1. Factors
A factor is a number or letter that will divide exactly into another number or expression without leaving a remainder
Examples
(a) Factors of 12…
(b) Factors of 3x2 12 12 12
3x2
3x2
3x2 1 12 2 6 3 4 1
3x2 3 x2 x 3x
1, 2, 3, 4, 6, 12
1, 3, x, 3x, x2, 3x2

2. Factors
(c) Factors of 18
(d) Factors of 6x2 1 2 3 6 1 2 3 9 18
6x2
3x2
2x2 x2 x 2x 3x 6x
(e) Factors of 9x 1 3 9 x 3x 9x

3. Highest Common Factors1 2 3 4 6 12 1 2 4 8 16 1 5 a 5a 1 a a2 1 2 4 8 x 2x 4x 8x
4x2 8x 1 2 x 2x 4x x2
2x2
4x2 12 16 5a a2
The highest common factor (HCF) is the largest factor that appears in both lists.

4. Putting The Brackets Back In
It is not only important to be able to multiply out brackets but also to be able to put the brackets back. This process is called FACTORISATION.
Consider the expression below :
Can you find a Highest Common Factor (HCF)?
6a + 12
= 6 ( a +
2 )
6 is the common factor
Now take 6 outside the bracket and work out what goes inside the bracket.

5. Now factorise the following expressions:
5 ( x +
2 )
(2) 7x + 21 =
7 ( x +
3 )
(3) 6x - 9 =
3 ( 2x -
3 )
(4) 15x - 20 =
5 ( 3x -
4 )
(5) 24x + 8 =
8 ( 3x +
1 )

6. (6) 6x + 12 = 6
( x +
2 )
(7) 9x - 18 = 9
( x -
2 )
(8) 8x + 12 = 4
( 2x +
3 )
(9) 7x - 21 = 7
( x -
3 )
(10) 10x + 15 = 5
( 2x +
3 )

7. Multiplying Out Brackets Reminder
(1) 3t( 2t + 5 ) =
6t2 +
15t
So 6t2 + 15t factorised = 3t(2t + 5)
(2) 4w ( 3w - 7 ) =
12w2 -
28w
So 12w2 – 28w factorised = 4w(3w - 7)
(3) 5a ( 2a + 9 ) =
10a2 +
45a
So 10a2 + 45a factorised = 5a(2a + 9)
(4) 2z ( 5z - 8 ) =
10z2 -
16z
So 10z2 – 16z factorised = 2z(5z - 8)

8. Factorise the following expressions:
(1) 5wg – 10wm
(3) 9ab + 12bc
= 5w
( g +
2m )
= 3b
( 3a -
4c )
(2) 16xy – 8xw
(4) 6x2 + 9 xy
= 8x
( 2y -
w )
= 3x
( 2x +
3y )

9. Set 1: Factorise each of the following
Question
Factorize 1
8x - 32 2
12y + 30 3
5x - 50 4
7m - 7 5
6t - 4 6
8 – 4n 7
15f - 30 8
12c + 16 9
2x2 + 4x 10
5y2 – 20y
8(x – 4)
6(y + 5)
5(x + 5)
7(m - 1)
2(3t - 2)
4(2 - n)
5(3f - 6)
4(3c + 4)
2x(x + 2)
5y(y - 4)

10. Question
Factorize 11
a2 + a 12
15m + 30m2 13
2x – 2x2 14
3t + 27t2 15
ab + 5b 16
pq – q2 17
x – 4x2 18
9y3 – 18y2 19
16x2 – 8x3 20
5x – 5x2
a(a + 1)
5m(3 + 6m)
2x(1 - x)
3t(1 + 9t)
b(a + 5)
q(p - q)
x(1 – 4x)
9y2(y – 2)
8x2(2 – x)
5x(1 – x)

11. Set 2: Factorise each of the following
Question
Factorize 1
5x - 20 2
9y + 81 3
2x + 2 4
6k - 60 5
16y - 24 6
18 + 6m 7
25f - 35 8
4c + 6 9
5x2 + 5x 10
6y2 – 24y
5(x - 4)
9(y + 9)
2(x + 1)
6(k - 10)
8(2y - 3)
6(3 + 2m)
5(5f - 7)
2(2c + 3)
5x(x + 1)
6y(y - 4)

12. Question
Factorize 11
2a2 + 4a 12
12m + 32m2 13
12x – 4x2 14
7t + 56t2 15
8b - 48b2 16
2pq – 4q2 17
4x – 4x2 18
8c3 + 20c2 19
4x2 + 32x3 20
x – 12x2
2a(a + 2)
4m(3 + 8m)
4x(3 - x)
7t(1 + 8t)
8b(1 - 6b)
2q(p – 2q)
4x(1 – x)
4c2(2c + 5)
4x2(1 + 8x)
x(1 - 12x)

13. Factorise the following
More Practice
Factorise the following
9p – 63
8x + 20
35p – 28
56y – 32
36t + 45v
33a – 18b
84c + 56d
81 – 54k
9. 24ab + 18a
10. 21x – 6xy
11. 4t2 + 8t
12. 8y y
13. 10b2 + 15b
14. 12k + 2k2
15. 6c2 + 9cd
16. 7v2 + 91v
17. 6ab – 3a2b
18. 4x2y + 8xy
19. 5p2q – 5pq2
20. 2c2d + 6cd2
21. a3b + ab3
22. 8x3 + 12xy
23. 9g2h – 18gh3
24. 48t2v2 – 8t2