Partial fractions of equal degree expressions

Today we will discuss how to get partial fractions of equal degree expressions. Saying that, I mean about those expressions where both numerators and denominators have the same degree.

In one of my earlier posts, I have already explained how to identify an expression where both the numerator and the denominator have the same degrees.

Let us choose a few examples.

Examples

1. Write in partial fraction form:

.

Now our aim is to remove from the top. For that we can write this expression as

Now is removed from the top and is the partial fraction of .

Let us choose another example.

2. Write in partial fraction form:

.

Here also our aim is to remove from the top. So we can write the expression as

Now we can say is the partial fraction of .

Let us solve one more problem where both the numerator and the denominator have the degree 2.

3. Write in partial fraction form:

.

In this case, we have to remove both and from the top. Like other examples above, here also we start in the same way.

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

Let’s look at the expression . Here numerator has a lower degree than the denominator. So we can use the same way of partial fraction decomposition.

To start with, first we factorize .

Now we rewrite as .

Let us assume

where are constants.

Now we can say

.

We compare coefficients on both sides. Therefore we have two simple equations.

(1)

and

(2)

At the next stage we solve equations (1) and (2) to get the values of and . From equation (1), we can say, . We substitute this value of in equation (2) and then solve for .

For , .

Therefore the partial fraction of is .

Thus we can say the partial fraction form of is . This is the answer to the problem no 3.

In my next post, I will discuss more on partial fractions. Our topic will be how to get partial fractions of expressions with higher degree numerators. Till then, I say you all good bye!!

Wish you all a happy and safe weekend!!

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