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Factorising

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 + 18x  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

Exam 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

Exam 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) + 2a + 6
 

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


b(a + 3) + 2(a + 3)
 

(a + 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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