National 5MathsSQARevision NotesAlgebraic SkillsAlgebraic FractionsSimplest Form

Simplest Form (SQA National 5 Maths): Revision Note

Exam code: X847 75

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

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. fraction numerator 2 x over denominator x squared plus 3 x end fraction equals fraction numerator 2 x over denominator x open parentheses x plus 3 close parentheses end fraction

  • Cancel common factors

    • This factor may be a single term

      • E.g. fraction numerator up diagonal strike x open parentheses 5 x minus 1 close parentheses over denominator 4 up diagonal strike x end fraction equals fraction numerator 5 x minus 1 over denominator 4 end fraction

    • It could also be a common bracket

      • E.g. fraction numerator x up diagonal strike open parentheses x plus 2 close parentheses end strike over denominator up diagonal strike open parentheses x plus 2 close parentheses end strike open parentheses x minus 1 close parentheses end fraction equals fraction numerator x over denominator x minus 1 end fraction

  • 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 fraction numerator 6 x over denominator x plus 1 end fraction 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 fraction numerator x squared minus 5 x over denominator x squared minus 2 x minus 15 end fraction.

Answer:

Factorise the top, by using x as a common factor

fraction numerator x squared minus 5 x over denominator x squared minus 2 x minus 15 end fraction equals fraction numerator x open parentheses x minus 5 close parentheses over denominator x squared minus 2 x minus 15 end fraction

Factorise the bottom using your preferred method

  • open parentheses x minus 5 close parentheses being a factor on the top is a hint that it may also be a factor on the bottom!

equals fraction numerator x open parentheses x minus 5 close parentheses over denominator open parentheses x plus 3 close parentheses open parentheses x minus 5 close parentheses end fraction

Cancel the common factors of open parentheses x minus 5 close parentheses from the top and bottom

equals fraction numerator x up diagonal strike open parentheses x minus 5 close parentheses end strike over denominator open parentheses x plus 3 close parentheses up diagonal strike open parentheses x minus 5 close parentheses end strike end fraction

fraction numerator x over denominator x plus 3 end fraction

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.