Simplifying rational expressions with factorising

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.

Example

Simplify \frac{3t + 6}{3t}.

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

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

Question

Simplify \frac{x^2 + 5x + 4}{4x + 16}.

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 x^2 + 5x + 4.

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

The numbers are +4 and +1 as 4 \times 1 = 4 and 4 + 1 = 5.

x^2 + 5x + 4 = (x + 4)(x + 1)

Factorise the denominator 4x + 16.

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

4x \div 4 = x and 16 \div 4 = 4.

This gives 4(x + 4).

The original expression was \frac{x^2 + 5x + 4}{4x + 16}. Factorising gives \frac{(x + 4)(x + 1)}{4(x + 4)}.

There is a common factor throughout the fraction of (x + 4). Cancelling out this factor will simplify the expression.

\frac{\cancel{(x + 4)}(x + 1)}{4 \cancel{(x + 4)}} = \frac{x + 1}{4}