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Factorising Expressions (Edexcel International A Level Maths): Revision Note

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Factorising Expressions

What is meant by factorising expressions?

  • Many expressions in mathematics are written as a sum of terms

    • e.g.  x squared plus 6 x minus 16 is the sum of three the terms x squared, 6 x and negative 16

  • Many expressions are written as a product of factors

    • e.g.  left parenthesis x plus 8 right parenthesis left parenthesis x minus 2 right parenthesis is the product of the two (linear) factors x plus 8 and x minus 2

  • Factorising is the process of rewriting the sum of terms as a product of factors

    • The other way round is expanding

How do I factorise an expression? 

  • This will depend on the nature of the expression you are dealing with

  • In all cases the first thing to consider is if there is a factor (number and/or letter) of all terms in the expression

    • e.g. 2 x cubed plus 4 x squared minus 8 x equals 2 x left parenthesis x squared plus 2 x minus 4 right parenthesis

  • A quadratic expression may be able to be factorised into two linear factors

    • Look out for special cases

      • No constant term: x squared plus 5 x equals x left parenthesis x plus 5 right parenthesis

      • Difference of two squares (no x  term and constant is square): x squared minus 36 equals left parenthesis x minus 6 right parenthesis left parenthesis x plus 6 right parenthesis

      • Perfect squares: x squared plus 8 x plus 16 equals open parentheses x plus 4 close parentheses open parentheses x plus 4 close parentheses equals open parentheses x plus 4 close parentheses squared

      • ‘Hidden’ quadratics: 3 to the power of 2 x end exponent minus 12 cross times 3 to the power of x plus 27 equals open parentheses 3 to the power of x close parentheses squared minus 12 open parentheses 3 to the power of x close parentheses plus 27 equals left parenthesis 3 to the power of x minus 3 right parenthesis left parenthesis 3 to the power of x minus 9 right parenthesis

      • More than one variable: x squared minus y squared equals left parenthesis x minus y right parenthesis left parenthesis x plus y right parenthesis

  • A cubic expression (at this level) will not contain a constant term

    • This means will x be a factor (and there might be a number as a factor too)

    • The remaining expression will be a quadratic

      • this quadratic may also be able to be factorised

      • e.g. 6 x cubed plus 3 x squared minus 9 x equals 3 x open parentheses 2 x squared plus x minus 3 close parentheses equals 3 x open parentheses 2 x plus 3 close parentheses open parentheses x minus 1 close parentheses

Remind me how to factorise a quadratic … 

  • There are many shortcuts to factorise quadratic expressions, but they often only apply under certain conditions (such as when a = 1)

    • the method below works for any quadratic expression

    • it is most useful when the coefficient of the x squared term is greater than 1 (and not prime)

  • Follow the steps:

    • STEP 1 Starting with a x squared plus b x plus c find the product a c

      • For example: for 6 x squared plus 7 x minus 3 a c equals 6 cross times negative 3 equals negative 18

    • STEP 2 Find two numbers m & n whose product is a c and sum is b

      • For example: 9 cross times negative 2 equals negative 18 equals a c & 9 plus left parenthesis negative 2 right parenthesis equals 7 equals b

      • So m equals 9 space & space n equals negative 2

    • STEP 3 Split the b x term into m x plus n x

      • For example: 6 x squared minus 2 x plus 9 x minus 3

    • STEP 4 Factorise the first two terms and the last two terms

      • For example: 2 x open parentheses 3 x minus 1 close parentheses plus 3 open parentheses 3 x minus 1 close parentheses

    • STEP 5 Factorise once more for the final answer

      • For example: left parenthesis 3 x minus 1 right parenthesis left parenthesis 2 x plus 3 right parenthesis 

  • If a and/or c are prime, factorising can be done “by inspection”

    • For example: the only way to split (prime) 3 into factors would be 3 and 1

Why does the 'ac' method work? 

  • Suppose a x squared plus b x plus c identical to open parentheses p x plus r close parentheses open parentheses q x plus s close parentheses

    • then expanding and simplifying gives

      •  a x squared plus b x plus c identical to blank p q x squared plus p s x plus q r x plus r s identical to p q x squared plus open parentheses p s plus q r close parentheses x plus r s

  • By comparing coefficients

    • a equals p q

    • b equals p s plus q r

    • c equals r s

  • Let m equals p s and n equals q r then:

    • m plus n equals p s plus q r equals b

    • m cross times n equals p s q r equals a c

    • Therefore these are the two numbers whose product is ac and sum is b

Worked Example

1-5-1-ial-fig1-we-solution-fact

Examiner Tips and Tricks

  • Do use your tried and tested shortcuts for factorising quadratics

    • We’ve explained it in full above to help you understand the process rather than to learn ‘tricks’

  • You don't need to learn why the 'ac' method works - but we thought you might think that the algebra is cool

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    Paul

    Author: Paul

    Expertise: Maths Content Creator (Previous)

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams.