Simplifying Algebraic Fractions (Cambridge (CIE) IGCSE Maths)

Revision Note

Written by: Mark Curtis

Reviewed by: Dan Finlay

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An algebraic fraction is a fraction with an algebraic expression on the top (numerator) and/or the bottom (denominator)

If possible, factorise fully the top and bottom
- E.g. $\frac{2 x}{x^{2} + 3 x} = \frac{2 x}{x (x + 3)}$
Cancel common factors
- This factor may be a single term
  - E.g. $\frac{x (5 x - 1)}{4 x} = \frac{5 x - 1}{4}$
- It could also be a common bracket
  - E.g. $\frac{x (x + 2)}{(x + 2) (x - 1)} = \frac{x}{x - 1}$
A common mistake is to cancel a factor that is not common to all terms in either the top or the bottom of a fraction
- E.g. The fraction $\frac{6 x}{x + 1}$ cannot be simplified
  - $x$ is not common to all terms in the bottom of the fraction
  - and the expression on the bottom cannot be factorised

When asked to simplify an algebraic fraction, factorise top and bottom
- It is very likely that one of the factors will be the same on the top and the bottom
- Use this fact to help you, if one of the expressions is difficult to factorise!

Simplify $\frac{4 x + 6}{2 x^{2} - 7 x - 15}$

Factorise the top, by using 2 as a common factor

$\frac{2 (2 x + 3)}{2 x^{2} - 7 x - 15}$

Factorise the bottom using your preferred method
Using the fact that the top factorised to $(2 x + 3)$ may help!

$\frac{2 (2 x + 3)}{(2 x + 3) (x - 5)}$

The common factors on the top and bottom reduce to 1 (cancel out)

$\frac{2 (2 x + 3)}{(2 x + 3) (x - 5)}$

$\frac{2}{(x - 5)}$

Last updated: 17 October 2024

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