Today I will discuss partial fractions of higher degree numerators. That means I will focus on the expressions with higher degree numerators. My aim will be to reduce its degree first. Then I will try to get partial fractions of the expression.
You can also check out how to determine the degrees of the numerator and the denominator of any expression.
Let me give you a few examples.
Partial fractions of higher degree numerators with examples
Write the following expression into the partial fraction form:
Here the given expression is: .
That means the denominator is . Also the numerator is .
So, I can say that the degree of the numerator is 3. The denominator’s degree is 2.
This means the numerator’s degree is higher than that of the denominator. Therefore this expression has a higher degree numerator.
So, my first job is to reduce the degree of the numerator.
First of all, I’ll divide the numerator by the denominator.
So I can rewrite the expression as
Now is reduced to two parts:
one is and
the other is .
The first part cannot be reduced any more.
But I can still reduce the second part. So let’s start then.
Here I start with the factorisation. Then I use the assumptions like
Now I have the same denominator on both left hand and right hand sides.
Therefore I can say,
I can compare the coefficients of on both sides.
Thus I get
I can also compare constants on both sides. From that I have
From equation (1), I have .
Next substitute this value of in equation (2).
Then I get
Now I substitute this value of in equation (1).
Therefore I have
So the partial fraction of is .
Hence I can conclude that the partial fraction of is .
This is the solution of the given example.
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!!
You might find the following posts on partial fractions quite interesting:)
- Partial fractions of equal degree expressions
- How to determine partial fractions of expressions with lower degree numerators
- Partial fractions of repeated roots
- Partial fractions of irreducible quadratic factors