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.

### 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 2. 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

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

##### Example

Write in partial fraction form

.

##### Solution

Here the given expression is

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

(1)

Here

Now I will try to get the values of

For that, I will start with the simplification of the equation (1).

Thus it will be

So at the end I get

Here the denominator is the same on both left- and right-hand sides.

So I can compare the numerators of both sides.

Therefore I can say

Now I compare the coefficients of

So it gives

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

That gives

(2)

Now I put

Thus it will be

Next I’ll simplify it to get the value of

Thus I can say the partial fraction of

Now 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

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

Here also I write this expression as

where

Now I can do simple arithmetic on the right-hand side of the equation.

Thus it becomes

Here denominators on both left- and right-hand sides are the same.

So I can say,

Now I compare coefficients of

Next, I compare coefficients of

Finally I will compare the constants on both sides.

Thus it will be

So now I have

Therefore the partial fraction form of

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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