Partial fractions of lower degree numerators. Today it’s all about the partial fractions of lower degree numerators. Here it goes!!

If you are looking for more in partial differentiation, do check out:

**Partial fractions of irreducible quadratic factors**

**Partial fractions of equal degree expressions**

**How to get partial fractions of repeated roots**

**Partial fractions of higher degree numerators**

**Partial fractions of lower degree numerators**

Ok, so, before I go further, let me clear a question: what is a lower degree numerator?

**What is a lower degree numerator?**

Good question!! Isn’t it?

Let me sort that out now. So have a look at the following three expressions:

Now in all these expressions,

- the bottom part of the fraction is called denominator and the top part is numerator.
- the highest degree of the numerator is . This means the highest power of in the numerator is .
- the highest degree of the denominator is . This is because here also the highest power of is .

So I can conclude that in all these expressions both the numerator and denominator have the same degree.

Don’t mix up coefficients and degrees!! Coefficients are the ones which come to the front of the variable. I know probably you guys won’t, but in my undergrad days, I did it several times in a hurry!!

Next, I’ll give you a few examples.

** Example 1**

Now have a look at this expression:

As I can see, in this expression,

- The highest power of in the numerator is 1. So the numerator has a degree 1.
- The highest power of in the denominator is 2. So the denominator’s degree is 2.

Therefore this expression has a lower degree numerator.

Next, I’ll give another example.

** Example 2**

Now I’ll take a look at the second expression. As I can see, here:

- The highest power of in the numerator is 2. Hence the numerator has a degree 2.
- The highest power of in the denominator is also 2. So the denominator also has a degree 2.

So the conclusion is: both the numerator and the denominator have the same degree.

Next comes my third example.

** Example 3**

Finally, I come to the third expression. As I can see, here:

- The highest power of in the numerator is 2. So the numerator has a degree 2.
- The highest power of in the denominator is 1. That means the denominator has a degree 1.

So the numerator has a higher degree than the denominator.

Okay, now I will talk about partial fractions of lower degree numerators.

So I’ll solve some examples.

**Example 1**

Write the following expression into a partial fraction form:

.

**Solution**

Now here the given expression is . So the numerator of this expression is . And the denominator is .

Thus I can say that the numerator has a lower degree than the denominator . In order to write this expression in partial fraction form, first of all, I have to factorize the denominator.

So I can write as

And this gives

Therefore I can rewrite the expression as

Let me assume

where are constants.

Now I’ll simplify as

So this means

Therefore I can say that

Now I’ll compare the numerators of both sides. That is

From here I get two equations with two unknowns and like

(1)

and

(2)

Next, I’ll solve equations (1) and (2) to get the values of and .

So I can rewrite equation (1) as

Then I’ll put this value of in equation (2). So this gives

Now I’ll simplify it to get

Thus it gives

So I can say

Therefore I can say,

.

Thus the partial fraction of is .

Hence I can conclude that this is the answer to this example.

Now I’ll give some other examples.

Disclaimer: None of these examples is mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

**Example 2**

According to Stroud and Booth (2013)*, “Express each of the following in partial fractions: .”

**Solution**

Now here the given expression is . As I can see, the numerator has a lower degree than the denominator . In order to write this expression in partial fraction form, first of all, I have to factorize the denominator.

**Step 1**

So I can write as

And this gives

Therefore I can rewrite the expression as

**Step 2**

Let me assume

where are constants.

Now I’ll simplify as

So this means

Therefore I can say that

Now I’ll compare the numerators of both sides. And that means

Thus from here I get two equations with two unknowns and . So these are

(3)

and

Now I can also rewrite the second equation as

(4)

Next, I’ll solve equations (3) and (4) to get the values of and .

So I can rewrite equation (3) as

Then I’ll put this value of in equation (4). So this gives

Now I’ll simplify it to get

Thus it gives

So I can say

Therefore I can say,

.

Thus the partial fraction of is .

Hence I can conclude that this is the answer to the given example.

Now I’ll give another example.

**Example 3**

According to Stroud and Booth (2013)*, “Express each of the following in partial fractions: .”

**Solution**

Now here the given expression is . And my task is to write it in partial fraction form.

As I can see, the numerator has a degree of . And the denominator has the degree . Thus this expression has a lower degree numerator. First of all, I have to factorize the denominator.

**Step 1**

So I can write as

And this gives

**Step 2**

Now let me assume

where are constants.

Next, I’ll simplify as

So this means

Therefore I can say that

Now I’ll compare the numerators of both sides. So that means

Thus from here I get two equations with two unknowns and . So these are

(5)

and

(6)

Next, I’ll solve equations (5) and (6) to get the values of and .

So I’ll subtract times equation (6) from equation (5) to get

Now I’ll simplify it to get

And this gives

Next, I’ll put back in equation (6) to get

So this means

Thus the partial fraction of is .

Hence I can conclude that this is the answer to the given example.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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