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 and are constants.

Now I will try to get the values of and .

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 on both sides.

So it gives

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

That gives

(2)

Now I put in equation (2).

Thus it will be

Next I’ll simplify it to get the value of as

Thus I can say the partial fraction of is . This is the answer to the problem.

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 and are constants.

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 on both sides to get

Next, I compare coefficients of on both sides to get

Finally I will compare the constants on both sides.

Thus it will be

So now I have and .

Therefore the partial fraction form of is:

.

Dear friends, this is the end of my 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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