Partial fraction of equal degree expressions. Today I will talk about a partial fraction of equal degree expressions.

**The partial fraction of equal degree expressions**

If you are looking for more in partial fractions, do check-in:

**Partial fractions of lower degree numerators**

**How to get the partial fractions of higher degree numerators**

**Partial fractions of repeated roots**

**Partial fractions of irreducible quadratic factors**

Ok, so I come back to the original topic now.

The partial fraction of equal degree expressions means the partial fraction of expressions where both numerators and denominators have the same degree.

Let me choose a few examples.

##### Example 1

Write in partial fraction form:

.

##### Solution

Now here the given expression is . As I can see, both the numerator and the denominator has the degree . So this is an equal degree expression.

Now in order to write this expression in partial fraction form, I have to remove from the top.

So I can write this expression as

Next, I’ll simplify it to get

Now this means

for .

So this brings

Now I can say is the partial fraction of .

Next, I’ll give other examples where both the numerator and the denominator have the degree 2.

##### Example 2

Write in partial fraction form:

.

##### Solution

Now here the given expression is . As I can see, both the numerator and the denominator has the degree . So this is an equal degree expression.

Now in order to write this expression in partial fraction form, I have to remove both and from the top. As example 1, here also I’ll start in the same way.

**Step 1**

So I can write this expression as

Then I’ll simplify it to get

Now I have two components of this expression: one is 2 and the other is . I cannot decompose 2 further. But I can still break down .

So let me look at the expression . Now here numerator has a lower degree than the denominator. Then I can use the same way of partial fraction decomposition as the partial fraction of a lower degree numerator.

**Step 2**

To start with, first I’ll factorize . Thus it will be

So this gives

Next, I’ll rewrite as .

Let me assume

where are constants. Now my task is to find out the values of and .

In order to do that, I’ll simplify this expression as

So this means

Now I can compare coefficients on both sides. And I get two different equations as

(1)

and

(2)

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

**Step 3**

Now from equation (1), I can say, . Then I substitute this value of in equation (2) to get the value of . So this gives

Thus for , .

Therefore the partial fraction of is .

Hence I can say that the partial fraction form of is . And this is the answer to the given 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 3

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, both the numerator and the denominator has the degree . So this is an equal degree expression.

Now in order to write this expression in partial fraction form, I have to remove both and from the top. As example 1, here also I’ll start in the same way.

**Step 1**

So I can write this expression as

Then I’ll simplify it to get

Now I have two components of this expression: one is and the other is . I cannot decompose further. But I can still break down .

So let me look at the expression . Now here numerator has a lower degree than the denominator. Then I can use the same way of partial fraction decomposition as the partial fraction of a lower degree numerator.

**Step 2**

To start with, first I’ll factorize . Thus it will be

So this gives

Next, I’ll rewrite as .

Let me assume

where are constants. Now my task is to find out the values of and .

In order to do that, I’ll simplify this expression as

So this means

Now I can compare coefficients on both sides. And I get two different equations as

(3)

and

(4)

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

**Step 3**

Now I’ll multiply equation (3) with and add it to the equation (4). And this gives

So this means

Next, I’ll substitute this value of in equation (3) to get

Therefore the partial fraction of is .

Hence I can say that the partial fraction form of is . And this is the answer to the given example.

Now I’ll give another example.

##### Example 4

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, both the numerator and the denominator has the degree . So this is an equal degree expression.

Now in order to write this expression in partial fraction form, I have to remove both and from the top. As example 1, here also I’ll start in the same way.

**Step 1**

So I can write this expression as

Then I’ll simplify it to get

Now I have two components of this expression: one is and the other is . I cannot decompose further. But I can still break down .

So let me look at the expression . Now here numerator has a lower degree than the denominator. Then I can use the same way of partial fraction decomposition as the partial fraction of a lower degree numerator.

**Step 2**

To start with, first I’ll factorize . Thus it will be

So this gives

Next, I’ll rewrite as .

Let me assume

where are constants. Now my task is to find out the values of and .

In order to do that, I’ll simplify this expression as

So this means

Now I can compare coefficients on both sides. And I get two different equations as

(5)

and

(6)

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

**Step 3**

Now I’ll multiply equation (5) with and add it to the equation (6). And this gives

So this means

Next, I’ll substitute this value of in equation (5) to get

Therefore the partial fraction of is .

Hence I can say that the partial fraction form of is . And 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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