AoPS Intermediate Algebra

Lessons

Module 1: Basic Techniques for Solving Equations, Functions Review, and Complex Numbers

Isolation and Substitution

Elimination

Larger Systems of Linear Equations

Function Basics and Graphing Functions

Composition and Inverse Functions

Arithmetic of Complex Numbers

The Complex Plane

Real and Imaginary Parts and Graphing in the Complex Plane

Test 1 Review

Test 1

Module 2: Quadratics, Conics, and Polynomial Division

Factoring Quadratics

Relating Roots and Coefficients and Completing the Square

The Discriminant

Quadratic Inequalities

Parabolas and Problem Solving with Parabolas

Maxima and Minima of Quadratics

Circles

Ellipses

Hyperbolas

Polynomial Review and Introduction to Polynomial Division

Synthetic Division

The Remainder Theorem

Test 2 Review

Test 2

Module 3: Polynomial Roots, Factoring Multivariable Polynomials, and Sequences and Series

The Factor Theorem and Integer Roots

Rational Roots and Bounds

Graphing and the Fundamental Theorem of Algebra

Algebraic Applications of the Fundamental Theorem

Irrational Roots and Nonreal Roots

Vieta's Formulas

Using Roots to Make Equations

Grouping/Sums and Differences of Powers

The Factor Theorem for Multivariable Polynomials

Arithmetic Sequences and Series

Geometric Sequences and Series

Sequence, Summation, and Product Notation

Nested Sums and Products

Module 4: Identities, Manipulations, and Induction, Inequalities, and Exponents and Logarithms

Brute Force and Ratios

Induction and Binomial Theorem

Manipulating Inequalities

The Trivial Inequality

AM-GM Inequality with Two Variables

AM-GM with More Variables

The Cauchy-Schwarz Inequality

Maxima and Minima

Exponential Function Basics and Introduction to Logarithms

Logarithmic Identities and Using Logarithm Identities

Switching Between Logs and Exponents

Natural Logarithms and Exponential Decay

Test 4 Review

Test 4

Module 5: Radicals, Special Classes of Functions, and Piecewise Defined Functions

Raising Radicals to Powers and Evaluating Expressions with Radicals

Radical Conjugates

Rational Functions and their Graphs

Rational Function Equations and Inequalities

Even and Odd Functions and Monotonic Functions

Introduction to Piecewise Defined Functions

Absolute Value and Graphing Absolute Value Functions

Floor and Ceiling Functions

Problem Solving with the Floor Function

Test 5 Review

Test 5

Module 6: More Sequences and Series, More Inequalities, Functional Equations, and Some Advanced Strategies

Algebra of Recursive Sequences

Telescoping

Sums of Polynomial Series

Arithmetico-Geometric Series

Finite Differences

Mean Inequality Chain and the Rearrangement Inequality

When Formulas Fail

Finding Values and Finding Functions with Substitution

Separation

Cyclic Functions

Symmetry and Substitution for Simplification

Method of Undetermined Coefficients

Constructing Polynomials from Roots

Common Divisors of Polynomials

Symmetric Sums Revisited

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