2. Factorizing expressions
Factorizing an expression is the opposite of expanding it.
Expanding or multiplying out
a(b + c)
ab + ac
Factorizing
When we expand an expression we remove the brackets.
When we factorize an expression we write it with brackets.

3. Factorizing expressions
Expressions can be factorized by dividing each term by a common factor and writing this outside of a pair of brackets.
For example, factorize the expression: 5x + 10.
The terms 5x and 10 have a common factor of 5.
Write the 5 outside of a set of brackets.
Mentally divide 5x + 10 by 5.
(5x + 10) ÷ 5 = x + 2
Teacher notes
Encourage pupils to check this by multiplying the expression out to 5x + 10.
This is written inside the bracket.
5( )
x + 2

4. Factorizing expressions
Teacher notes
Point out that we do not normally show the line involving division. This is done mentally. We can check the answer by multiplying out the bracket.

5. Factorization Teacher notes
Start by asking pupils to give you the value of the highest common factor of the two terms. Reveal this and then ask pupils to give you the values of the terms inside the brackets.

6. Factorization by pairing
Some expressions containing four terms can be factorized by regrouping the terms into pairs that share a common factor.
For example, factorize the expression: 4a + ab b.
Two terms share a common factor of 4 and the remaining two terms share a common factor of b.
4a + ab b = 4a ab + b
= 4(a + 1) + b(a + 1)
4(a + 1) and b(a + 1) share a common factor of (a + 1) so we can write this as:
(a + 1)(4 + b)

7. The difference between two squares
A quadratic expression in the form x2 – a2 is called the difference between two squares.
The difference between two squares can be factorized as follows:
x2 – a2 = (x + a)(x – a)
For example,
Teacher notes
Pupils should be encouraged to spot the difference between two squares whenever possible.
9x2 – 16 = (3x + 4)(3x – 4)
25a2 – 1 = (5a + 1)(5a – 1)
m4 – 49n2 = (m2 + 7n)(m2 – 7n)

8. The difference between two squares
Teacher notes
Select an expression involving the difference between two squares and ask a volunteer to find the corresponding factorization.