National 5MathsSQARevision NotesAlgebraic SkillsFactorisationCommon Factors

Common Factors (SQA National 5 Maths): Revision Note

Exam code: X847 75

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

Factorising expressions with a common factor

What is factorisation?

  • A factorised expression is one written as the product (multiplication) of two or more terms (factors)

    • 3 left parenthesis x plus 2 right parenthesis is factorised

      • It is 3 cross times left parenthesis x plus 2 right parenthesis

    • 3 x plus 6  is not factorised

    • 3 x y  is factorised

      • It is 3 cross times x cross times y

    • Numbers can also be factorised

      • 12 equals 2 cross times 2 cross times 3

  • In algebra, factorisation is the reverse of expanding brackets

    • It's putting it into brackets, rather than removing brackets

How do I factorise two terms?

  • To factorise 12 x squared plus 18 x

    • Find the highest common factor of the number parts

      • 6

    • Find the highest common factor of the algebra parts

      • x

    • Multiply both to get the overall highest common factor

      • 6 x

    • 12 x squared plus 18 x  is the same as 6 x cross times 2 x plus 6 x cross times 3

      • Using the highest common factor

    • Take out the highest common factor

      • Write it outside a set of brackets

      • Put the remaining terms, 2 x plus 3,  inside the brackets

    • This gives the answer

      • 6 x left parenthesis 2 x plus 3 right parenthesis

  • To factorise an expression containing multiple variables, e.g. 2 a cubed b minus 4 a squared b squared

    • Use the same approach as above

    • Find the highest common factor of the number parts

      • 2

    • Find the highest common factor of the algebra parts

      • a and b appear in both terms

        • The highest common factor of a cubed and a squared is a squared

        • The highest common factor of b and b squared is b

      • a squared b

    • Multiply both to get the overall highest common factor

      • 2 a squared b

    • 2 a cubed b minus 4 a squared b squared  is the same as 2 a squared b cross times a minus 2 a squared b cross times 2 b

      • Using the highest common factor

    • Take out the highest common factor

      • Write it outside a set of brackets

      • Put the remaining terms, a minus 2 b,  inside the brackets

    • This gives the answer

      • 2 a squared b left parenthesis a minus 2 b right parenthesis

Examiner Tips and Tricks

In the exam, check that your factorisation is correct by expanding the brackets!

Also make sure you factorise expressions fully:

  • x left parenthesis 6 x plus 10 right parenthesis is a partially factorised form of 6 x squared plus 10 x

  • But the fully factorised form is 2 x left parenthesis 3 x plus 5 right parenthesis

Worked Example

(a) Factorise 5 x plus 15.

(b) Factorise fully 30 x squared minus 24 x.

Answer:

Part (a)

Find the highest common factor of 5 and 15 

5

There is no x in the second term, so no highest common factor in x  is needed
Think of each term as 5 times something

5 x plus 15 equals 5 cross times x plus 5 cross times 3

Take out the 5 and put x + 3 in brackets

equals 5 left parenthesis x plus 3 right parenthesis

5 left parenthesis x plus 3 right parenthesis

Part (b)

Find the highest common factor of 30 and 24 

6

Find the highest common factor of x2  and x

x

Find the overall highest common factor by multiplying these together

6 x

Think of each term as 6x  times something

30 x squared minus 24 x equals 6 x cross times 5 x minus 6 x cross times 4

Take out the 6x  and put 5x  - 4 in brackets

equals 6 x left parenthesis 5 x minus 4 right parenthesis

6 x left parenthesis 5 x minus 4 right parenthesis

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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.