Download a + 2 - Official Mathematics Revision Website

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…
12
1
(b) Factors of 3x2
12
12
2
6
1, 2, 3, 4, 6, 12
3
3x2
3x2
12
4
1
3x2
3
x2
3x2
x
1, 3, x, 3x, x2, 3x2
3x
Factors
(c) Factors of 18
1 2 3 6
9 18
(d) Factors of 6x2
1
2 3
6x2 3x2 2x2 x2
x 2x 3x 6x
(e) Factors of 9x
1
3
x
3x
9
9x
Highest Common Factors
8x 4x2
12
16
5a
a2
1
1
1
1
1
1
2
2
5
a
2
2
3
4
a
a2
4
x
4
8
5a
8
2x
6
16
x
4x
2x
x2
12
The highest common factor
(HCF) is the largest factor
that appears in both lists.
4x 2x2
8x 4x2
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
6a + 12
Common Factor (HCF)?
=6( a + 2)
6 is the common factor
Now take 6 outside the bracket
and work out what goes inside
the bracket.
Now factorise the following expressions:
(1) 5 x + 10 = 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) 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 )
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)
Factorise the following expressions:
(1) 5wg – 10wm
= 5w( g + 2m )
(2) 16xy – 8xw
= 8x ( 2y - w )
(3) 9ab + 12bc
= 3b ( 3a - 4c )
(4) 6x2 + 9 xy
= 3x ( 2x + 3y )
Set 1: Factorise each of the following
Question
Factorize
Question
Factorize
1
8x - 32
8(x – 4)
5x - 50
5(x + 5)
6t - 4
2(3t - 2)
15f - 30
5(3f - 6)
2x2 + 4x
2x(x + 2)
2
12y + 30
6(y + 5)
7m - 7
7(m - 1)
8 – 4n
4(2 - n)
12c + 16
4(3c + 4)
5y2 – 20y
5y(y - 4)
3
5
7
9
4
6
8
10
Question
11
13
15
17
19
Factorize
a2 + a
a(a + 1)
2x – 2x2
2x(1 - x)
ab + 5b
b(a + 5)
x – 4x2
x(1 – 4x)
16x2 – 8x3
8x2(2 – x)
Question
12
14
16
Factorize
15m + 30m2
5m(3 + 6m)
3t + 27t2
3t(1 + 9t)
pq – q2
q(p - q)
18
9y3 – 18y2
9y2(y – 2)
20
5x – 5x2
5x(1 – x)
Set 2: Factorise each of the following
Question
Factorize
Question
Factorize
1
5x - 20
2
9y + 81
3
5(x - 4)
2x + 2
2(x + 1)
4
9(y + 9)
6k - 60
6(k - 10)
5
16y - 24
8(2y - 3)
6
18 + 6m
6(3 + 2m)
7
25f - 35
8
4c + 6
9
5(5f - 7)
5x2 + 5x
5x(x + 1)
10
2(2c + 3)
6y2 – 24y
6y(y - 4)
Question
Factorize
Question
Factorize
11
2a2 + 4a
12
12m + 32m2
13
2a(a + 2)
12x – 4x2
14
4m(3 + 8m)
7t + 56t2
4x(3 - x)
15
17
7t(1 + 8t)
8b - 48b2
8b(1 - 6b)
16
2pq – 4q2
4x – 4x2
18
2q(p – 2q)
8c3 + 20c2
4x(1 – x)
19
4x2 + 32x3
4x2(1 + 8x)
4c2(2c + 5)
20
x – 12x2
x(1 - 12x)
More Practice
Factorise the following
1. 9p – 63
9. 24ab + 18a
17. 6ab – 3a2b
2. 8x + 20
10. 21x – 6xy
18. 4x2y + 8xy
3. 35p – 28
11. 4t2 + 8t
19. 5p2q – 5pq2
4. 56y – 32
12. 8y2 - 32y
20. 2c2d + 6cd2
5. 36t + 45v
13. 10b2 + 15b
21. a3b + ab3
6. 33a – 18b
14. 12k + 2k2
22. 8x3 + 12xy
7. 84c + 56d
15. 6c2 + 9cd
23. 9g2h – 18gh3
8. 81 – 54k
16. 7v2 + 91v
24. 48t2v2 – 8t2