c ha p
ter
Factors
2
Key words
factor   highest common factor   grouping terms   difference of two squares
area model    quadratic expressions    simplifying fractions
Section 2.1 Factorising with common factors 
Since 9 3 5 5 45, we say that 9 and 5 are factors of 45.
15 and 3 are also factors of 45.
The factors of 24 are 1, 2, 3, 4, 6, 12 , 24.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12 , 18, 36.
The highest common factor is 12.
Here are two algebraic terms: 6xy and 12x.
The highest common factor of the numbers is 6.
The highest common factor of the variables is x.
So the highest common factor of the two terms is 6 3 x, i.e. 6x.
Similarly, the highest common factors of
(i) 3a and 6a2 5 3a
(iii) 5a2b 2 15ab 5 5ab
(ii) 6x2 2 12xy 5 6x
(iv) 4x2 1 16xy2 5 4x
Take the expression 5x 1 10.
5x 1 10 5 5(x 1 2)
5 and (x 1 2) are called the factors of 5x 1 10.
To factorise an algebraic expression:
> Find the highest common factor and write it outside the brackets.
> Divide each term by this factor and write the results inside the brackets.
> Check your result by expanding the brackets.
Here are some expressions that have been factorised:
(i) x2 1 7x 5 x(x 1 7)
(iii) 3xy 2 12x 5 3x(y 2 4)
(ii) 3x2 2 9x 5 3x(x 2 3)
(iv) 12x2y2 2 6xy 5 6xy(2xy 2 1)
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Exercise 2.1
  1. Write down the highest common factor of each of these:
(i) 9 and 12
(ii) 12 and 18
(iii) 14 and 21
(iv) 21 and 35
  2. Write down the highest common factor of each of these:
(i) 4x and 12x
(ii) 3n and 9n
2
(iii) 10x and 15x
2
(iv) 3a and 6a
(vi) 2a2b and 6ab
(v) 3xy and 12x   3. Copy and complete each of these:
(i) 7x 1 14y 5 7( )
(ii) 16a 1 24b 5 8( )
(iii) ab 1 bc 5 b( )
(iv) 3a2 1 6a 5 3a( )
(v) 5x2 2 15xy 5 5x( )
(vi) 12xy 2 18yz 5 6y( )
3
2
2
(viii) 6a2b 2 8ab2 1 4ab 5 2ab( )
(vii) 15x 1 10x y 5 5x ( )
Factorise each of the following:
  4. 6x 1 18y
  5. 3ab 1 3bc
  6. 6ax 2 12ay
  7. 6a2 2 12a
  8. 7x2 2 28x
  9. 15x2 1 25xy
10. 3x2 2 6x2y
11. 3ab2 2 6ab
12. 3p2 2 6pq
13. 2x2y 2 6x2z
14. 6y2z 1 10y2
15. 10p2q 1 5pq2
16. 2a3 2 4a2 1 8a
17. 4x2 2 6xy 1 8xz
18. 5xy2 2 20x2y
19. 4x2y2 2 8xy
20. 5x3 2 10x2 1 15x
21. 2a2b 2 4ab2 1 12abc
22.
E
H
P
S
O
A
5
2a
3a
2b
7b
a2
I
L
ab 3b2
G
a1b
R
T
U
N
a 2 5b 2a 2 b ab 1 1 2a 1 3b
Fully factorise each expression below as the product of two factors.
Use the code above to find a letter for each factor.
Rearrange each set of letters to spell a bird.
(i) 3a2 2 15ab
2a3 2 a2b
7ab 2 35b2
(ii) 4a2 2 2ab
2a2b 1 2a
2ab 2 10b2
(iii) 7ab 1 7b2
5a 2 25b
2ab2 1 2b
(iv) 4ab 2 2b2
3b2a 1 3b3
a3 2 5a2b
2a2b 1 3ab2
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Section 2.2 Factorising by grouping terms Some four-termed expressions do not have an overall common factor, but can be factorised
by pairing the four terms.
For example,
{
{
ab 1 ac 1 bd 1 dc
5 a(b 1 c) 1 d(b 1 c)
… factorising each pair separately
5 (b 1 c)(a 1 d)
… removing common factor (b 1 c)
Example 1
Find the factors of
(i) 2ab 1 2ac 1 3bx 1 3cy
(ii) 3ax 2 bx 2 3ay 1 by
(i)2ab 1 2ac 1 3bx 1 3cx5 2a(b 1 c) 1 3x(b 1 c)
5 (b 1 c)(2a 1 3x)
(ii)3ax 2 bx 2 3ay 1 by 5 x(3a 2 b) 2y(3a 2 b)
5 (3a 2 b)(x 2 y)
Note: Sometimes it may be necessary to reorder terms before the method shown above
can be used.
Example 2
Factorise 6x2 1 2a 2 3ax 2 4x
Regroup:
6x2 2 3ax 1 2a 2 4x
3x(2x 2 a) 1 2(a 2 2x)   …
5 3x(2x 2 a) 2 2(2x 2 a)
5 (2x 2 a)(3x 2 2)
Here we had no
common factor, so we
changed 2(a 2 2x)
to 22(2x 2 a). (2x 2 a) is
now a common factor.
∴ 6x2 1 2a 2 3ax 2 4x 5 (2x 2 a)(3x 2 2)
Note: Be careful when dealing with negative terms.
For example,
(i) 23ax 2 6ay 5 23a(x 1 2y)
(ii) 25x2 1 10xy 5 25x(x 2 2y)
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Exercise 2.2
Factorise fully each of the following:
  1. 2a(x 1 y) 1 3(x 1 y)
  2. 3x(2a 2 b) 2 4(2a 2 b)
  3. 3a(2b 2 c) 2 4(2b 2 c)
  4. 2x(5y 2 z) 1 b(5y 2 z)
  5. 2a(x 2 2y) 2 (x 2 2y)
  6. a2 1 ab 1 ac 1 bc
  7. x2 2 ax 1 3x 2 3a
  8. ab 1 ac 2 5b 2 5c
  9. ab 1 5b 1 3a 1 15
10. 3x2 2 3xz 1 4xy 2 4yz
11. 2c2 2 4cd 1 c 2 2d
12. 2ax 2 6ay 2 3x 1 9y
13. 2ac 2 4ad 1 bc 2 2bd
14. 3xy 2 3xyz 1 2z 2 2z2
15. 8ax 1 4ay 2 6bx 2 3by
16. 6ax2 1 9a 2 8x2 2 12
17. x(2y 2 z) 2 2y 1 z
18. an 2 5a 2 5b 1 bn
19. 2x2y 2 2xz 2 3xy 1 3z
20. 7y2 2 21by 1 2ay 2 6ab
21. 4a2b 2 3b 2 6a 1 2ab2
22. 12a2 2 8ab 1 9ac 2 6bc
23. 10ab 2 5ac 2 2bd 1 cd
24. 4x2 1 3ay 2 2ax 2 6xy
25. 6a2 1 15xy 2 10ay 2 9ax
26. 6xy 1 12yz 2 8xz 2 9y2
27. 3abx2 2 5axy 2 3bxy 1 5y2
28. 6a2c 2 6ab 2 4bc 1 9a3
29. x2 2 x(2a 2 b) 2 2ab
30. 6x2 2 3y(3x 2 2a) 2 4ax
Section 2.3 Difference of two squares Numbers such as 1, 4, 9, 16, 25, … are called perfect squares as they are obtained by
multiplying some whole number by itself, e.g., 4 5 22, 9 5 32, …
Similarly, in algebra, 4x2 5 (2x)2 and 9y2 5 (3y)2.
Expressions such as 102 2 42, x2 2 y2 and 4x2 2 9 are known as the difference of
two squares.
When you multiply (x 1 y)(x 2 y), you get x2 2 y2.
Thus, the factors of x2 2 y2 are (x 1 y)(x 2 y).
x2 2 y2 5 (x 1 y)(x 2 y)
In words: (first)2 2 (second)2 5 (first 1 second)(first 2 second)
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Here is an area model of the difference of two squares.
x
Area of large square 5 x2
y2
Area of shaded square 5 y2
x
B
y
A
Unshaded area 5 x2 2 y2 …… ➀
Area of B 5 x(x 2 y)
x�y
y
x�y
y
Area of A 5 y(x 2 y)
Area of B 1 A 5 x(x 2 y) 1 y(x 2 y)
5 (x 2 y)(x 1 y) …… ➁
Equating ➀ and ➁ we get: x2 2 y2 5 (x 2 y)(x 1 y)
Example 1
Factorise
(i) 9x2 2 4
(ii) 25a2 2 81b2
(iii) x2 y2 2 4a2b2
(i) 9x2 2 4 5 (3x)2 2 (2)2 5 (3x 1 2)(3x 2 2)
(ii) 25a2 2 81b2 5 (5a)2 2 (9b)2 5 (5a 1 9b)(5a 2 9b)
(iii) x2 y2 2 4a2b2 5 (xy)2 2 (2ab)2 5 (xy 1 2ab)(xy 2 2ab)
Example 2
(i) Factorise 12x2 2 75y2
(ii) Evaluate 512 2 492 without using a calculator.
(i) Here it is not immediately obvious that 12x2 2 75y2 involves the difference
of two squares.
However, if we take out the common factor 3 we get 3(4x2 2 25y2).
12x2 2 75y25 3(4x2 2 25y2)
5 3[(2x)2 2 (5y)2]
5 3(2x 1 5y)(2x 2 5y)
(ii)512 2 4925 (51 1 49)(51 2 49) … using the difference of two squares
5 (100)(2)
5 200
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Exercise 2.3
Factorise each of the following:
  1. x2 2 y2
  2. a2 2 b2
  3. x2 2 4y2
  4. x2 2 16y2
  5. 4x2 2 y2
  6. 9x2 2 16y2
  7. 4a2 2 25b2
  8. 36x2 2 49y2
  9. 64x2 29y2
10. 16x2 2 25
11. 25x2 2 1
12. 36x2 2 25
13. 15x2 2 64y2
14. 1 2 36x2
15. 1 2 81y2
16. 36 2 121y2
17. 49a2 2 4b2
18. (xy)2 2 4
19. (ab)2 2 25
20. x2y2 2 16
21. a2b2 2 49
22. (5xy)2 2 36
23. 16a2b2 2 25
24. 9x2y2 2 1
25. 4a2b2 2 49c2d2
26. 121a2 2 64b2c2
27. 81h2k2 2 25p2q2
28. First take out the highest common factor and then factorise each of the following:
(i) 3x2 2 27y2
(ii) 12x2 2 3y2
(iii) 27x2 2 3y2
(iv) 45 2 5x2
(v) 45k2 2 20
(vi) 4a2x2 2 36y2
29. Challenge: factorise a4 2 b4 as far as possible.
30. Use the difference of two squares to evaluate each of these:
(i) 962 2 42
(ii) 232 2 172
(iii) (7.9)2 2 (2.1)2
(iv) (9.4)2 2 (0.6)2
31. Simplify (3x 1 b)(6x 2 2b) 2 (2y 1 b)(4y 2 2b).
Now factorise fully the simplified expression.
32. Simplify and hence factorise (3x 2 2y)2 2 y(5y 2 12x).
Section 2.4 Factorising quadratic expressions An expression of the form ax2 1 bx 1 c, where a, b and c are numbers is called a quadratic
expression since the highest power of x is 2.
Since (x 1 5)(x 1 2) 5 x2 1 7x 1 10, we say that (x 1 5) and (x 1 2) are the factors
of x2 1 7x 1 10.
outside terms
In the product (x 1 5)(x 1 2) 5 x2 1 7x 1 10,
inside terms
2
(i) x is obtained from the product x 3 x
(ii) 10 is the product of 5 and 2, the two number terms
(iii) 7x is obtained by adding the product of the outside
terms to the product of the inside terms
i.e. 2x 1 5x 5 7x.
An expression in the form
ax2 1 bx 1 c is generally
called a quadratic trinomial
as it contains 3 terms.
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We factorise a quadratic expression by ‘trial and error’ to find numbers such that the product
of the outside terms added to the product of the inside terms gives the middle term.
outside terms
3x 1 5x 5 8x
Here are the factors of x2 1 8x 1 15: (x 1 5)(x 1 3) 3 3 5 5 15
inside terms
Example 1
Factorise 3x2 1 10x 1 8.
The factors of 3x2 1 10x 1 8 will take the form (3x 1 ?)(x 1 ?)
The factors of 8 are 8 3 1 or 4 3 2
Investigate 4 and 2:
(3x 1 2)(x 1 4)
Try again:
(3x 1 4)(x 1 2)
The factors of 3x2 1 10x 1 8 are (3x 1 4)(x 1 2)
Outside 1 inside terms
5 12x 1 2x
5 14x …… incorrect
Outside 1 inside terms
5 6x 1 4x 5 10x
…… correct
Final term positive If the third term of a quadratic expression is positive and the middle term is negative,
e.g. x2 2 8x 1 15, the factors will take the form
(x 2 ?)(x 2 ?)
shown on the right.
Example 2
Find the factors of 2x2 2 11x 1 12.
The factors will take the form (2x 2 ?)(x 2 ?)
28x
(2x 2 3)(x 2 4) … correct factors
23x
Factors of 12:
  632
  433
12 3 1
∴ 2x 2 11x 1 12 5 (2x 2 3)(x 2 4)
2
Final term negative If the final term is negative, the factors will take either
of the forms shown on the right.
(x 1 ?)(x 2 ?)
or
(x 2 ?)(x 1 ?)
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Example 3
Factorise (i) 8x2 1 10x 2 3 (ii) 7x2 2 19xy 2 6y2
(i)8x2 1 10x 2 3 5 (4x 2 1)(2x 1 3)
12x 2 2x 5 10x
(correct)
∴ 8x 1 10x 2 3 5 (4x 2 1)(2x 1 3)
2
(ii)7x2 2 19xy 2 6y2 5 (7x 1 3y)(x 2 2y)
214xy 1 3xy 5 211xy
(incorrect)
(7x 1 2y)(x 2 3y)
221xy 1 2xy 5 219xy
(correct)
∴ 7x2 2 19xy 2 6y2 5 (7x 1 2y)(x 2 3y)
Exercise 2.4
Factorise each of the following:
  1. x2 1 5x 1 6
  2. x2 1 8x 1 12
  3. x2 1 9x 1 14
  4. x2 1 11x 1 24
  5. x2 1 12x 1 20
  6. x2 1 12x 1 27
  7. x2 1 11x 1 30
  8. x2 1 15x 1 44
  9. x2 1 20x 1 36
10. 2x2 1 5x 1 2
11. 2x2 1 11x 1 14
12. 5x2 1 21x 1 4
13. x2 2 7x 1 12
14. x2 2 9x 1 18
15. x2 2 9x 1 20
16. x2 2 14x 1 24
17. x2 2 12x 1 27
18. x2 2 13x 1 36
19. 2x2 2 7x 1 3
20. 3x2 2 17x 1 10
21. 5x2 2 17x 1 6
22. 3x2 2 17x 1 20
23. 5x2 1 27x 2 18
24. 3x2 2 14x 1 15
25. x2 2 4x 2 12
26. x2 2 3x 2 10
27. x2 1 7x 2 18
28. x2 1 7x 2 30
29. x2 2 13x 2 30
30. x2 2 18x 2 40
31. 2x2 2 7x 2 15
32. 3x2 1 11x 2 20
33. 5x2 2 12x 2 9
34. x2 2 6x 2 72
35. 8x2 1 10x 2 3
36. 2x2 2 19x 1 9
37. 12x2 2 11x 2 5
38. 6x2 1 x 2 15
39. 8x2 2 14x 1 3
40. 3x2 1 13x 2 10
41. 9x2 1 24x 1 16
42. 5x2 2 31x 1 6
43. 3x2 2 x 2 14
44. 6x2 2 11x 1 3
45. 12x2 2 23x 1 10
46. 9x2 1 25x 2 6
47. 6x2 1 x 2 22
48. 9x2 2 x 2 10
49. 4x2 2 11x 1 6
50. 10x2 2 17x 2 20
51. 36x2 2 7x 2 4
52. 12x2 2 17x 1 6
53. 15x2 2 14x 2 8
54. 24x2 1 2x 2 15
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Section 2.5 Using factors to simplify algebraic
fractions 10  ​ can be simplified by dividing above and The fraction ​ ___
15
below by the common factor 5.
2
10  ​5 ​ __
2  ​
​ ___
15
3
3
x2 2 4 ​  can be simplified by dividing above and below by a common factor.
Similarly, ​ ______
x12
1
(x 1 2)(x 2 2)
______
  
​ 
 ​ 
5x22
​ x 2 4 ​ 5 ____________
2
x12
(x 1
2)
1
Example 1
2 
3x2 2 5x 2
 
(ii)​ ___________
 ​
x22
3n 2 12
Simplify (i) ​ _______
 ​ 
n24
1
1
3(n 2 4)
3n 2 12
  _______
(i)​ _______
 ​5
​ 
 ​ 
n24
(n 21 4)
(3x 1 1)(x 2 2)
2 
3x2 2 5x 2
 
  
(ii)​ ___________
 ​
5 _____________
​ 
 ​ 
x22
(x 2
2)
1
5 35 3x 1 1
Exercise 2.5
  1. Simplify each of the following:
14  ​
(i)​ ___
35
7x  ​ 
(ii)​ ___
14
9x2 ​ 
(iii)​ ___
3x
  2. Simplify each of the following:
4x 1 4y
12(a 1 b)
(i)​ _______
 
 ​ 
(ii)​ ________ ​ 
4
3(a 1 b)
8p2
(iv)​ ___ ​ 
2p
3x 1 12 ​ 
(iii)​ _______
x(x 1 4)
9x2y
(v)​ ____ ​ 
3xy
4a 2 8b 
(iv)​ ________
 ​ 
3(a 2 2b)
Simplify each of the following, using factors where necessary:
2(y 2 1)(y 1 3)
(x 2 1)(x 1 3)
7 
x2 1 8x 1
  3.​ ____________
  
 
  
 
 
 ​ 
4.​ _____________
 ​ 
5.​ __________
 ​
x13
y21
x11
3x 2 3  ​ 
x 2 2   ​ 
x 2 4   ​ 
 
 
7.​ ___________
8.​ __________
  6.​ __________
x2 2 6x 1 8
x2 1 5x 2 14
x2 2 2x 1 1
2x 2 6  ​ 
  9.​ __________
x2 1 x 2 12
30 
x2 1 x 2 ​
 
10.​ __________
x25
a2 1 2ab ​ 
 
11.​ ________
3a 1 6b
x2 2 9 ​ 
12.​ ______
x23
a2 2 16 
13.​ _______
 ​ 
3a 2 12
n 1 9  
   ​
14.​ _____________
n2 1 18n 1 81
4x 2 8 ​ 
15.​ ______
x2 2 4
3 
2x2 1 5x 2
 
16.​ ___________
 ​
2x 2 1
1
15  
2x2 1 11x  
17.​ _____________
 ​
2x 1 5
ab 2 ac
 ​ 
18.​ _______
b2c
5 2 x  ​ 
19.​ _____
x25
3a 1 9 ​ 
20.​ ______
4 _____
​ a 1 3 
 ​
a2 2 1 a 2 1
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Test yourself 2
  1. (a)(i) Complete the following: 6x2 2 18xy 5 6x( )
(ii) Factorise x2 2 10x 1 24
(b) Factorise each of these:
(i) 7a 1 7b 1 xa 1 xb
(ii) 25a2 2 81
4x2 2 7x 2 ​
2 
(c) Factorise and hence simplify ​ ___________
 
4x2 2 8x
  2. (a) Factorise each of these:
(i) 6x2 2 x 2 2
(b) Factorise each of these:
(i) 6a2x 1 3ax2 2 9ax
(ii) 6a2 1 2ab 1 3ac 1 bc
(ii) 3x2 2 48
6x2 2 11x  
2 ​
10  
(c) Factorise fully and simplify ​ _____________
4x2 2 25
  3. (a) Factorise each of these:
(i) 8a2b 1 2ab2
(ii) 3x2 2 16x 1 21
(b) Factorise fully each of these:
(i) 2x2 2 8y2
(ii) 2xy 2 xz 2 2y 1 z
(c) Simplify (2x 2 z)(6x 1 3z) 2 (6a 2 3z)(2a 1 z) and factorise fully
the simplified expression.
  4. (a) Factorise each of the following:
(i) 15bc 2 3c2
(ii) 24x2 1 x 2 3
(b) Factorise fully each of these:
(i) 8x2 2 2y2
(ii) ax 2 2ay 1 2by 2 bx
10x2 2 29x  
1 ​
10  
(c) Factorise fully and simplify ​ ______________
4x2 2 25
  5. (a) Factorise each of the following:
(ii) a2 1 ab 2 2a 2 2b
(i) 3x2 1 2x 2 8
(b) Factorise fully each of these:
(i) 5a2 2 125b2
(ii) 2x3 1 3x2 2 2xy2 2 3y2
1 ​
18  
2x2 2 15x  
(c) Factorise and simplify ​ _____________
x3 2 36x
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