THEORY OF EQUATIONS

2002 Edition

These notes are now available in PDF format.

Table of Contents

Introduction (HTML)

Chapter 1 College Algebra

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.

References (HTML)

OLD 1999

The 1999 edition is posted here . This is a combination of HTML, GIF and DVI files. There are two additional chapters:

Chapter 6 Analysis -- Elliptic Functions
Chapter 7 Computational Algebra -- Groebner Bases

Comments?

Please send comments to Barry Dayton at

b-dayton@neiu.edu