Factorising Expressions (Edexcel International A Level Maths: Pure 1)

What is meant by factorising expressions?

• Many expressions in mathematics are written as a sum of terms
  • e.g.   is the sum of three the terms , and
• Many expressions are written as a product of factors
  • e.g.   is the product of the two (linear) factors and 
• 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. 
• A quadratic expression may be able to be factorised into two linear factors
  • Look out for special cases
    • No constant term: 
    • Difference of two squares (no x  term and constant is square): 
    • Perfect squares: 
    • ‘Hidden’ quadratics: 
    • More than one variable: 

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

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  term is greater than 1 (and not prime)
• Follow the steps:
  • STEP 1 Starting with find the product 
    • For example: for  
  • STEP 2 Find two numbers m & n whose product is and sum is 
    • For example: &
    • So 
  • STEP 3 Split the term into 
    • For example: 
  • STEP 4 Factorise the first two terms and the last two terms
    • For example:
  • STEP 5 Factorise once more for the final answer
    • For example:  

• 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
  • then expanding and simplifying gives
    •  

• By comparing coefficients
  • 
  • 
  • 
• Let  and then:
  • 
  • 
  • Therefore these are the two numbers whose product is ac and sum is b