What is a Partial Fraction?

Date: 01/08/99 at 06:57:16
From: Axel Ishmael
Subject: What is a Partial Fraction: Polynomial?

Hello, 

I was just surfing the Net for math, and I saw this definition in  
Webster's Dictionary on-line:

Main Entry: partial fraction
Function: noun
Date: 1816

One of the simpler fractions into the sum of which the quotient of two 
polynomials may be decomposed.

What does the above sentence mean? I have a basic understanding of 
polynomials, but I have never heard of a partial fraction.

Thank you.

Date: 01/08/99 at 08:23:37
From: Doctor Jerry
Subject: Re: What is a Partial Fraction: Polynomial?

Hi Axel,

For some purposes (such as integration, in calculus) it is useful to 
"decompose" a fraction like 1/(x^2-a^2). Specifically, notice that if 
we have:

   1/(x^2-a^2) = A/(x-a) + B/(x+a)

and if A = 1/(2a) and B = -1/(2a), this equation is satisfied. Thus 
one can deal more easily with the sum of the simpler "partial 
fractions" than with 1/(x^2-a^2).

There is a theorem in algebra that says that any ratio of polynomials 
p(x)/q(x) where the degree of p is less than that of q can be 
decomposed into a sum of partial fractions. To put this theorem into 
practice, one must factor q into products of linear factors, or powers 
of linear factors and irreducible quadratic factors, or powers of 
irreducible quadratic factors (irreducible means no real roots). After 
all of this, one then must solve for the unknowns (A and B, above, are 
unknowns).  

The decomposition of a rational function into partial fractions is 
computationally labor-intensive. For examples of how to decompose 
fractions into partial fractions, please see:

  Partial Fractions
  http://mathforum.org/dr.math/problems/duke6.8.97.html   

  Partial Fractions
  http://mathforum.org/dr.math/problems/yanzhen1.29.98.html   

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/