Today we will discuss partial fractions of repeated roots. These repeated roots always come at the bottom, i.e., the denominator. They can also be repeated several times.

### 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).

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

Now we can say

.

Therefore we compare the coefficients of on both sides.

Now we compare the constants on both sides.

So we can say the partial fraction of is . This is the answer to the problem.

Now we try 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 we can write this expression as

where and are constants.

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

Here denominators on both sides are the same. So we can say,

.

Now we compare coefficients of on both sides.

Next we compare coefficients of on both sides.

At the end we compare the constants on both sides.

So now we 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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