# What does partial-fraction decomposition mean?

##### 1 Answer

Transforming a rational polynomial into a sum of simpler rational polynomials due to the factorization of the denominator.

#### Explanation:

Let me try and explain this to you. :)

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**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.)

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**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

holds.

++++++++++++++++++++++

3) Non-linear factors

Last but not least, there might be non-linear factors in your factorized denominator.

Example:

Unfortunately, you can't factorize

In this case, your partial-fraction decomposition needs to look like follows:

Find

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**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: