5 Factorisation

A number is said to be factorised when it is written as a product. For example, 21 can be factorised into 7 × 3 . We say that 7 and 3 are factors of 21.

Algebraic expressions can also be factorised. Consider the expression 7 ( 2 x + 1 ) . Removing the brackets we can rewrite this as

7 ( 2 x + 1 ) = 7 ( 2 x ) + ( 7 ) ( 1 ) = 14 x + 7.

Thus 14 x + 7 is equivalent to 7 ( 2 x + 1 ) . We see that 14 x + 7 has factors 7 and ( 2 x + 1 ) . The factors 7 and ( 2 x + 1 ) multiply together to give 14 x + 7 . The process of writing an expression as a product of its factors is called factorisation . When asked to factorise 14 x + 7 we write

14 x + 7 = 7 ( 2 x + 1 )

and so we see that factorisation can be regarded as reversing the process of removing brackets.

Always remember that the factors of an algebraic expression are multiplied together.

Example 43

Factorise the expression 4 x + 20 .

Solution

Both terms in the expression 4 x + 20 are examined to see if they have any factors in common. Clearly 20 can be factorised as ( 4 ) ( 5 ) and so we can write

4 x + 20 = 4 x + ( 4 ) ( 5 )

The factor 4 is common to both terms on the right; it is called a common factor and is placed at the front and outside brackets to give

4 x + 20 = 4 ( x + 5 )

Note that the factorised form can be checked by removing the brackets again.

Example 44

Factorise z 2 − 5 z .

Solution

Note that since z 2 = z × z we can write

z 2 − 5 z = z ( z ) − 5 z

so that there is a common factor of z . Hence

z 2 − 5 z = z ( z ) − 5 z = z ( z − 5 )

Example 45

Factorise 6 x − 9 y .

Solution

By observation, we see that there is a common factor of 3. Thus 6 x − 9 y = 3 ( 2 x − 3 y )

Task!

Factorise 14 z + 21 w .

  1. Find the factor common to both 14 z and 21 w :

    7

  2. Now factorise 14 z + 21 w :

    7 ( 2 z + 3 w )

    Note : If you have any doubt, you can check your answer by removing the brackets again.

Task!

Factorise 6 x − 12 x y .

First identify the two common factors:

6 and x

Now factorise 6 x − 12 x y :

6 x ( 1 − 2 y )

Exercises
  1. Factorise
    1. 5 x + 15 y ,
    2. 3 x − 9 y ,
    3. 2 x + 12 y ,
    4. 4 x + 32 z + 16 y ,
    5. 1 2 x + 1 4 y .

      In each case check your answer by removing the brackets again.

  2. Factorise
    1. a 2 + 3 a b ,
    2. x y + x y z ,
    3. 9 x 2 − 12 x
  3. Explain why a is a factor of a + a b but b is not. Factorise a + a b .
  4. Explain why x 2 is a factor of 4 x 2 + 3 y x 3 + 5 y x 4 but y is not.

    Factorise 4 x 2 + 3 y x 3 + 5 y x 4 .

    1. 5 ( x + 3 y ) ,
    2. 3 ( x − 3 y ) ,
    3. 2 ( x + 6 y ) ,
    4. 4 ( x + 8 z + 4 y ) ,
    5. 1 2 ( x + 1 2 y )
    1. a ( a + 3 b ) ,
    2. x y ( 1 + z ) ,
    3. 3 x ( 3 x − 4 ) .
  3. a ( 1 + b ) .
  4. x 2 ( 4 + 3 y x + 5 y x 2 ) .