What does partial-fraction decomposition mean?
Transforming a rational polynomial into a sum of simpler rational polynomials due to the factorization of the denominator.
Let me try and explain this to you. :)
What is a partial-fraction decomposition?
We are all familiar with the process of adding and subtracting fractions, e.g.
Now, partial-fraction decomposition does exactly the reverse thing.
It takes a rational polynomial, so a fraction like
from my example, and tries to decompose it into a sum of "simpler" fractions (to be more precise: into a sum of fractions which denominators are factors of the original fractions's denominator.)
How to compute a partial-fraction decomposition?
1) Linear and unique factors
Let's stick with my example:
The first thing to do is a always to find a complete factorization of the denominator:
Here, all the factors are linear and unique, this is the simple case.
The goal is to to find
To do so, first we should multiply both sides with the denominator:
Now, in order to solve this equation, we need to "group" the
Thus, we can form an equation based on the "red" terms and an equation based on the "blue" terms:
The solution of this linear equation system is inded
2) Non-unique factors
The computation is more complicated in case that the complete factorization of the original denominator doesn't consist just of linear and unique factors.
Just a short example:
Let's say we could factorize our rational polynomial like follows:
As the factors
In this case, we would need to find
3) Non-linear factors
Last but not least, there might be non-linear factors in your factorized denominator.
Unfortunately, you can't factorize
In this case, your partial-fraction decomposition needs to look like follows:
When can a partial-fraction decomposition be useful?
Finally, I'd like to note that a very common application of this method is integration of rational polynomials:
If you need to compute
and know that
it's easy to integrate by "splitting" the original fraction into a sum of simpler fractions and integrating each one of them: