Introduction to Algebra

Algebra at the High-School Level

日本語版

Overview

This introductory course systematically covers the algebra taught in high-school mathematics. Starting from the basics of numbers and expressions, we progress through quadratic equations and functions, and on to complex numbers and higher-degree equations.

Learning Objectives

  • Freely use algebraic expressions, expansion, and factoring
  • Solve quadratic equations by various methods
  • Understand graphs and properties of quadratic functions
  • Master the concept and arithmetic of complex numbers
  • Understand methods for solving higher-degree equations
  • Master proof techniques for identities and inequalities

Table of Contents

Part 1: Foundations of Numbers and Expressions

  1. Ch. 1 Algebraic Expressions and Polynomials

    Meaning of algebraic expressions, definition and ordering of polynomials

  2. Ch. 2 Expansion Formulas

    Distributive law, product formulas, cubic expansions

  3. Ch. 3 Factoring

    Common factors, factoring formulas, cross-multiplication

  4. Ch. 4 Real Numbers and Square Roots

    Rationals, irrationals, square root arithmetic, nested radicals

  5. Ch. 5 Linear Inequalities

    Properties of inequalities, systems of inequalities, absolute value

  6. Ch. 6 Exercises (Numbers & Expressions)

    Comprehensive exercises for Part 1

Part 2: Quadratic Equations and Functions

  1. Ch. 7 Quadratic Equations

    Solving by factoring and completing the square, existence of solutions

  2. Ch. 8 The Quadratic Formula

    Derivation of the formula, discriminant and nature of roots

  3. Ch. 9 Quadratic Functions

    Graphing, vertex and axis, translations

  4. Ch. 10 Maxima and Minima of Quadratic Functions

    Max/min on a restricted domain, applications

  5. Ch. 11 Quadratic Inequalities

    Solving using graphs, conditions for solutions

  6. Ch. 12 Exercises (Quadratics)

    Comprehensive exercises for Part 2

Part 3: Complex Numbers and Higher-Degree Equations

  1. Ch. 13 Complex Numbers

    The imaginary unit, arithmetic of complex numbers, conjugates

  2. Ch. 14 Cubic Equations and Cardano's Formula

    Tschirnhaus transformation, derivation of Cardano's formula, discriminant

  3. Ch. 15 Quartic Equations and Ferrari's Method

    Ferrari's method, resolvent cubic, quartic discriminant

  4. Ch. 16 Higher-Degree Equations

    Factor theorem, solving higher-degree equations, Fundamental Theorem of Algebra

  5. Ch. 17 The Remainder Theorem

    Remainder theorem, relation to the factor theorem

  6. Ch. 18 Identities

    Properties of identities, determining coefficients

  7. Ch. 19 Expressions and Proofs

    AM–GM inequality, Cauchy–Schwarz inequality

  8. Ch. 20 Exercises (Complex & Higher-Degree)

    Comprehensive exercises for Part 3

Prerequisites

  • Middle-school arithmetic (four operations, fractions, decimals)
  • Basic concepts of algebraic expressions
  • Solving linear equations