Associated Topics || Dr. Math Home || Search Dr. Math

### 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/
```
Associated Topics:
High School Basic Algebra
Middle School Algebra

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search