On factoring polynomials

Introduction

Polynomials form one of the oldest classes of mathematical functions in mathematics with an important history in science, engineering, and otherquantitative fields [Pan, 1997] , [Pan, 1998] , [Traub, 1998] , [Derbyshire, 2006] , [Sitton, 2003] . Part of the polynomial's appeal comes from the fact that it may be numerically evaluatedusing a finite number of multiplications and additions. Another advantage it presents is its use in modeling physical processes or complicated mathematicalfunctions. Polynomials are used to build expansion systems or basis sets in various vector spaces. Segments of polynomials are used to create splines.It is worth studying the basic ideas and the variety of applications.

Polynomials are introduced early in the teaching of algebra as a means of demonstrating basic principles and methods, e.g. substitution, simplification,and factoring. Most people are aware that there exist formulas for factoring quadratic (2nd degree), cubic (3rd degree), and quartic (4th degree) polynomials.In 1824 the mathematician Niels Abel proved that there is no possible "closed form" solution in terms of basic operations for the general pentic or quintic (5th degree) polynomial or those of higher degrees. This fact requires the development of effective numerical methodsfor iteratively factoring polynomials above degree 4.

The results reported in these modules are from a group at Rice University called the "Polynomial Club" (Jim Fox, Sidney Burrus, Gary Sitton, and Sven Treitel). The program was designed and written by Jim Fox.

Basic definitions

The definition of an $N$ th degree polynomial is

where both $f\left(z\right)$ and $z$ are complex valued and the coefficients $a_k$ can be complex but are often real valued. $N$ , which is the highest power of $z$ in the polynomial, is called the degree (or sometimes the order) of the polynomial. The number ofcoefficients is $N+1$ .

From the various forms of the Fundamental Theorem of Algebra , one can show that all polynomials can also be expressed in a “factored" formby

where the possibly complex valued $z_r$ are called the “zeros" of the polynomial since $f\left(z_k\right)=0$ . Because the zeros are not necessarily distinct, a unique form of [link] is given by

where $M_k$ is the multiplicity of the $k^{th}$ zero, $L$ is the number of distinct zeros, and ${\sum }_k{M}_k=N$ . The first degree polynomials $(z-z_k)$ in [link] and [link] are called the factors of $f\left(z\right)$ . Note the analogy between polynomials and integers. Indeed, multiplying two polynomials is the same operation as multiplyingtwo integers (except for carrying) or convolving two number sequences.

Creating [link] from [link] or [link] is fairly straight forward and requires only a finite number of arithmetic operations, butfinding [link] or [link] from [link] is difficult. That process is called “factoring" the polynomial and is the topic of thesenotes. Actually, the effects of finite precision arithmetic sometimes make the "unfactoring" process of calculating [link] from [link] or [link] poorly defined because it depends on the sequence order of combining the zeros.