9 EXPANDING BRACKETS AND
FACTORISING
Algebra is regularly used by Formula One teams to maximise the performance of their
cars when racing. For example, new rules introduced into Formula One in 2009 have
given teams a booster button which gives the car extra power and can be pushed for a
maximum of 6.7 seconds during a race. In order to maximise the benefit of the ‘boost’, F1
teams use algebra to work out the best moment for the driver to use it.
Objectives
In this chapter you will:
expand brackets
factorise algebraic expressions
simplify complicated algebraic expressions.
132
Before you start
You should be able to:
simplify algebraic expressions by collecting
like terms
use the index law xm  xn  xm  n
add, subtract, multiply and divide directed
numbers.
9.1 Expanding brackets
9.1 Expanding brackets
Objective
You can expand expressions which have a single pair of brackets.
Get Ready
1. Simplify
a 5  2x
d 4x  5  3x  1
b 3x  (4x)
e x²  2x  3x  6
c (x)  (2x)
f x²  x  2x  2
Key Points
When there is a number outside a bracket there is a hidden multiplication sign. So 20(n  3)  20  (n  3).
In algebra, expand usually means multiply out.
To expand a bracket you multiply each term inside the bracket by the term outside the bracket.
Example 1
Expand 20(n  3).
20(n  3)  20  n  20  3
 20n  60
Example 2
Expand 3(2x  1).
Remember to multiply both terms inside the bracket by 20.
Write your answer in its simplest form.
Multiply both 2x and 1 by 3.
3(2x  1)  3  2x  3  1
 6x  3
Example 3
Expand p(p  q  5).
p(p  q  5)  p  p  p  q  p  5
 p²  pq  5p
Example 4
Expand 2x(3x  1).
2x(3x  1)  2x  3x  2x  1
 6x²  2x
Expand
a 2(x  3)
e 2(2x  y  3)
Multiply both terms by 2x.
For each term, negative  positive  negative.
Questions in this chapter are targeted at the grades indicated.
Exercise 9A
1
p  5 is usually written as 5p.
D
b 3(p  2)
f 5(2c  1)
c 4(m  n)
g 4(x²  2)
d 3(5  q)
h 3(n²  2n  1)
expand
133
Chapter 9 Expanding brackets and factorising
C
2
3
Expand
a y(y  2)
e a(b  c)
b g(g  3)
f s(3s  4)
c 2x(x  5)
g 3t(2t  1)
d n(4  n)
h 4x²(x  3)
Expand
a 2(m  3)
e 5(p  2)
b 3(2x  2)
f 3q(1  q)
c m(m  5)
g 2s(s  3)
d 4y(2y  3)
h 3n(4m  n  5)
Example 5
Expand and simplify 3(2a  1)  2(3a  5).
Expand each bracket separately.
3(2a  1)  2(3a  5)  6a  3  6a  10
 12a  13
Example 6
Collect like terms.
Expand and simplify 3x(y  2)  2y(x  3).
3x(y  2)  2y(x  3)  3xy  6x  2xy  6y
 xy  6x  6y
Example 7
Expand and simplify 6p  3p(2p  7)  4.
6p  3p(2p  7)  4  6p  6p²  21p  4
 6p²  15p  4
For the last term,
negative  negative  positive.
Watch Out!
You must multiply out the
brackets before you collect
like terms.
Check your signs.
Exercise 9B
C
1
2
3
4
134
Expand and simplify
a 3(t  1)  5t
d 3(d  2)  4(d  2)
b 6p  3(p  2)
e 3a  b  2(a  b)
c 6(w  1)  5w
f 2(5x  y)  5(y  x  1)
Expand and simplify
a 3(y 10)  2(y  5)
d q(q  3)  3(q  1)
b 6(2a  1)  3(a  4)
e 2n(n  2)  n(2n  1)
c x  5(x  3)
f 3m(2  5m)  4m(1  m)
Expand and simplify
a 5(t  4)  4(t  1)
d 6c(2c  3)  c(4  c)
b 3(x  3)  2(x  5)
e 4s(s  3)  2(1  s)
c 2g(g  1)  g(g  1)
f p(p  q)  q(p  q)
Expand and simplify
a 7s  4(s  1)
d 5n  n(n  1)
b 12m  3(m  2)
e 2x  x(x  y)
c 8f 2  3f( f  1)
f 7p  2p(1  p)
9.2 Factorising by taking out common factors
9.2 Factorising by taking out common factors
Objectives
You can take out common factors.
You can factorise expressions by taking out common factors.
Get Ready
1. What is the HCF (Highest Common Factor) of the following pairs?
a 6 and 8
b 10 and 25
c 8x and 12
d 9y and 15y
Key Points
Factorising is the opposite of expanding brackets, as you will need to put brackets in.
To factorise an expression, find a common factor of the terms, take this factor outside the brackets, then decide
what is needed inside the brackets.
You can check your answer by expanding the brackets.
Common factors are not always single terms such as 2, 5x, 3a²b.
Sometimes a common factor can have more than one term, for example x  2 or 2a  b.
Example 8
Factorise 12b  8.
12b  8  4(
)
 4(3b  2)
The common factor of 12b and 8 is 4.
Note that you would not usually write the 4(
) but it
is there to remind you to find the common factor first.
Check this multiplies out to give 12b  8.
Example 9
Factorise 2  6y.
2  6y  2(
)
 2(1  3y)
Example 10
Factorise x²  3x.
x²  3x  x(
)
 x(x  3)
Example 11
Pick out the common factor first.
1 is needed as the first term in the bracket.
The common factor of x² and 3x is x.
Remember to check by multiplying out.
Factorise 15p  10q  20pq.
15p  10q  20pq  5(
)
 5(3p  2q  4pq)
Find the common factor of all three terms.
factorising
135
Chapter 9 Expanding brackets and factorising
Example 12
Factorise completely 6a²b  9ab².
6a²b  9ab²  3ab(
)
 3ab(2a  3b)
The common factor of
6  a  a  b and 9  a  b  b
is 3  a  b.
Exercise 9C
D
C
1
2
Factorise
a 3x  6
e 8s  2t
i ac  c
m 4x²  3x
b
f
j
n
2y  2
9a  18b
6x²  9x  3
2h  5h²
c
g
k
o
5p  10q
15u 5v  10w
2p²  2p
p³  2p
d
h
l
p
14t  7
xt  yt
q²  q
s²  s3
Factorise completely
a 5xy  5xt
e 4pq  2ps  8pt
i 6f ²  2f 3
m 8pqr  10prs
b
f
j
n
3ad  6ac
mn  kmn
y4  y²
14a²b  7ab²  21ab
c
g
k
o
6pq  4hp
2x² 4x
3cd²  5c²d
15x²y  35x²y²
d
h
l
p
8xy  4y
12s²  24s
a³b  ab³
(3y)²  3y
Example 13
Factorise 5(x  2)²  3(x  2).
5(x  2)²  3(x  2) (x  2)[
]
 (x  2)[5(x  2)  3]
 (x  2)[5x  10  3]
 (x  2)(5x  7)
Example 14
(x  2) is a common factor.
Simplify the expression inside
the square bracket.
Factorise completely 12(s  2t)  4(s  2t)².
The common factor is 4(s  2t).
12(s  2t)  4(s  2t)²  4(s  2t)[
]
 4(s  2t)[3  (s  2t)]
 4(s  2t)(3  s  2t)
This cannot be simplified further.
Exercise 9D
B
1
2
136
Factorise
a (x  3)²  2(x  3)
b x(x  y)  y(x  y)
d (2t  s)(2t  s)  (2t  s) e (a  5)²  2(a  5)
c p(p  4)  3p
f (2d  1)²  (2d  1)
Factorise completely
a 2(y  2)²  4(y  2)
d 9(q  1)  6(q  1)²
c 8(p  5)²  10(p  5)
f 4x²(x  1)  6x(x  1)
b 15(x  1)² 10(x  1)
e 7(a  b)(a  b)  14(a  b)
9.3 Expanding the product of two brackets
9.3 Expanding the product of two brackets
Objective
You can multiply out the product of two brackets.
Get Ready
1. Work out the area of this
rectangle.
6 cm
2. Write down an expression, in terms of x,
for the area of this rectangle.
4 cm
x�2
x
Key Points
c�d
a�b
Area of rectangle P  (a  b)  (c  d)
 (a  b)(c  d)
P
c�d
a
Q
b
R
Area of rectangle Q  a(c  d)
Area of rectangle R  b(c  d)
Area of rectangle P  Area of rectangle Q  Area of rectangle R
(a  b)(c  d)
 a(c  d)  b(c  d)
 ac  ad  bc  bd
To multiply out the product of two brackets:
multiply each term in the first bracket by the second bracket
expand the brackets
simplify the resulting expression.
An alternative method is the grid method (see Example 15).
Example 15
Expand and simplify (x  2)(x  3).
Method 1
(x  2)(x  3)  x(x  3)  2(x  3)
 x2  3x  2x  6
 x2  5x  6
Take each term in the first bracket, in turn, and
multiply it by the second bracket.
Expand the brackets.
Collect the like terms.
grid method
137
Chapter 9 Expanding brackets and factorising
Method 2  the grid method
x
x
x
2
3
2
2x
Each term in the first bracket is multiplied
by each term in the second bracket.
3x
6
(x  2)(x  3)  x2  3x  2x  6
Add the four terms highlighted.
 x2  5x  6
Example 16
Collect like terms.
Expand and simplify (m  2)2.
(m  2)2  (m  2)(m  2)
 m(m  2)  2(m  2)
 m2  2m  2m  4
 m2  4m  4
Example 17
Write out (m  2)2
in full.
Watch Out!
Note that (a  b)2 is not equal to
a2  b2.
Expand and simplify (2t  1)(3t  2).
Method 1
(2t  1)(3t  2)  2t(3t  2)  1(3t  2)
 6t2  4t  3t  2
 6t2  7t  2
Check your signs are correct.
Method 2
2t
1
3t
2
6t2
4t
2
3t
(2t  1)(3t  2)  6t2  4t  3t  2
 6t2  7t  2
Exercise 9E
B
138
1
Expand and simplify
a (x  3)(x  4)
b (x  1)(x  2)
c (x  2)(x  5)
d (y  2)(y  3)
e (y  1)(y  2)
f (x  2)(x  3)
g (a  4)(a  5)
h (x  2)2
i (p  4)2
j (k  7)2
k (a  b)2
l (a  b)2
9.4 Factorising quadratic expressions
2
3
A
Expand and simplify
a (x  1)(2x  1)
d (y  3)(3y  1)
g (3s  2)(2s  5)
j (2a  1)(3a  2)
b
e
h
k
Expand and simplify
a (x  y)(x  2y)
d (x  y)(x  2y)
g (2a  3b)2
b (x  y)(x  2y)
e (2p  3q)(3p  q)
h (2a  3b)2
(x  1)(3x  1)
(2p  1)(p  3)
(2x  3)(2x  5)
(3x  2)2
c
f
i
l
(2x  3)(x  4)
(2t  1)(3t  2)
(3y  2)(4y  1)
(2k  1)2
c (x  y)(x  2y)
f (3s  2t)(2s  t)
9.4 Factorising quadratic expressions
Objectives
You can factorise quadratic expressions of the form x2  bx  c.
You can recognise and factorise the difference of two squares.
You can factorise quadratic expressions of the form ax2  bx  c.
Get Ready
1. Write down all possible pairs
of numbers whose product is
a 6
b 15.
2. Find a pair of numbers
whose product is 10 and
whose sum is 7.
3. Find a pair of numbers
whose product is 15 and
whose sum is 8.
Key Points
Factorising is the reverse process to expanding brackets so, for example, factorising
x2  5x  6 gives (x  2)(x  3).
To factorise the quadratic expression x2  bx  c
find two numbers whose product is c and whose sum is b
use these two numbers, p and q, to write down the factorised form (x  p)(x  q).
To factorise the quadratic expression ax2  bx  c
work out the value of ac
find a pair of numbers whose product is ac and sum is b
rewrite the x term in the expression using these two numbers
factorise the first two terms and the last two terms
pick out the common factor and write as the product of two brackets.
Any expression which may be written in the form a2  b2, known as the difference of two squares, can be
factorised using the result a2  b2  (a  b)(a  b).
139
Chapter 9 Expanding brackets and factorising
Example 18
Factorise x2  7x  12.
Examiner’s Tip
The pairs of numbers whose product is 12 are:
1  12
1  12
2  6
2  6
3  4
3  4
3  4  12
3  4  7
x2  7x  12  (x  3)(x  4)
Example 19
You may find it helpful to start
by writing down all the pairs of
numbers whose product is 12.
Find two numbers whose product
is 12 and whose sum is 7.
Put into factorised form using
the numbers 3 and 4.
Factorise x2  10x  25.
Examiner’s Tip
The pairs of numbers whose product is 25 are:
1   25
1  25
5   5
5  5
5  5  25
5  5  10
2
x  10x  25  (x  5)(x  5)
You can check your answer by
expanding the brackets.
This may also be written as (x  5)2.
Exercise 9F
A
1
Write down a pair of numbers:
a whose product is 15 and whose sum is 8
b whose product is 24 and whose sum is 10
c whose product is 18 and whose sum is 9
d whose product is 8 and whose sum is 2
e whose product is 8 and whose sum is 2
f whose product is 9 and whose sum is 0.
2
Factorise
a x2  8x  15
d x2  6x  9
g x2  3x  18
j x2  x  12
m x2  81
Example 20
b
e
h
k
x2  8x  7
x2  6x  5
x2  3x  18
x2  2x  24
c
f
i
l
x2  9x  20
x2  2x  1
x2  3x  28
x2  4
Factorise x2  n2.
x2  n2  x2  0x  n2
The pair of numbers whose product is n2 and whose sum is 0 is n  n:
x2  n2  (x  n)(x  n)
140
9.4 Factorising quadratic expressions
Example 21
x2  100
Example 22
Factorise x2  100.
Examiner’s Tip
Substitute
a  x and b  10 into
a2  b2  (a  b)(a  b).
 x2  102
 (x  10)(x  10)
It will help you in the
examination if you learn
a2  b2  (a  b)(a  b).
a Factorise p2  q2.
b Hence, without using a calculator, find the value of 1012  992.
a p2  q2  (p  q)(p  q)
b 1012  992
Use the result a2  b2  (a  b)(a  b).
Substitute p  101 and q  99 in the answer to part (a).
 (101  99)(101  99)
 200  2
Work out each bracket.
 400
Example 23
Factorise (x  y)2  4(x  y)2.
 (x  y)2  [2(x  y)]2
Write (x  y)2  4(x  y)2 in
the form a2  b2.
 [(x  y)  2(x  y)][(x  y)  2(x  y)]
 [x  y  2x  2y][x  y  2x  2y]
(x  y)2  4(x  y)2  (3x  y)(x  3y)
Substitute a  (x  y)
and b  2(x  y) into
a2  b2  (a  b)(a  b).
Expand and simplify
the expression in
each square bracket.
Note that alternatively this answer
may be written as (3x  y)(3y  x).
Exercise 9G
1
2
Factorise
a x2  36
d 25  y2
g (x  1)2  4
A
b x  49
e w2  2500
h 81  (9  y)2
2
Without using a calculator, find the value of:
a 642  362
b 7.52  2.52
c y  144
f 10 000  a2
i (a  b)2  (a  b)2
2
c 0.8752  0.1252
d 10052  9952
AO2
141
Chapter 9 Expanding brackets and factorising
A
Factorise these expressions, simplifying your answers where possible.
3
4
a 4x2  49
b 9y2  1
c 121t2  400
d 1  (q  2)2
e (2t  1)2  (2t  1)2
f (p  q  1)2  (p  q  1)2
g 100(p  _12 )2  4(q  _12 )2
h 25(s  t)2  25(s  t)2
Factorise completely
a 3x2  12
d 4p2  64q2
b 5y2  125
e 12a2  27b2
Example 24
c 10w2  1000
f 2(x  1)2  2(x  1)2
Factorise 3x 2  7x  4.
a  3, b  7, c  4
ac  12, b  7
Find two numbers whose product is 12 and whose sum is 7.
3  4  12
3  4  7
Replace 7x with 3x  4x.
3x2 7x  4  3x2  3x  4x  4
Factorise by grouping.
 3x(x  1)  4(x  1)
Pick out the common factor and write
as the product of two brackets.
 (x  1)(3x  4)
3x  7x  4  (x  1) (3x  4)
2
Exercise 9H
A
1
2
A
3
Factorise
a 5x2  16x  3
e 6x2  13x  6
i 8x2  2x  3
m 4y2  12y  5
b
f
j
n
2x2  11x  5
6x2  7x  1
2x2  7x  15
6y2  13y  2
c
g
k
o
3x2  4x  1
5x2  7x  2
7x2  19x  6
6y2  25y  25
d 8x2  6x  1
h 12x2  11x  2
l 3x2  10x  8
Factorise completely
a 6x2  14x  8
b 6y2  15y  6
c 5x2  5x  10
Factorise
a x2  xy  2y2
b 2x2  7xy  5y2
c 6x2  5xy  6y2
Chapter review
When there is a number outside a bracket there is a hidden multiplication sign.
In algebra, expand usually means multiply out.
To expand a bracket you multiply each term inside the bracket by the term outside the bracket.
142
Chapter review
Factorising is the opposite of expanding brackets, as you will need to put brackets in.
To factorise an expression, find the common factor of the terms, take this factor outside the brackets, decide
what is needed inside the brackets.
You can check your answer by expanding the brackets.
Common factors are not always single terms.
To multiply out the product of two brackets:
multiply each term in the first bracket by the second bracket
expand the brackets
simplify the resulting expression.
An alternative method is the grid method.
Factorising is the reverse process to expanding brackets.
To factorise the quadratic expression x2  bx  c
find two numbers whose product is c and whose sum is b
use these two numbers, p and q, to write down the factorised form (x  p)(x  q).
Any expression which may be written in the form a2  b2, known as the difference of two squares, can be
factorised using the result a2  b2  (a  b)(a  b).
Review exercise
1
C
Expand and simplify 2(x  4)  3(x  2)
Exam Question Report
87% of students answered this sort of question
well because they remembered all of the
necessary multiplications.
June 2009
2
a Factorise 5m  10
b Factorise y2  3y
Exam Question Report
50% of students answered this question poorly
because they did not put factors in the right place.
Nov 2008
3
Factorise a ax  by  bx  ay
4
Expand and simplify (x  4)(x  3)
5
Expand a (a  2)2
6
Expand and simplify
a (x  5)(x  10) b (y  9)²
f (x  4)(2x  3) g (3p  1)(2p  1)
b (c  3)2
b Factorise ac  bd  ad  bc
June 2009
c (d  1)2
d (x  y)2
B
c (x  4)(x  2)
d (x  2)(x  3)
h (2c  d)(2c  d) i (4y  1)²
e (t  1)(t  6)
143
Chapter 9 Expanding brackets and factorising
B
7
AO2
AO3
Factorise
a t²  11t  30
d y²  12y  36
b x²  14x  49
e x²  5x  4
8
a Factorise x2  8x  7
9
Jamie is planting flowers in a local park.
c p²  2p  15
f s²  64
b Express 187 as the product of 2 prime numbers.
R
R
R
R
R
R
Y
Y
Y
R
R
R
R
R
R
When he plants three yellow flowers, he surrounds them with twelve red flowers, as shown in the diagram.
a How many red flowers (R) does he plant with ten yellow flowers (Y)?
b Write your answer in both factorised and unfactorised forms.
A
10
AO2
AO3
AO3
AO3
A
AO3
Factorise
a x²  400
b 9t²  4
c 100  y²
d 25  4p²
11
Use some of your answers to question 7 to work out the values of the following expressions without
using a calculator.
a 21²  20²
b 10²  9.9²
c 5²  3²
12
Work out 10022  9982 using algebra.
13
For any three consecutive numbers show that the difference between the product of the first and
second and the product of the second and third is equal to double the second number.
14
Factorise fully 3(x  2)2  3x(x  2)
15
a In the group stage of the Champions League, four teams play each other both at home and away.
Prove that this requires 12 matches in total.
b Similarly, in the Premier League all the teams play each other twice.
There are 20 teams. How many games are there altogether?
c How many games are there in a league with a teams? Write your answer in a factorised and
unfactorised form.
16
Factorise
a 2x²  5x  2
d 30z²  23z  2
17
b 2w²  5w  3
e 8y²  23y  3
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
c 3a²  14a  8
f 6p²  pq  q²
3  8  2  9  6 and 12  17  11  18  6
Show that for any 2 by 2 square of numbers from the grid, the difference of the products of numbers
from opposite corners is always 6.
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