A proper fraction with the product of irreducible polynomials in the denominator can be factored into a sum of fractions of the form , where and are some polynomials.
Generally this can be written as:
Given a fraction of the form where .
Where and are some polynomial.
Procedure
For each irreducible polynomial in the denominator a set of fractions is added to the sum based on one of three cases.
Case 1 - Single Linear Factor
Given a factor of the form it's term in the partial fraction decomposition is a term of the form:
Where is some constant.
Case 2 - Linear Factor to a Power
Given a factor of the form it's term in the partial fraction decomposition is a sum of the following form:
Where is some constant.
Example
Given the linear factor of it's terms assuming it's irreducible, are
Where , , , and are some constants.
Case 3 - Irreducible Non-Linear Factor
Given a factor that is an irreducible polynomial of degree , the corresponding term in the partial fraction decomposition is of the form:
where is a polynomial of degree one less than
Example
For a denominator containing the irreducible cubic factor , the corresponding term in the decomposition would be:
where , and are some constants.
Case 4 - Irreducible Non-Linear Factor to a Power
If the denominator contains a repeated irreducible polynomial factor , the corresponding terms in the partial fraction decomposition take the form:
where each is a polynomial of degree one less than
Example
For a denominator containing , the corresponding terms in the partial fraction decomposition would be:
where , , , , , and are some constants.