Factorising (Cambridge O Level Maths)

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

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