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2 Manipulating algebra
This unit will help you to manipulate algebraic expressions.
AO1 Fluency check
1 Simplify
a x2 × x3 
2 Expand
​x​ 5​
 ​ 
b ___
​  3 
​x​  ​
c2a + 3b − 2b2 + 4a 
b x(x2 + 3x) 
a2x(4x − 1) 
d−3(x2 + 4) 
c(x + 1)(x - 4) 
e(x + 5)2 
f(x + 2)(x - 2) 
b x2 − 64 
c x2 + 5x + 6 
3
2
b __
​   ​× __
​   ​ 
7
5
3 Factorise
a x2 − 9 
4 Work out
2
1
 ​   ​ 
a ​ __ ​+ __
5
2
3
1
c __
​   ​÷ __
 ​   ​ 
4
8
Number sense
5
Find factors of 24 that sum to
a25 
b11 
c−10 
Key points
To simplify expressions, expand any brackets
and collect like terms.
To expand an expression like (2x + 3)(3x + 1),
multiply both terms in the second bracket by
both terms in the first bracket.
These
1
skills boosts will help you to manipulate algebraic expressions.
Expanding
double
brackets
2
Factorising
quadratic
expressions
of the form
ax 2 + bx + c
3
Simplifying
expressions with
brackets and
powers
4
Simplifying
expressions
involving
algebraic
fractions
You might have already done some work on manipulating algebraic expressions. Before starting the first
skills boost, rate your confidence with these questions.
1
Expand
(2x + 3)(3x + 1)
2
Factorise
4x 2 + 8x – 5
3
4
Simplify
x(x 2 + 5x) – 2x 2 + 3
x
x
Simplify __
​    ​+ __
​    ​
4
3
How
confident
are you?
Unit 2 Manipulating algebra     9
Skills boost
1
Expanding double brackets
To expand double brackets, split them into two expansions.
For example, (2x + 3)(3x + 2) = 2x(3x + 2) + 3(3x + 2)
Guided practice
Worked
exam
question
Expand (2x + 3)(3x + 2)
3x + 2
3x
Split (2x + 3)(3x + 2)
into= 2x(3x + 2) + 3(3x + 2)
Expand the brackets.
2x(3x + 2)
= 6x2 +
+ 9x +
Collect like terms.
6x2
4x
2x + 3
+
3(3x + 2)
= 6x2 + 13x + 6
2x
2
9x
3
6
1 Expand
a(3x + 4)(2x + 1)
b(2x + 5)(4x − 2)
c(3x − 2)(4x + 1)
2 Expand
a(5x + 3)(2x − 1)
b(4x − 5)(3x − 1)
c(5x − 4)(2x − 3)
3 Expand
a(5x + 3)(5x − 3)
b(4x + 7)(4x − 7)
c(3x − 1)(3x + 1)
4 Show that the area of the square is 16x2 − 24x + 9
4x – 3
Exam-style question
5 Expand and simplify (2x + 3)2   
  (2 marks)
Reflect
Why do you think the answers in Q3 are called ‘the difference of two squares’?
10     Unit 2 Manipulating algebra
Skills boost
2
Factorising quadratic expressions of the form ax 2
+ bx + c
A quadratic expression, like 2x2 + 7x + 3, may factorise into two brackets.
Guided practice
Worked
exam
question
Factorise 2x2 + 9x + 10
The expression is of the form ax2 + bx + c
Write down the values of a, b and c
2 × 10 = 20
2x2 + 9x + 10
2x2  +  9x  +  10
a = 2   b = 9   c = 10
4+5
Work out ac.
ac = 2 × 10 = 20
Writing the x terms in the reverse order gives
the same result.
2x2 + 5x + 4x + 10
= x(2x + 5) + 2(2x + 5)
= (x + 2)(2x + 5)
⎫
⎪
⎬
⎪
⎭
⎫
⎪
⎪
⎬
⎪
⎪
⎭
Find the factors of ac that sum to b.
Split bx.
4and5
↓↓
2x2 + 4x + 5x + 10
Factorise pairs of terms.
) + 5(x +
)
= 2x(x +
= (2x + 5)(x + 2)
1 Factorise
a3x2 + 14x + 8  b2x2 + 7x + 6 
2 Factorise
c10x2 + 9x + 2 
Hint ac = −12
−6 + 2 = −4
b3x2 + 13x – 10 
⎫
⎪
⎪
⎬
⎪
⎪
⎭
a2x2 − 4x − 6
2x2 − 6x +
−6
3 Factorise
a3x2 − 14x + 8  b4x2 − 16x + 15  4 Factorise
Hint 9x2 – 4 is the difference
a9x2 − 4  of two squares.
(3x −
c8x2 − 10x + 3 
)(3x +
b144x2 − 49 
)
Exam-style question
5 Factorise 6x2 + 17x + 5
  (2 marks)
Reflect
How can you tell that a factorisation will have negative numbers in it?
Unit 2 Manipulating algebra     11
Skills boost
3
Simplifying expressions with brackets and powers
To expand a bracket, multiply every term inside the bracket by the term outside the bracket.
Guided practice
Worked
exam
question
Expand and simplify x(x2 + 2x + 5) − 3x + 1
Expand the bracket.
x(x2 + 2x + 5) − 3x + 1
= x3 + 2x2 +
Collect like terms.
− 3x + 1
Multiply x(x2 +2x +5)
= x × x2 + x × 2x + x × 5
= x3 +2x2 +5x
= x3 + 2x2 + 2x + 1
1 Expand and simplify
a4(a + 3b) + 2(a + b)
c3(d − 2e) − (e + 3d)
b5(x + 3y) + 4(2x − y)
d6(z − 2t) − 2(z − 3t)
2 Expand
a x(x2 + 3x + 4)
b x(x2 − 2x + 1)
c a2(a2 + 2a − 3)
d y3(4 − y2 + 2y)
3 Expand and simplify
a x(x2 + 2x − 1) + 4x c x(x2 − 3x + 5) + 4(x2 − 3)
b m(m2 − 3m + 2) + 5m – 7
d y(y2 + 5y − 2) − 7(y2 − 2y + 3)
4 Expand and simplify
a(x + 3)2 − 2x
b(x − 4)2 + 3x − 2
Hint Square the
c(2a + 3)2 + 4a
d(3p + 1)2 − (2p + 3)
12     Unit 2 Manipulating algebra
bracket first.
Skills boost
5 Expand
a x(x + 1)(x + 2)
b y(y − 1)(y + 3)
Hint Expand the
double brackets
first.
6 Expand and simplify
a y(y + 3)2 b x(x − 2)2
c a(a + 4)2 − 3a d x(x + 1)2 − (2x + 3)
7 Expand and simplify
Hint (x + 1)(x2 + 2x + 3) = x(x2 + 2x + 3) + 1(x2 + 2x + 3)
a(x + 1)(x2 + 2x + 3)
b(x − 2)(x2 + x − 1) c(x + 4)(x2 − 3x + 2)
8 Expand and simplify
Hint (x + 1)(x + 4)(x − 2)
= (x + 1)(x2 −
+
)
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
a(x + 1)(x + 4)(x − 2)
Expand these first.
b(x + 2)(x − 3)(x + 1) c(y − 1)(y + 2)(y + 3)
Exam-style question
9 Expand and simplify (x − 1)(x + 3)2
  (3 marks)
Reflect
In Q8, does it matter which pair of brackets you expand first?
Unit 2 Manipulating algebra     13
Skills boost
4
Simplifying expressions involving algebraic fractions
To simplify expressions with algebraic fractions:
• factorise the numerator and the denominator, if possible
• cancel common factors.
Guided practice
3x + 3
Simplify fully _____________
  
​   
 ​
2
x + 5x + 4
Factorise the numerator and the denominator.
+
)
3(
3x + 3
____________
_______________
​   
   ​= ​   
  
 ​
x2 + 5x + 4
(x + 4)(x + 1)
x2 + 5x + 4 = (x + 4)(x + 1)
Cancel common factors.
(x + 1)
_______
3(x + 1)
= _____________
​   
   ​
(x + 4)(x + 1)
​ 
 
 ​= 1
(x + 1)
3
= ______
​      
​
x+4
1 Simplify by cancelling the common factors.
6 ​x​ 2​ y
a ______
​ 
  
​  
3x ​y​ 2​
2​x​ 3​
b ____
​     
 
​×
y
Hint
2 Simplify
x
x
 ​   ​ 
a __
​    ​+ __
2
6
x
___
2
 
 ​    
​ 
4
8
c ​ __
x  ​÷
4y
 ​___
​  
x ​ 
×3
x
__
3x
​    ​= ___
​   ​ 
​ 
2
6
xy ​ ​ 
____
x
​   ​ 
 + ​ __  ​= 
6
6
3x
___
×3
x
__
b ​     
​+  ​   ​ 
5
10
3x
___
x
__
c ​   ​ 
 +  ​   ​ 
4
3
Hint
_____
 
 
​     
​+ _____
​     
​=
12
12
3 Simplify
x+5
x−3
 
 
​ 
b ____
​     
​+ _____
 ​    
4
8
x + 6 __
x
2(x + 6 )
 
​− ​    ​_____
a _____
​     
x+6
_______
2
4
   
​  =  ​ 
   
​ 
Hint ​ 
 
2
4
2(x + 6 ) __
x
Expand, then collect like terms.
= _______
​ 
   
​ 
− ​    ​ 
4
4
4 Simplify fully
​x​ 2​+ 2x
   
​  
a ________
​ 
x
Hint Factorise
then cancel.
x+3
b ______
​  2
 
  
​ 
​x​  ​− 9
2(x + 4 )
c ​ __________
  
  
 ​ 
2
​x​  ​+ 5x + 4
Exam-style question
​x​ 2​+ x − 2
__________
​   
 ​
5 Simplify fully   
2
​x​  ​− 6x + 5
  (3 marks)
Reflect
How have you used common factors and common multiples in these questions?
14     Unit 2 Manipulating algebra
Get back
on track
Practise the methods
Answer this question to check where to start.
Check up
Tick the correct expansion of (2x + 1)2
A
B
4x2
4x2
+1
C
+ 4x + 1
If you ticked B go to Q2.
4x2 + 2x + 1
If you ticked A or C go to Q1 for more
practice.
1 Expand and simplify
a(2x + 3)(2x + 1)
b(2x + 5)(2x + 5)
c(2x + 4)2
2 Expand and simplify
a6(a − 2b) − 3(a + b)
b4x(x − 3) + 2(x − 5)
3 Expand
a x(x2 + 3x − 4)
by(y2 − 2y + 5)
4 Expand
a x(x + 1)(x + 5)
bx(x − 3)(x + 2)
5 Expand and simplify
a(x − 2)(x − 3)(x + 4)
b(x − 5)(x + 1)2
6 Simplify
m m
a __
​   ​  −  ​__ ​ ​  
5
3
2x
x
b ​ __  ​+ ___
 ​  ​ 
​  
6
9
5x
2x
c ​ ___ ​ 
 +  ​___ ​ 
​  
7
3
7 Simplify fully
​x​ 2​+ 5x
a ________
​ 
   
​  
2x
3x + 6
b​​ ___________
  
 
 ​ 
2
​x​  ​− x − 6
Exam-style question
8 Simplify
4 ​x​ 2​
 ÷
a ____
​   ​ 
3y
2x
  
​ 
 ​___
​y​ 3​
(2n + 1 ​)​ 2​
  
 ​
b ______________
​   
4 ​n​ 2​+ 8n + 3
  (2 marks)
  (2 marks)
Unit 2 Manipulating algebra     15
Get back
on track
Problem-solve!
1 Show that the area of the rectangle is 6x2 + x − 15
3x + 5
2x – 3
 
Exam-style questions
2 Show that (2x + 1)2 − 2(x + 1)2 ​≡​2x2 − 1
   (3 marks)
3 Expand and simplify (x + 2)3    (3 marks)
4 Expand
a(x − 1)(x + 1)(x − 1)
b(x − 3)(x2 − 25)
5 Find an expression for the shaded area.
Simplify your answer as much as possible.
x
2x + 5
x+4
 
3x – 1
6 Write an expression for the volume of the cuboid.
x–1
Simplify your answer as much as possible.
x+1
x+2
 
Exam-style questions
​x​ 2​− 4
________
 
  
​
2
7 Simplify fully ​ 
(x + 2​)​  ​
   (3 marks)
8 a Factorise 2x2 + 7x + 3
  (2 marks)
2 ​x​ 2​+ 7x + 3
  
  
bSimplify fully ______________
​ 
 
​
​x​ 2​− 9
  (2 marks)
Now that you have completed this unit, how confident do you feel?
1
Expanding
double
brackets
2
Factorising
quadratic
expressions
of the form
ax 2 + bx + c
16     Unit 2 Manipulating algebra
3
Simplifying
expressions with
brackets and
powers
4
Simplifying
expressions
involving
algebraic
fractions