A partial fraction is a rational fraction of one of the following two types:
where A, B, C, a, b and c are constants and n is a positive integer.
A rational expression is a fraction whose numerator and denominator are both polynomials. A rational expression is proper if the degree of the numerator is lower than the degree of the denominator; otherwise it is improper. An improper rational expression is the sum of a polynomial and a proper rational expression. For example
where x2 - 2 x + 5 is the quotient and - 10 is the remainder upon division of the denominator into the numerator.
Partial fraction decomposition is the process of rewriting a rational expression as the sum of a quotient polynomial plus partial fractions. If the rational expression is proper, the quotient will be zero.
After finding the quotient, the next step in decomposing a rational expression is factoring the denominator. In principle, every polynomial with real coefficients can be factored into factors of the following two forms:
where a, b and c are constants and n is a positive integer.
Example:
The next step is find the appropriate form of the partial fraction decomposition.
If all the factors are of the form ( x - c ), then each of
the partial fractions will be of the form 
.
To complete the process, the values of the constants must be found.
For terms of the form 
the constant may be found using Heaviside's Method.
First, multiply the original proper fraction by ( x - c ) and evaluate the result at x = c to get the value of the constant.
Thus
and the partial fraction decomposition is complete.
Find the partial fraction decomposition of
When the rational expression contains a factor of the form ( x - c )n with n 1 , the procedure is different. For example
There must be a partial fraction present for each power of ( x - c) less than or equal to n. Furthermore, only the constant corresponding to ( x - c )n may be found by Heaviside's method.
So
To find the other constant, multiply both sides by the least common denominator to clear the fractions:
The polynomial on the left of the equal sign must have the same coefficients as the polynomial on the right of the equal sign. Thus
2 = A
and
- 1 = 1 - A.
In either case, A = 2. So
Find the partial fraction decomposition of
Now let� decompose
Using Heaviside's method,
So
Clearing the fractions yields
Rather than simplify the expression on the right, pick some value of
x (not
- 1 or - 2, lest B vanish!) and solve for B. For, example, let x
= 0.
Then the equation reduces to
10 = - 2 + 2 B + 10
Thus, B = 1. So the decomposition is
Decompose the following
What about factors in the denominators of the form 
?
For one thing, we cannot use Heaviside's method to find the constants associated
with quadratic terms.
Decompose the following:
Clear the fractions to get
Thus, 7 = 4 + B, so B = 3. And since C - 3B = -10, then C = -1.
So the decomposition is
Decompose
One last example: Decompose
Clearing the fractions yields
So A = 1 and B = 2. Since A + C = 3, C = 2. And since B + D = 1, D = -1.
Thus the partial fraction decomposition is
Decompose into partial fractions
