Today we will discuss about partial fractions of lower degree numerators.
Before we go further, let us clear a question.
What is a lower degree numerator?
Good question!! Isn’t it?
Let’s sort that out now. So have a look at the following three expressions:
In all these expressions,
The bottom part of the fraction is called denominator and the top part is numerator.
In all these expressions the highest degree of the numerator is 1. This means the highest power of in the numerator is 1.
In all these expressions the highest degree of the denominator is 1. This is because here also the highest power of is 1.
That means in all these expressions both the numerator and denominator have the same degrees.
Don’t mix up coefficients and degrees!! Coefficients are the ones which come to the front of the variable. I know probably you guys won’t, but in my undergrad days I did it several times in a hurry!!
Let us choose a few more examples.
In this expression,
The highest power of in the numerator is 1. So the numerator has a degree 1.
The highest power of in the denominator is 2. So the denominator’s degree is 2.
Therefore this expression has a lower degree numerator.
Now we take a look at the second expression. Here:
The highest power of in the numerator is 2. Hence the numerator has a degree 2.
The highest power of in the denominator is also 2. So the denominator also has a degree 2.
The conclusion is both the numerator and the denominator have the same degrees.
Finally we come to the third expression. Here:
the highest power of in the numerator is 2. So the numerator has a degree 2.
The highest power of in the denominator is 1. That means the denominator has a degree 1.
So the numerator has a higher degree than the denominator.
So, now we talk about partial fractions of lower degree numerators.
Partial fractions of lower degree numerators
Let us solve an example.
Write the following expression into a partial fraction form:
Note that here the numerator has a lower degree than the denominator .
First of all, we try to factorize . Here I have already explained how we can factorize any expression. So we can write as
Now we can rewrite the expression as
Let us assume
where A, B are constants.
Therefore we can say,
Now we compare numerators of both sides. That is
From here we get two equations with two unknowns and like
From equation (1), we can write
We substitute this value of in equation (2). Then we use simple arithmetic to get the value of .
Therefore we can say, .
Thus the partial fraction of is .
In my upcoming posts, I will write more on partial fractions. Later on we will use partial fractions in integral calculus.
So bye for today!! Hope you have enjoyed reading my post. Please leave a comment. I would appreciate very much. Thank you for your time.