National 5MathsSQARevision NotesAlgebraic SkillsAlgebraic FractionsSimplest Form

Simplest Form (SQA National 5 Maths): Revision Note

Exam code: X847 75

Written by: Roger B

Reviewed by: Dan Finlay

Updated on 1 December 2025

Reducing an algebraic fraction to its simplest form

What is an algebraic fraction?

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

How do you simplify an algebraic fraction?

If possible, fully factorise 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

Examiner Tips and Tricks

When asked to simplify an algebraic fraction, start by factorising the 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 seems difficult to factorise!

Worked Example

Simplify $\frac{x^{2} - 5 x}{x^{2} - 2 x - 15}$ .

Answer:

Factorise the top, by using x as a common factor

$\frac{x^{2} - 5 x}{x^{2} - 2 x - 15} = \frac{x (x - 5)}{x^{2} - 2 x - 15}$

Factorise the bottom using your preferred method

$(x - 5)$ being a factor on the top is a hint that it may also be a factor on the bottom!

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

Cancel the common factors of $(x - 5)$ from the top and bottom

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

$\frac{x}{x + 3}$

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