Today I will talk about partial fractions of repeated roots. And, these repeated roots always come at the bottom, i.e., the denominator. They can also be repeated several times as well.

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

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

**Partial fractions of higher degree numerators**

**How to get the partial fractions of expressions with irreducible quadratic factors**

**Partial fractions of equal degree expressions**

**READ, DOWNLOAD & PRINT – Partial fractions of expressions with repeated roots**

**Partial fractions of expressions with repeated roots**

Here I will give you two examples. In the first one, the roots will be repeated twice. So the degree of the denominator will be . In the second example, the roots will be repeated thrice (three times).

You can read more about the** degrees of the numerator and denominator from here**.

Let’s start then.

**Partial fractions of repeated roots of degree 2**

Now I will give an example of partial fractions of an expression where roots are repeated twice.

**Example **

Write in partial fraction form

.

**Solution**

So the given expression is . First of all, I will rewrite the expression in partial fraction form. Thus it will be

(1)

where and are constants. Now I’ll get the values of and .

First of all, I’ll simplify the equation (1).

**Step 1**

As I can see, there’s nothing to do on the left-hand side. So I’ll simplify the right-hand side. Thus it will be

And that gives

Next, I’ll separate the coefficient of and the constant terms. So that means

Now here the denominator is the same on both the left- and right-hand sides. Therefore I can compare the numerators of both sides. And that gives

**Step 2**

Thus I compare the coefficients of on both sides. So I can say that

Next, I’ll compare the constants on both sides. And that means

(2)

Then I put in equation (2) to get

So that gives

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

Next, I will show you how to find out the partial fraction form of an expression where the roots come thrice at the bottom.

**Partial fractions of repeated roots of degree 3**

Now here I will give an example of partial fractions of an expression where roots are repeated thrice, that is, three times.

**Example **

Write in partial fraction form: .

**Solution**

First of all, I’ll rewrite this expression in partial fraction form. So that gives

where and are constants.

Now I’ll simplify it.

**Step 1**

So I’ll simplify the right-hand side of the equation. Thus it becomes

Next, I’ll expand to get

Then I’ll simplify it a bit more to get

Finally, I’ll separate the coefficients of and the constant terms. And that gives

As I can see, here the denominator on both left- and right-hand sides are the same.

So I can say,

.

Now I’ll get the values of and .

**Step 2**

First of all, I’ll compare the coefficient of on both sides. And that gives

So that means

Next, I’ll compare the coefficients of on both sides to get

Then I’ll substitute to get

Now I’ll simplify it. So that means

Thus I’ll simplify it a bit more to get the value of as

So I can say that the value of is

Finally I will compare the constants on both sides.

**Step 3**

And that gives

Now I’ll substitute and to get

Then I’ll simplify it. And that gives

So I can say that

Therefore the value of will be

Now I have and .

Therefore the partial fraction form 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 on the partial fractions of repeated roots. 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!!

Jesulayomi says

What a well arranged work, neat and easy to understand thanks for this piece of information

Dr. Aspriha Peters says

Thank you very much. Your word means a lot to me.