A level revision

Partial fractions

Posted in Maths, Core 4

Partial fractions involve splitting up a larger fraction into two or more parts. This technique is required to both make expressions easier to expand and make expressions easier to integrate.

There are 3 types of algebraic fractions that you need to be able to write as partial fractions.

Denominators in the form (ax + b)(cx + d)

Express as partial fractions

First step is to write the equation as where A and B are just some constant

Now multiply both sides by the denominator, in this case to give

From here there are two methods to find the values of A and B, both methods have their merits and their uses and you'll find one is easier for you. In this example we'll use both to demonstrate how they work.

By substitution

The substitution method works by substituting a value for x into the equation to get rid of one of the constants. Basically put you want to make one of the brackets equal to 0.

Firstly to find the value for A we need to remove B which can be done by substituting x = 1 into the equation:

Now to find B we need to remove A by substituting in x = -2:

Now, returning to the original expression we wrote in step 1 we can substitute in values for A and B:

By equating coefficients

This method is slightly more complex but is far more versatile. It involves equating the coefficients of x and the constant terms. Firstly expand the expression from step 2:

Then equation the coefficients of x (the numbers and constants in front of the x) and the constants:

Now it's just a case of solving these simultaneous equations:

So we get the same values from substitution.

Denominators in the form (ax + b)(cx² + d)

Express as partial fractions

The process is similar to above, firstly write the equation as where A, B and C are just some constants

And again multiply both sides by the denominator to give:

By substituting x = -1 into the equation a value for A can be found:

We can find B and C by equating the coefficients of x², x and equating the constants. Firstly expand the right hand side of the equation:

And equate the coefficients and constants:

We know the value of A so we can find the value of B using the first equation:

Now, substituting the value of B into the second equation we can find C:

Finally we can check to see if we're right by substituting A and C into the last equation:

Lastly we put our values of A, B and C into the original equation:

Denominators in the form (ax + b)(cx + d)²

Express as partial fractions

This time write the equation as where A, B and C are just some constants

And again multiply both sides by the denominator to give:

By substituting x = 1 we can find a value for C:

By substituting x = -1 we can find a value for A:

By equating the coefficients of x² and using the value for A above the value of B can be found:

Putting these values in the original equation: