1 Basic Techniques for Solving Equations

  1.1 Isolation                         

  1.2 Substitution                       

  1.3 Elimination                       

  1.4 Larger Systems of Linear Equations         

  1.5 Summary                        

2 Functions Review

  2.1 Function Basics   

  2.2 Graphing Functions

  2.3 Composition    

  2.4 Inverse Functions 

  2.5 Summary      

3 Complex Numbers

  3.1 Arithmetic of Complex Numbers

  3.2 The Complex Plane        

  3.3 Real and Imaginary Parts     

  3.4 Graphing in the Complex Plane 

  3.5 Summary              

4 Quadratics

  4.1 Factoring Quadratics        

  4.2 Relating Roots and Coe cients  

  4.3 Completing the Square       

  4.4 The Discriminant           

  4.5 Quadratic Inequalities        

  4.6 Summary               

5 Conics

  5.1 Parabolas                 

  5.2 Problem Solving With Parabolas    

  5.3 Maxima and Minima of Quadratics  

  5.4 Circles                   

  5.5 Ellipses                  

  5.6 Hyperbolas                

  5.7 Summary                 

6 Polynomial Division

  6.1 Polynomial Review          

  6.2 Introduction to Polynomial Division 

  6.3 Synthetic Division           

  6.4 The Remainder Theorem       

  6.5 Summary                

7 Polynomial Roots Part I

  7.1 The Factor Theorem                   

  7.2 Integer Roots                       

  7.3 Rational Roots                      

  7.4 Bounds                          

  7.5 Graphing and the Fundamental Theorem of Algebra

  7.6 Algebraic Applications of the Fundamental Theorem

  7.7 Summary                         

8 Polynomial Roots Part II

  8.1 Irrational Roots              

  8.2 Nonreal Roots              

  8.3 Vieta’s Formulas             

  8.4 Using Roots to Make Equations    

  8.5 Summary                 

9 Factoring Multivariable Polynomials

  9.1 Grouping                       

  9.2 Sums and Di↵erences of Powers          

  9.3 The Factor Theorem for Multivariable Polynomials

  9.4 Summary                       

10 Sequences and Series

  10.1 Arithmetic Sequences              

  10.2 Arithmetic Series                 

  10.3 Geometric Sequences              

  10.4 Geometric Series                 

  10.5 Sequence, Summation, and Product Notation

  10.6 Nested Sums and Products           

  10.7 Summary                     

11 Identities, Manipulations, and Induction

  11.1 Brute Force                          

  11.2 Ratios                             

  11.3 Induction                           

  11.4 Binomial Theorem                      

  11.5 Summary                           

12 Inequalities

  12.1 Manipulating Inequalities     

  12.2 The Trivial Inequality       

  12.3 AM-GM Inequality with Two Variables

  12.4 AM-GM with More Variables

  12.5 The Cauchy-Schwarz Inequality

  12.6 Maxima and Minima   

  12.7 Summary          

13 Exponents and Logarithms

  13.1 Exponential Function Basics      

  13.2 Introduction to Logarithms       

  13.3 Logarithmic Identities          

  13.4 Using Logarithm Identities       

  13.5 Switching Between Logs and Exponents

  13.6 Natural Logarithms and Exponential Decay

  13.7 Summary                 

14 Radicals

  14.1 Raising Radicals to Powers     

  14.2 Evaluating Expressions With Radicals

  14.3 Radical Conjugates         

  14.4 Summary               

15 Special Classes of Functions

  15.1 Rational Functions and Their Graphs 

  15.2 Rational Function Equations and Inequalities

  15.3 Even and Odd Functions       

  15.4 Monotonic Functions         

  15.5 Summary                

16 Piecewise Defined Functions

  16.1 Introduction to Piecewise Defined Functions

  16.2 Absolute Value             

  16.3 Graphing Absolute Value       

  16.4 Floor and Ceiling            

  16.5 Problem Solving with the Floor Function

  16.6 Summary                

17 More Sequences and Series

  17.1 Algebra of Recursive Sequences

  17.2 Telescoping            

  17.3 Sums of Polynomial Series   

  17.4 Arithmetico-Geometric Series 

  17.5 Finite Di↵erences         

  17.6 Summary             

18 More Inequalities

  18.1 Mean Inequality Chain 

  18.2 The Rearrangement Inequality

  18.3 When Formulas Fail  

  18.4 Summary        

19 Functional Equations

  19.1 Finding Values       

  19.2 Finding Functions with Substitution

  19.3 Separation          

  19.4 Cyclic Functions      

  19.5 Summary          

20 Some Advanced Strategies

  20.1 Symmetry              

  20.2 Substitution for Simplification  

  20.3 Method of Undetermined Coe cients

  20.4 Constructing Polynomials From Roots

  20.5 Common Divisors of Polynomials

  20.6 Symmetric Sums Revisited    

  20.7 Summary