How to do Partial Fractions
Partial Fractions are a way of turning a single gruesome algebraic fraction involving multiple ‘Linear Factors’. Into a set of added fractions each with the denominator corresponding to one of the linear factors. Let me give you an example.
Not so difficult to see what is going on but how do we do this?
Method 1
For method 1 we must first write an expression for the fraction as so.
As you may be able to see I have split the denominator into its two linear factors x and 2x+3. For the Denominators of these fractions (which I don’t yet know), I have used two variables A and B. Next, we multiply everything by the denominator of the original fraction and then expand:
Now we can see that the terms in front of the ‘x’s must be equal, and the constant terms must be equal. So, we now have two simultaneous equations which we can solve to find A and B.
There isn’t that so pleasing, a horrible fraction turned into something that is far easier to work with.
Method 2
Method two is a faster method but doesn’t always work and you must pay close attention to it. First, we start out by stating the equation again involving A and B.
And again, we multiply by the denominator:
Now instead of expanding we think, ‘what value for x would make the A term disappear… -3/2?
If we then plug that in, we get:
We can then do the same to calculate A as 2/3 by saying that x=0.
Method 3 – cover up
This method is faster than the other two but also causes the most errors when being carried out. We start out with our original equation.
And then we look at the big fraction, we then ask what makes each factor on the bottom 0, so for 2x+3 that is -3/2. We then subsidise that into the remain part of the faction covering up the term we just made zero. This resulting expression is the term on top of our factor we covered up.
We can then do the same for the other term. And get exactly what we got before.
If the fraction has a factor that is in the denominator multiple times, then we are not able to use the 3rd method anymore. This is because we need to write the multiple fractions that add to it as all the powers of that factor up to what it is in the denominator.
This can then be solved for A, B and C.
Overall, partial fractions don’t need to be difficult as they are one of the most useful tools in a mathematician’s arsenal. They help with differentiation, integration and far more areas of maths. If you want more practice Kahn Academy has lots of great resources on this very topic. And as always,
Have fun and never stop solving.
