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Last exams 2024

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Factorising (CIE IGCSE Maths: Extended)

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Basic Factorising

What is factorisation?

  • A factorised expression is one written as the product (multiplication) of two, or more, terms (factors)
    • 3(x + 2) is factorised, as it is 3 × (x + 2)
    • 3x + 6  is not factorised as it is "something" + "something"
    • 3xy is factorised as it is 3 × x × y
    • 12 can also be factorised: 12 = 2 x 2 x 3
  • In algebra, factorisation is the opposite of expanding brackets
    • it's "putting it into" brackets

  

How do I factorise two terms?

  • To factorise 12x2 + 18x  
    • The highest common factor of 12 and 18 is 6
    • The highest common factor of x2 and x is x
      • this is the largest letter that divides both x2 and x 
    • Multiply both to get the common factor
      • 6x
    • Rewrite each term in 12x2 + 18 as "common factor × something"
      • 6x × 2x + 6x × 3
    • "Take out" the common factor by writing it outside brackets
    • Put the remaining 2x + 3 inside the brackets
      • Answer: 6x(2x + 3)
      • Check this expands to give 12x2 + 18x

Examiner Tip

  • You can always check that your factorisation is correct by simply expanding the brackets in your answer!

Worked example

(a)

Factorise 5x + 15
 

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 needed
Write each term as 5 × "something"

 

5 × x + 5 × 3
 

"Take out" the 5
 

5(x + 3)

5(x + 3)

(b)
Factorise fully 30x2 - 24x
 
Find the highest common factor of 30 and 24
 
6
 
Find the highest common factor of x2 and x
 
x
 
Find the common factor (by multiplying these together)
  
6x

 

Write each term as 6x × "something"
 

6x × 5x - 6x × 4
 

"Take out" the 6x
 

6x(5x - 4)

6x(5x - 4)

Factorising by Grouping

How do I factorise expressions with common brackets?

  • To factorise 3x(t + 4) + 2(t + 4), both terms have a common bracket, (t + 4)
    • the whole bracket, (t + 4), can be "taken out" like a common factor
      • (t + 4)(3x + 2)
    • this is like factorising 3xy + 2y to y(3x + 2)
      • y represents (t + 4) above

 

How do I factorise by grouping?

  • Some questions may require you to form the common bracket yourself
    • for example, factorise xy + px + qy + pq
      • "group" the first pair of terms, xy + px, and factorise, x(y + p)
      • "group" the second pair of terms, qy + pq, and factorise, q(y + p),
    • now factorise x(y + p) + q(y + p) as above
      • (y + p)(x + q)
    • This is called factorising by grouping
  • The groupings are not always the first pair of terms and the second pair of terms, but two terms with common factors

Examiner Tip

  • As always, once you have factorised something, expand it by hand to check your answer is correct.

Worked example

Factorise ab + 3b + 2a + 6

 

Method 1
Notice that
ab and 3b have a common factor of b
Notice that 2a and 6 have a common factor of 2

Factorise the first two terms, using b as a common factor
 

b(a + 3) + 2+ 6
 

Factorise the second two terms, using 2 as a common factor 


b(a + 3) + 2(a + 3)
 

(+ 3) is a common bracket 
We can factorise using (a + 3) as a factor

(a + 3)(b + 2)

Method 2
Notice that
ab and 2a have a common factor of a
Notice that 3b and 6 have a common factor of 3

Rewrite the expression grouping these terms together 
 

ab + 2a + 3b + 6
 

Factorise the first two terms, using a as a common factor 
 

a(b + 2) + 3b + 6
 

Factorise the second two terms, using 3 as a common factor 


a(b + 2) + 3(b + 2)
 

(b + 2) is a common bracket
 
We can factorise using (b + 2) as a factor

(b + 2)(a + 3)

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Mark

Author: Mark

Expertise: Maths

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.