Factorising | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Basic Factorising

What is factorisation?

• A factorised expression is one that is written as the product (multiplication) of two or more terms (factors). For example:
  • 3(x + 2) is factorised as it is 3 x (x + 2).
  • 3x + 6 is not factorised because it is a single term.
  • 3xy is factorised as 3 x x x y.
  • 12 can also be factorised as 12 = 2 x 2 x 3.
• In algebra, factorisation is the opposite of expanding brackets; it involves putting an expression into brackets.

How do I factorise two terms?

To factorise an expression like 12x² - 18x, we follow these steps:

• Identify the highest common factor of the coefficients and the variables separately. 
• For 12 and 18, the HCF is 6. 
• For x² and x, it is x.
• Multiply these common factors to get 6x.
• Rewrite each term as the common factor multiplied by something else: 6x x 2x and 6x x 3.
• Take out the common factor (6x) by writing it outside the brackets and put the remaining terms inside. 
• The answer is 6x(2x - 3).
• Check this expands to give 12x² + 18x

Factorising by Grouping

How to Factorise Expressions with Common Brackets 

• When you are asked to factorise expressions like 3x(t - 4) + 2(t - 4), you may notice that both terms share a common bracket, (t - 4). 
• This common bracket can be "taken out" as a common factor, resulting in (t - 4)(3x + 2). 
• This process is similar to factorising expressions like 3xy + 2y, where you factor out the common factor y to get y(3x + 2). 
• Here, y represents (t - 4).

How do I factorise by grouping?

• When factorising by grouping, you may need to create a common bracket.
  • For instance, consider the expression: xy + px + qy + pq.
  • Group the first pair of terms, xy and px, and factorise to get x(y + p).
  • Next, group the second pair of terms, qy and pq, and factorise to obtain q(y + p).
  • Now, factorise x(y + p) and q(y + p) together to get (y + p)(x + q).
• This method is known as factorizing by grouping.
• The groupings don't always consist of the first and second pairs of terms but rather any two terms that share common factors.