Some rational expressions do not have obvious common factors. In these cases, it is necessary to factorise either the numerator or the denominator, or both, to find common factors.

Simplify .

The numerator of this fraction will factorise as there is a common factor of 3.

This gives . Now, there is clearly a common factor of 3 between the numerator and denominator. Cancelling this through the fraction gives . There are no more common factors in this expression. Note cannot be cancelled as there is no term in the +2 in the numerator.

- Question
Simplify .

In this example, both the numerator and the denominator can be factorised. The numerator is a quadratic with no common factors which will therefore factorise into two brackets. The denominator does contain a common factor of 4 so will factorise into one bracket.

Factorise the numerator .

Find two numbers with a product of +4 and a sum of +5.

The numbers are +4 and +1 as and .

Factorise the denominator .

The highest common factor of and 16 is 4. Put this number in front of a bracket and then divide each term by 4 to complete the sum.

and .

This gives .

The original expression was . Factorising gives .

There is a common factor throughout the fraction of . Cancelling out this factor will simplify the expression.