# THEORY OF EQUATIONS

## 2002 Edition

Table of Contents

**Chapter 1** College Algebra

Lesson 1 (PDF 123KB)

1.1 The Real Number System

1.2 A little History of Real and Complex Numbers

1.3 Complex Numbers
Lesson 2 (PDF 105KB)
1.4 Polynomial Arithmetic

1.5 Degree

1.6 The division Algorithm

1.7 Factors and Roots
Lesson 3 (PDF 146KB)
1.8 Greatest Common Divisor

1.9 Unique Factorization

1.10 Formal Differentiation of Polynomials

1.11 Test for Multiple Roots

1.12 Partial Fraction Decomposition

1.13 The Resultant
**New*** Polynomial GCDs by Linear Algebra*

Appendix to Chapter 1, (PDF 57KB)
**Chapter 2** Numerical Analysis

Lesson 4 (PDF 165KB)

2.1 Exact and Approximate Numbers

2.2 Numerical Algorithms

2.3 Maple Programming and Evaluation of Polynomials

2.4 Taylor's Series and Horner's Process

2.5 Synthetic Division

Lesson 5 (PDF 124KB)

2.6 Graphing

2.7 Descartes' Rule and Newton's Bound

2.8 The Bisection Method

2.9 Horner's Method

2.10 Iteration Method
Lesson 6 (PDF 135KB)

2.11 Newton's Method, Real Case

2.12 Newton's Method, Complex Case

2.13 The Newton-Barstow Algorithm

2.14 Other root finding algorithms

2.15 Polynomial Interpolation
**Chapter 3** Topology
Lesson 7 (PDF 146KB)

3.1 History of the Fundamental Theorem of Algebra

3.2 Gauss' Fourth Proof

3.3 Topological Proof

3.4 Analysis Proof

Lesson 8 (PDF 197KB)
3.5 Another Curve Proof

3.6 Connection between the FTA and Newton's method

3.7 Where Newton's method does not converge

3.8 Real Newton's method revisited

3.9 Iteration of Quadratic Polynomials
**Chapter 4:** Ancient and Modern Algebra
Lesson 9 (PDF 132KB)

4.1 Solutions of Quadratic Equations

4.2 Omar Khayyam and Viete's Solutions of the cubic

4.3 History of the Cubic and Biquadratic

4.4 Algebraic solution of the cubic

4.5 Algebraic solution of the biquadratic

Lesson 10 (PDF 151KB)
4.6 Newton's Identities

4.7 More on Newton's Indentities

4.8 Symmetric Polynomials

4.9 Lagrange's Solution of the Biquadratic

4.10 Insolvability of the Quintic

**Chapter 5** Number Theory
Lesson 11 (PDF 142KB)

5.1 Rational Polynomials and Algebraic Numbers

5.2 Integer Polynomials

5.3 Rational Roots and Factors

5.4 Eisenstein's Irreducibility Criterion

5.5 Hand Factoring methods

5.6 Computer Factoring

5.7 Comparison of factoring algorithms
Lesson 11A (PDF 106KB)

Appendix to Chapter 5 on Polynomials over the integers mod p.
1. Modular Arithmetic

2. Polynomials over `Z`_{p}

3. Factoring in `Z`_{p}[x]

4. Roots of Polynomials in `Z`_{n}

**Chapter 6** Analysis -- Elliptic Functions

Chapter 6 Text only (PDF 212KB)

Chapter 6 Text and Figures (zip 750KB)

6.1 Review of trigonometric and hyperbolic functions

6.2 Historical Background

6.3 The Jacobi Elliptic Functions

6.4 The inverse Jacobi Elliptic Functions

6.5 Elliptic Integrals

6.6 The Complex Theory

For an updated look at relevant *Mathematica* software for Elliptic Functions and integrals please see wolfram.com

### Comments?

Please send comments to Barry Dayton
barry@barryhdayton.us