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