https://youtu.be/5QQP2kqE_ss

In this video I go over further into partial fraction decomposition and this time explain why the techniques shown in my earlier videos for decomposing proper fractions don't work for improper fractions. Thus to decompose improper fractions we first need to perform polynomial long division to obtain proper fractions.

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# Partial Fraction Decomposition: Why it doesn't work with Improper fractions

![Partial Fractions Why Improper Dont Work.jpeg](https://cdn.steemitimages.com/DQmaZ3Moap5tocB2VYhKELezECbyEAotVwU1osKX7qqLDDN/Partial%20Fractions%20Why%20Improper%20Dont%20Work.jpeg)

In my earlier videos I stated that partial fraction decomposition only worked for proper fractions and not improper fractions and thus improper fractions first needed to be broken down to proper fractions using polynomial long division:

![](https://cdn.steemitimages.com/DQmWfFHRKpwQLvUQWqaCah8pCv4BKwn3GACh53geUxfbhp9/image.png)

**To recap: **

- Proper fractions have a denominator with higher degree than the numerator.

- Improper fractions have a denominator with smaller than or equal to degree as with the numerator.
	
Let's see what happens if we apply the same method as for proper fractions:

![](https://cdn.steemitimages.com/DQmbjSRU6uYpns6wgGQdBwDJiQrDJa4kTUedmYxxBwfn8jg/image.png)

Now we multiply both sides by (x<sup>2</sup> + 1)(x - 2):

![](https://cdn.steemitimages.com/DQmbpio5d4SJYMRzMeSa5jnYwvWMVc3uYfjh9aSKXSZAgYR/image.png)

What if the numerator had the same degree as the denominator?

![](https://cdn.steemitimages.com/DQmZjz58YcVwvo7MUcgDFVZugsUahQyKZkEBcCRrLigzhWg/image.png)

Thus we can't decompose improper fractions in the same way we would decompose proper fractions. 

Like I should in my previous video, we first need to perform polynomial long division on the improper fraction to obtain a proper fraction and then decompose it accordingly.