Partial Fractions

MathsDirect

The expression on the right is called an identity. This is because the left and right hand sides of the expression would be identical, regardless of the value of x.

For this reason the two are completely interchangeable.

Ever since you started working with algebraic fractions, you have been told that the expression on the right should be sought, as it is simplified.

There is however, one case where the expression on the left is more useful, Integration.

We therefore need to know how to un-simplify the fraction. This process is called splitting into Partial Fractions.

Clearly, each bracket on the bottom will get it's own fraction, but we do not know what the numerators will be, so we call them A, B, etc...

Having split into two fractions, each with an unknown numerator, we recombine them by cross-multiplying, and collect terms on the numerator.

Now, we have stated that the left and right hand side are identical, whatever value x takes. This must mean that, since the denominators are the same, so the numerators must be identical.