Today we will discuss about partial fractions of expressions with repeated roots. These repeated roots always come at the bottom, i.e., the denominator. They can also be repeated several times.
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
Write in partial fraction form
We can rewrite the expression as
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
Write in partial fraction form:
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:
This is the end for today’s post. In the next post I will discuss one more instances of partial fraction where the denominator cannot be factorized.