Real Numbers: Algebra Essentials

Introduction to Dividing Polynomials

Introduction to Real Numbers: Algebra Essentials

Use long division to divide polynomials

Classifying a Real Number

Use synthetic division to divide polynomials

Performing Calculations Using the Order of Operations

Use polynomial division to solve application problems

Using Properties of Real Numbers

Key Concepts & Glossary

Evaluating Algebraic Expressions

Section Exercises

Formulas

Solutions

Simplifying Algebraic Expressions

Zeros of Polynomial Functions

Key Concepts & Glossary

Introduction to Zeros of Polynomials

Section Exercises

Evaluate a polynomial using the Remainder Theorem

Solutions

Use the Factor Theorem to solve a polynomial equation

Linear Equations in One Variable

Use the Rational Zero Theorem to find rational zeros

Introduction to Linear Equations in One Variable

Find zeros of a polynomial function

Solving Linear Equations in One Variable

Use the Fundamental Theorem of Algebra

Solving a Rational Equation

Use the Linear Factorization Theorem to find polynomials with given zeros

Finding a Linear Equation

Use Descartes’ Rule of Signs

Determining Whether Graphs of Lines are Parallel or Perpendicular

Solve real-world applications of polynomial equations

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Models and Applications

Rational Functions

Introduction: Models and Applicaitons

Introduction to Rational Functions

Setting up a Linear Equation to Solve a Real-World Application

Use arrow notation

Using a Formula to Solve a Real-World Application

Solve applied problems involving rational functions

Key Concepts & Glossary

Find the domains of rational functions

Section Exercises

Identify vertical and horizontal asymptotes

Solutions to Selected Exercises

Identify horizontal asymptotes

Complex Numbers

Graph rational functions

Introduction to Complex Numbers

Key Concepts & Glossary

Express square roots of negative numbers as multiples of  i

Section Exercises

Plot complex numbers on the complex plane

Solutions

Add and subtract complex numbers

Inverses and Radical Functions

Multiply and divide complex numbers

Introduction to Inverses and Radical Functions

Key Concepts & Glossary

Find the inverse of a polynomial function

Section Exercises

Restrict the domain to find the inverse of a polynomial function

Quadratic Equations

Key Concepts & Glossary

Introduction: Quadratic Equations

Section Exercises

Solving Quadratic Equations by Factoring

Solutions

Using the Square Root Property

Exponential Functions

Completing the Square

Introduction to Exponential Functions

Using the Quadratic Formula

Evaluate exponential functions

Key Concepts & Glossary

Find the equation of an exponential function

Section Exercises

Use compound interest formulas

Solutions

Evaluate exponential functions with base e

Other Types of Equations

Key Concepts & Glossary

Introduction: Other Types of Equations

Section Exercises

Solving Equations Involving Rational Exponents

Solutions

Solving Equations Using Factoring

Graphs of Exponential Functions

Solving Radical Equations

Introduction to Graphs of Exponential Functions

Solving an Absolute Value Equation

Graph exponential functions

Solving Other Types of Equations

Graph exponential functions using transformations

Key Concepts & Glossary

Key Concepts

Section Exercises

Section Exercises

Solutions

Solutions

Linear Inequalities and Absolute Value Inequalities

Logarithmic Functions

Introduction: Linear Inequalities and Absolute Value Inequalities

Introduction to Logarithmic Functions

Using Interval Notation

Convert from logarithmic to exponential form

Using the Properties of Inequalities

Convert from exponential to logarithmic form

Understanding Compound Inequalities

Evaluate logarithms

Solving Absolute Value Inequalities

Use common logarithms

Key Concepts & Glossary

Use natural logarithms

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

Functions and Function Notation

Solutions

Introduction to Functions and Function Notation

Graphs of Logarithmic Functions

Determine whether a relation represents a function

Introduction to Graphs of Logarithmic Functions

Find the input and output values of a function

Identify the domain of a logarithmic function

Determine whether a function is one-to-one

Graph logarithmic functions

Use the vertical line test to identify functions

Graphing Transformations of Logarithmic Functions

Graph the functions listed in the library of functions

Key Concepts

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Logarithmic Properties

Domain and Range

Introduction to Logarithmic Properties

Introduction to Domain and Range

Use the product rule for logarithms

Find the domain of a function defined by an equation

Use the quotient and power rules for logarithms

Use notations to specify domain and range

Expand logarithmic expressions

Find domain and range from graphs

Condense logarithmic expressions

Find domains and ranges of the toolkit functions

Use the change-of-base formula for logarithms

Graph piecewise-defined functions

Key Terms & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Exponential and Logarithmic Equations

Rates of Change and Behavior of Graphs

Introduction to Exponential and Logarithmic Equations

Introduction to Rates of Change and Behaviors of Graphs

Use like bases to solve exponential equations

Find the average rate of change of a function

Use logarithms to solve exponential equations

Use a graph to determine where a function is increasing, decreasing, or constant

Use the definition of a logarithm to solve logarithmic equations

Use a graph to locate the absolute maximum and absolute minimum

Use the one-to-one property of logarithms to solve logarithmic equations

Key Concepts & Glossary

Solve applied problems involving exponential and logarithmic equations

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

Composition of Functions

Solutions

Introduction to Composition of Functions

Exponential and Logarithmic Models

Combine functions using algebraic operations

Introduction to Exponential and Logarithmic Models

Create a new function by composition of functions

Model exponential growth and decay

Evaluate composite functions

Use Newton’s Law of Cooling

Find the domain of a composite function

Use logistic-growth models

Decompose a composite function into its component functions

Choose an appropriate model for data

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Transformation of Functions

Systems of Linear Equations: Two Variables

Introduction to Transformation of Functions

Introduction to Systems of Linear Equations: Two Variables

Graph functions using vertical and horizontal shifts

Solving Systems of Equations by Graphing

Graph functions using reflections about the x-axis and the y-axis

Solving Systems of Equations by Substitution

Determine whether a function is even, odd, or neither from its graph

Solving Systems of Equations in Two Variables by the Addition Method

Graph functions using compressions and stretches

Identifying and Expressing Solutions to Systems of Equations

Combine vertical and horizontal shifts

Using Systems of Equations to Investigate Profits

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Absolute Value Functions

Systems of Linear Equations: Three Variables

Introduction to Absolute Value Functions

Introduction to Systems of Linear Equations: Three Variables

Graph an absolute value function

Solving Systems of Three Equations in Three Variables

Solve an absolute value equation

Inconsistent and Dependent Systems in Three Variables

Solve an absolute value inequality

Key Concepts & Glossary

Key Concepts & Glossary

Section Exercises

Section Exercises

Solutions

Solutions

Systems of Nonlinear Equations and Inequalities: Two Variables

Inverse Functions

Introduction to Systems of Nonlinear Equations and Inequalities: Two Variables

Introduction to Inverse Functions

Solving a System of Nonlinear Equations Using Substitution

Verify inverse functions

Solving a System of Nonlinear Equations Using Elimination

Determine the domain and range of an inverse function

Graphing Nonlinear Inequalities and Systems of Nonlinear Inequalities

Find or evaluate the inverse of a function

Key Concepts & Glossary

Use the graph of a function to graph its inverse

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

Matrices and Matrix Operations

Solutions

Introduction: Matrices and Matrix Operations

Linear Functions

Finding the Sum and Difference of Two Matrices

Introduction to Linear Functions

Finding Scalar Multiples of a Matrix

Represent a linear function

Finding the Product of Two Matrices

Determine whether a linear function is increasing, decreasing, or constant

Key Concepts & Glossary

Calculate and interpret slope

Section Exercises

Write the point-slope form of an equation

Solutions

Write and interpret a linear function

Solving Systems with Gaussian Elimination

Key Concepts & Glossary

Introduction to Solving Systems with Gaussian Elimination

Section Exercises

The Augmented Matrix of a System of Equations

Solutions

Performing Row Operations on a Matrix

Graphs of Linear Functions

Solving a System of Linear Equations Using Matrices

Introduction to Graphs of Linear Functions

Key Concepts & Glossary

Graph linear functions

Section Exercises

Write the equation for a linear function from the graph of a line

Solutions

Given the equations of two lines, determine whether their graphs are parallel or perpendicular

Solving Systems with Inverses

Write the equation of a line parallel or perpendicular to a given line

Introduction to Solving Systems with Inverses

Solve a system of linear equations

Finding the Inverse of a Matrix

Key Concepts & Glossary

Solving a System of Linear Equations Using the Inverse of a Matrix

Section Exercises

Key Concepts & Glossary

Solutions

Section Exercises

Complex Numbers

Solutions

Introduction to Complex Numbers

Practice Tests

Express square roots of negative numbers as multiples of  i

Functions Practice Test

Plot complex numbers on the complex plane

Solutions: Functions Practice Test

Add and subtract complex numbers

Linear Functions Practice Test

Multiply and divide complex numbers

Solutions: Linear Functions Practice Test

Key Concepts & Glossary

Polynomial and Rational Functions Practice Test

Section Exercises

Solutions: Polynomial and Rational Functions Practice Test

Solutions

Exponential and Logarithmic Functions Practice Test

Dividing Polynomials

Solutions: Exponential and Logarithmic Functions Practice Test