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

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

3. Factorizing expressions
Writing 5x + 10 as 5(x + 2) is called factorizing the expression.
Factorize 6a + 8
Factorize 12n – 9n2
The highest common factor of 6a and 8 is
The highest common factor of 12n and 9n2 is 2.
3n.
(6a + 8) ÷ 2 =
3a + 4
(12n – 9n2) ÷ 3n =
4 – 3n
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.
6a + 8 =
2(3a + 4)
12n – 9n2 =
3n(4 – 3n)

4. Factorizing expressions
Writing 5x + 10 as 5(x + 2) is called factorizing the expression.
Factorize 3x + x2
Factorize 2p + 6p2 – 4p3
The highest common factor of 3x and x2 is
The highest common factor of 2p, 6p2 and 4p3 is x.
2p.
(2p + 6p2 – 4p3) ÷ 2p =
(3x + x2) ÷ x =
3 + x
1 + 3p – 2p2
3x + x2 =
x(3 + x)
2p + 6p2 – 4p3 =
2p(1 + 3p – 2p2)

5. Factorization
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 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. Factorization by pairing
Factorize xy – 6 + 2y – 3x
We can regroup the terms in this expression into two pairs of terms that share a common factor.
When we take out a factor of –3, – 6 becomes + 2
xy – 6 + 2y – 3x = xy + 2y – 3x – 6
= y(x + 2) – 3(x + 2)
This expression could also be written as xy – 3x + 2y – 6 to give x(y – 3) + 2(y – 3) = (y – 3)(x + 2)
y(x + 2) and – 3(x + 2) share a common factor of (x + 2) so we can write this as
(x + 2)(y – 3)