Expressions

• 1.4 Exponents
  • 1.4.1 An Introduction to Exponents
  • 1.4.2 Evaluating Exponential Expressions
  • 1.4.3 Applying the Rules of Exponents
  • 1.4.4 Evaluating Expressions with Negative Exponents

• 1.5 Converting between Notations
  • 1.5.1 Converting between Decimal and Scientific Notation
  • 1.5.2 Converting Rational Exponents and Radicals

• 1.6 Radical Expressions
  • 1.6.1 Simplifying Radical Expressions
  • 1.6.2 Simplifying Radical Expressions with Variables
  • 1.6.3 Rationalizing Denominators

• 1.7 Polynomial Expressions
  • 1.7.1 Determining Components and Degree
  • 1.7.2 Adding, Subtracting, and Multiplying Polynomials
  • 1.7.3 Multiplying Big Products
  • 1.7.4 Using Special Products

• 1.8 Factoring
  • 1.8.1 Factoring Using the Greatest Common Factor
  • 1.8.2 Factoring by Grouping
  • 1.8.3 Factoring Trinomials Completely
  • 1.8.4 Factoring Trinomials: The Grouping Method

• 1.9 Factoring Patterns
  • 1.9.1 Factoring Perfect Square Trinomials
  • 1.9.2 Factoring the Difference of Two Squares
  • 1.9.3 Factoring Sums and Differences of Cubes
  • 1.9.4 Factoring by Any Method

• 1.10 Rational Expressions
  • 1.10.1 Rational Expressions and Domain
  • 1.10.2 Working with Fractions
  • 1.10.3 Writing Rational Expressions in Lowest Terms

• 1.11 Working with Rationals
  • 1.11.1 Multiplying and Dividing Rational Expressions
  • 1.11.2 Adding and Subtracting Rational Expressions
  • 1.11.3 Rewriting Complex Fractions

• 1.12 Complex Numbers
  • 1.12.1 Introducing and Writing Complex Numbers
  • 1.12.2 Rewriting Powers of i
  • 1.12.3 Adding and Subtracting Complex Numbers
  • 1.12.4 Multiplying Complex Numbers
  • 1.12.5 Dividing Complex Numbers

2. Equations and Inequalities

• 2.1 Linear Equations
  • 2.1.1 An Introduction to Solving Equations
  • 2.1.2 Solving a Linear Equation
  • 2.1.3 Solving a Linear Equation with Rationals
  • 2.1.4 Solving a Linear Equation That Has Restrictions

• 2.2 Word Problems with Linear Equations: Math Topics
  • 2.2.1 An Introduction to Solving Word Problems
  • 2.2.2 Solving for Perimeter
  • 2.2.3 Solving a Linear Geometry Problem
  • 2.2.4 Solving for Consecutive Numbers
  • 2.2.5 Solving to Find the Average

• 2.3 Word Problems with Linear Equations: Applications
  • 2.3.1 Solving for Constant Velocity
  • 2.3.2 Solving a Problem about Work
  • 2.3.3 Solving a Mixture Problem
  • 2.3.4 Solving an Investment Problem
  • 2.3.5 Solving Business Problems

• 2.4 Quadratic Equations: Some Solution Techniques
  • 2.4.1 Solving Quadratics by Factoring
  • 2.4.2 Solving Quadratics by Completing the Square
  • 2.4.3 Completing the Square: Another Example

• 2.5 Quadratic Equations and the Quadratic Formula
  • 2.5.1 Proving the Quadratic Formula
  • 2.5.2 Using the Quadratic Formula
  • 2.5.3 Predicting the Type of Solutions Using the Discriminant

• 2.6 Quadratic Equations: Special Topics
  • 2.6.1 Solving for a Squared Variable
  • 2.6.2 Finding Real Number Restrictions
  • 2.6.3 Solving Fancy Quadratics

• 2.7 Word Problems with Quadratics: Math Topics
  • 2.7.1 An Introduction to Word Problems with Quadratics
  • 2.7.2 Solving a Quadratic Geometry Problem
  • 2.7.3 Solving with the Pythagorean Theorem
  • 2.7.4 The Pythagorean Theorem: Another Example

• 2.8 Word Problems with Quadratics: Applications
  • 2.8.1 Solving a Motion Problem
  • 2.8.2 Solving a Projectile Problem
  • 2.8.3 Solving Other Problems

• 2.9 Radical Equations
  • 2.9.1 Determining Extraneous Roots
  • 2.9.2 Solving an Equation Containing a Radical
  • 2.9.3 Solving an Equation with Two Radicals
  • 2.9.4 Solving an Equation with Rational Exponents

• 2.10 Variation
  • 2.10.1 An Introduction to Variation
  • 2.10.2 Direct Proportion
  • 2.10.3 Inverse Proportion

• 2.11 Solving Inequalities
  • 2.11.1 An Introduction to Solving Inequalities
  • 2.11.2 Solving Compound Inequalities
  • 2.11.3 More on Compound Inequalities
  • 2.11.4 Solving Word Problems Involving Inequalities

• 2.12 Inequalities: Quadratics
  • 2.12.1 Solving Quadratic Inequalities
  • 2.12.2 Solving Quadratic Inequalities: Another Example

• 2.13 Inequalities: Rationals and Radicals
  • 2.13.1 Solving Rational Inequalities
  • 2.13.2 Solving Rational Inequalities: Another Example
  • 2.13.3 Determining the Domains of Expressions with Radicals

• 2.14 Absolute Value
  • 2.14.1 Matching Number Lines with Absolute Values
  • 2.14.2 Solving Absolute Value Equations
  • 2.14.3 Solving Equations with Two Absolute Value Expressions
  • 2.14.4 Solving Absolute Value Inequalities
  • 2.14.5 Solving Absolute Value Inequalities: More Examples

3. Relations and Functions

• 3.1 Graphing Basics
  • 3.1.1 Using the Cartesian System
  • 3.1.2 Thinking Visually

• 3.2 Relationships between Two Points
  • 3.2.1 Finding the Distance between Two Points
  • 3.2.2 Finding the Second Endpoint of a Segment

• 3.3 Relationships among Three Points
  • 3.3.1 Collinearity and Distance
  • 3.3.2 Triangles

• 3.4 Circles
  • 3.4.1 Finding the Center-Radius Form of the Equation of a Circle
  • 3.4.2 Decoding the Circle Formula
  • 3.4.3 Finding the Center and Radius of a Circle
  • 3.4.4 Solving Word Problems Involving Circles

• 3.5 Graphing Equations
  • 3.5.1 Graphing Equations by Locating Points
  • 3.5.2 Finding the x- and y-Intercepts of an Equation

• 3.6 Function Basics
  • 3.6.1 Functions and the Vertical Line Test
  • 3.6.2 Identifying Functions
  • 3.6.3 Function Notation and Finding Function Values

• 3.7 Working with Functions
  • 3.7.1 Determining Intervals Over Which a Function Is Increasing
  • 3.7.2 Evaluating Piecewise-Defined Functions for Given Values
  • 3.7.3 Solving Word Problems Involving Functions

• 3.8 Function Domain and Range
  • 3.8.1 Finding the Domain and Range of a Function
  • 3.8.2 Domain and Range: One Explicit Example
  • 3.8.3 Satisfying the Domain of a Function

• 3.9 Linear Functions: Slope
  • 3.9.1 An Introduction to Slope
  • 3.9.2 Finding the Slope of a Line Given Two Points
  • 3.9.3 Interpreting Slope from a Graph
  • 3.9.4 Graphing a Line Using Point and Slope

• 3.10 Equations of a Line
  • 3.10.1 Writing an Equation in Slope-Intercept Form
  • 3.10.2 Writing an Equation Given Two Points
  • 3.10.3 Writing an Equation in Point-Slope Form
  • 3.10.4 Matching a Slope-Intercept Equation with Its Graph
  • 3.10.5 Slope with Parallel and Perpendicular Lines

• 3.11 Linear Functions: Applications
  • 3.11.1 Constructing Linear Function Models of a Set of Data
  • 3.11.2 Linear Cost and Revenue Functions

• 3.12 Graphing Functions
  • 3.12.1 Graphing Some Important Functions
  • 3.12.2 Graphing Piecewise-Defined Functions
  • 3.12.3 Matching Equations with Their Graphs

• 3.13 The Greatest Integer Function
  • 3.13.1 The Greatest Integer Function
  • 3.13.2 Graphing the Greatest Integer Function

• 3.14 Composite Functions
  • 3.14.1 Using Operations on Functions
  • 3.14.2 Composite Functions
  • 3.14.3 Components of Composite Functions
  • 3.14.4 Finding Functions That Form a Given Composite
  • 3.14.5 Finding the Difference Quotient of a Function
  • 3.14.6 Calculating the Average Rate of Change

• 3.15 Quadratic Functions: Basics
  • 3.15.1 Deconstructing the Graph of a Quadratic Function
  • 3.15.2 Nice-Looking Parabolas
  • 3.15.3 Using Discriminants to Graph Parabolas
  • 3.15.4 Maximum Height in the Real World

• 3.16 Quadratic Functions: The Vertex
  • 3.16.1 Finding the Vertex by Completing the Square
  • 3.16.2 Using the Vertex to Write the Quadratic Equation
  • 3.16.3 Finding the Maximum or Minimum of a Quadratic
  • 3.16.4 Graphing Parabolas

• 3.17 Manipulating Graphs: Shifts and Stretches
  • 3.17.1 Shifting Curves along Axes
  • 3.17.2 Shifting or Translating Curves along Axes
  • 3.17.3 Stretching a Graph
  • 3.17.4 Graphing Quadratics Using Patterns

• 3.18 Manipulating Graphs: Symmetry and Reflections
  • 3.18.1 Determining Symmetry
  • 3.18.2 Reflections
  • 3.18.3 Reflecting Specific Functions

4. Polynomial and Rational Functions

• 4.1 Polynomials: Long Division
  • 4.1.1 Using Long Division with Polynomials
  • 4.1.2 Long Division: Another Example

• 4.2 Polynomials: Synthetic Division
  • 4.2.1 Using Synthetic Division with Polynomials
  • 4.2.2 More Synthetic Division

• 4.3 The Remainder Theorem
  • 4.3.1 The Remainder Theorem
  • 4.3.2 More on the Remainder Theorem

• 4.4 The Factor Theorem
  • 4.4.1 The Factor Theorem and Its Uses
  • 4.4.2 Factoring a Polynomial Given a Zero

• 4.5 The Rational Root Theorem
  • 4.5.1 Presenting the Rational Zero Theorem
  • 4.5.2 Considering Possible Solutions

• 4.6 Zeros of Polynomials
  • 4.6.1 Finding Polynomials Given Zeros, Degree, and One Point
  • 4.6.2 Finding all Zeros and Multiplicities of a Polynomial
  • 4.6.3 Finding the Real Zeros for a Polynomial
  • 4.6.4 Using Descartes' Rule of Signs
  • 4.6.5 Finding the Zeros of a Polynomial from Start to Finish
  • 4.6.6 The Fundamental Theorem of Algebra
  • 4.6.7 The Conjugate Pair Theorem

• 4.7 Graphing Polynomials
  • 4.7.1 Matching Graphs to Polynomial Functions
  • 4.7.2 Sketching the Graphs of Basic Polynomial Functions
  • 4.7.3 Graphing Polynomial Functions: Another Example

• 4.8 Rational Functions
  • 4.8.1 Understanding Rational Functions
  • 4.8.2 Basic Rational Functions

• 4.9 Graphing Rational Functions
  • 4.9.1 Vertical Asymptotes
  • 4.9.2 Horizontal Asymptotes
  • 4.9.3 Graphing Rational Functions
  • 4.9.4 Graphing Rational Functions: More Examples
  • 4.9.5 Oblique Asymptotes
  • 4.9.6 Oblique Asymptotes: Another Example

5. Exponential and Logarithmic Functions

• 5.1 Function Inverses
  • 5.1.1 Understanding Inverse Functions
  • 5.1.2 The Horizontal Line Test
  • 5.1.3 Are Two Functions Inverses of Each Other?
  • 5.1.4 Graphing the Inverse

• 5.2 Finding Function Inverses
  • 5.2.1 Finding the Inverse of a Function
  • 5.2.2 Finding the Inverse of a Function with Higher Powers

• 5.3 Exponential Functions
  • 5.3.1 An Introduction to Exponential Functions
  • 5.3.2 Graphing Exponential Functions: Useful Patterns
  • 5.3.3 Graphing Exponential Functions: More Examples

• 5.4 Applying Exponential Functions
  • 5.4.1 Using Properties of Exponents to Solve Exponential Equations
  • 5.4.2 Finding Present Value and Future Value
  • 5.4.3 Finding an Interest Rate to Match Given Goals

• 5.5 The Number e
  • 5.5.1 e
  • 5.5.2 Applying Exponential Functions

• 5.6 Logarithmic Functions
  • 5.6.1 An Introduction to Logarithmic Functions
  • 5.6.2 Converting between Exponential and Logarithmic Functions

• 5.7 Solving Logarithmic Functions
  • 5.7.1 Finding the Value of a Logarithmic Function
  • 5.7.2 Solving for x in Logarithmic Equations
  • 5.7.3 Graphing Logarithmic Functions
  • 5.7.4 Matching Logarithmic Functions with Their Graphs

• 5.8 Properties of Logarithms
  • 5.8.1 Properties of Logarithms
  • 5.8.2 Expanding a Logarithmic Expression Using Properties
  • 5.8.3 Combining Logarithmic Expressions

• 5.9 Evaluating Logarithmic Functions
  • 5.9.1 Evaluating Logarithmic Functions Using a Calculator
  • 5.9.2 Using the Change of Base Formula

• 5.10 Applying Logarithmic Functions
  • 5.10.1 The Richter Scale
  • 5.10.2 The Distance Modulus Formula

• 5.11 Solving Exponential and Logarithmic Equations
  • 5.11.1 Solving Exponential Equations
  • 5.11.2 Solving Logarithmic Equations
  • 5.11.3 Solving Equations with Logarithmic Exponents

• 5.12 Applying Exponents and Logarithms
  • 5.12.1 Compound Interest
  • 5.12.2 Predicting Change

• 5.13 Word Problems Involving Exponential Growth and Decay
  • 5.13.1 An Introduction to Exponential Growth and Decay
  • 5.13.2 Half-Life
  • 5.13.3 Newton's Law of Cooling
  • 5.13.4 Continuously Compounded Interest

6. Systems of Equations

• 6.1 Linear Systems of Equations
  • 6.1.1 An Introduction to Linear Systems
  • 6.1.2 Solving Systems with Substitution
  • 6.1.3 Solving Systems by Elimination

• 6.2 Linear Systems in Three Variables
  • 6.2.1 An Introduction to Linear Systems in Three Variables
  • 6.2.2 Solving Linear Systems in Three Variables
  • 6.2.3 Solving Inconsistent Systems
  • 6.2.4 Solving Dependent Systems
  • 6.2.5 Solving Systems with Two Equations

• 6.3 Applying Linear Systems
  • 6.3.1 Investments
  • 6.3.2 Solving with Partial Fractions
  • 6.3.3 Partial Fractions: Another Example

• 6.4 Nonlinear Systems of Equations
  • 6.4.1 Solving Nonlinear Systems Using Elimination
  • 6.4.2 Solving Nonlinear Systems with Substitution

• 6.5 Matrices
  • 6.5.1 An Introduction to Matrices
  • 6.5.2 The Arithmetic of Matrices
  • 6.5.3 Multiplying Matrices by a Scalar
  • 6.5.4 Multiplying Matrices

• 6.6 Solving Systems Using the Gauss-Jordan Method
  • 6.6.1 Using the Gauss-Jordan Method
  • 6.6.2 Using Gauss-Jordan: Another Example
  • 6.6.3 Using the Gauss-Jordan Method with Three Equations
  • 6.6.4 Gaussian Elimination with Three Equations

• 6.7 Evaluating Determinants
  • 6.7.1 Evaluating 2x2 Determinants
  • 6.7.2 Evaluating nxn Determinants
  • 6.7.3 Evaluating a Determinant Using Elementary Row Operations
  • 6.7.4 Finding a Determinant using Expanding by Cofactors
  • 6.7.5 Applying Determinants

• 6.8 Cramer's Rule
  • 6.8.1 Using Cramer's Rule
  • 6.8.2 Using Cramer's Rule in a 3x3 Matrix

• 6.9 Inverses and Matrices
  • 6.9.1 An Introduction to Inverses
  • 6.9.2 Inverses: 2x2 Matrices
  • 6.9.3 Another Look at 2x2 Inverses
  • 6.9.4 Inverses: 3x3 Matrices
  • 6.9.5 Solving a System of Equations with Inverses

• 6.10 Systems of Inequalities
  • 6.10.1 An Introduction to Graphing Linear Inequalities
  • 6.10.2 Graphing Linear and Nonlinear Inequalities
  • 6.10.3 Graphing the Solution Set of a System of Inequalities

• 6.11 Linear Programming
  • 6.11.1 Solving for Maxima-Minima
  • 6.11.2 Applying Linear Programming

7. Conic Sections

• 7.1 Parabolas
  • 7.1.1 An Introduction to Conic Sections
  • 7.1.2 An Introduction to Parabolas
  • 7.1.3 Determining Information about a Parabola from Its Equation
  • 7.1.4 Writing an Equation for a Parabola

• 7.2 Ellipses
  • 7.2.1 An Introduction to Ellipses
  • 7.2.2 Finding the Equation for an Ellipse
  • 7.2.3 Applying Ellipses: Satellites
  • 7.2.4 The Eccentricity of an Ellipse

• 7.3 Hyperbolas
  • 7.3.1 An Introduction to Hyperbolas
  • 7.3.2 Finding the Equation for a Hyperbola
  • 7.3.3 Applying Hyperbolas: Navigation

• 7.4 Conic Sections
  • 7.4.1 Identifying a Conic
  • 7.4.2 Name That Conic

8. Further Topics in Algebra

• 8.1 The Binomial Theorem
  • 8.1.1 Using the Binomial Theorem
  • 8.1.2 Binomial Coefficients
  • 8.1.3 Finding a Term of a Binomial Expansion

• 8.2 Sequences
  • 8.2.1 General and Specific Terms
  • 8.2.2 Understanding Sequence Problems
  • 8.2.3 Series Notation, Definitions and Evaluating
  • 8.2.4 Solving Problems Involving Arithmetic Sequences
  • 8.2.5 Finding the Sum of an Arithmetic Sequence
  • 8.2.6 Solving Problems Involving Geometric Sequences
  • 8.2.7 Finding the Sum of a Geometric Sequence

• 8.3 Induction
  • 8.3.1 Proving Formulas Using Mathematical Induction
  • 8.3.2 Examples of Induction

• 8.4 Combinations and Probability
  • 8.4.1 Solving Problems Involving Permutations
  • 8.4.2 Solving Problems Involving Combinations
  • 8.4.3 Independent Events
  • 8.4.4 Inclusive and Exclusive Events

9. Conclusion

• 9.1 Conclusion
  • 9.1.1 Final Close