Worksheet 4.0 (page 1 of 4)
Factorising linear expressions
The number 8 can be rewritten (or factorised) as 8 = 2 × 4. The numbers 2 and 4
are referred to as factors of 8, as they divide into 8 exactly.
Similarly, when a linear expression, e.g. 4x + 10 is factorised as
4x + 10 = 2(2x + 5), the terms 2 and (2x + 5) are factors of the expanded
expression 4x + 10.
Factorising linear expressions is exactly the reverse process to expanding them.
The algebra blocks need to be arranged into a rectangle. The width and length of
the rectangle formed represent the factors of the expanded expression.
Example 1.
Factorise 2x + 6
Two solutions are possible:
The factors are 1 and (2x + 6), as the width is 1 and the length is (2x + 6).
2x + 6 = 1(2x + 6)
This solution is a trivial solution as it is always possible. This is the same as
factorising 8 as 1 × 8. For this reason this solution will be left out in future
examples.
The second solution:
The factors are 2 and (x + 3), as the width is 2 and the length (x + 3).
2x + 6 = 2(x + 3)
or
2x + 6 is 2 lots of (x + 3)
CSF Sample Units: Years 8 to 10 Maths – Working with linear expressions
Worksheet 4.0 (page 2 of 4)
Factorise 4x – 6
Example 2.
First attempt:
This diagram does not have the pieces
arranged as a rectangle, therefore this is
not a solution. This is the diagram of
4x – 6 = 4(x – 1) – 2
Trying a different arrangement,
The factors are 2 and (2x – 3).
Therefore, 4x – 6 = 2(2x – 3)
1. Use algebra blocks to factorise the following expressions.
a) 2x + 8
f) 5x – 10
b) 4x + 12
g) 4x – 4
c) 3x + 9
h) 4x – 10
d) 2x – 10
i) 6x + 9
e) 3x – 12
j) 4x – 14
CSF Sample Units: Years 8 to 10 Maths – Working with linear expressions
Worksheet 4.0 (page 3 of 4)
2. Use algebra blocks to factorise the following expressions.
a) –2x – 4
f) –4x + 6
b) –4x – 12
g) –2x – 10
c) –3x – 6
h) –3x – 15
d) –3x + 6
i) –4x + 10
e) –5x + 15
j) –4x + 16
Example 3.
The following expressions are all equal.
They are referred to as equivalent expressions. Not all equivalent expressions
below represent factors of 4x – 6.
2(x – 3) is the only equivalent expression which represents factors of
4x – 6, the other expressions do not form complete rectangles.
4x – 6
=
2(2x – 3)
=
4(x – 1) – 2
=
4(x – 2) + 2
CSF Sample Units: Years 8 to 10 Maths – Working with linear expressions
Worksheet 4.0 (page 4 of 4)
3. Use your algebra blocks to find three equivalent expressions for each of the
expressions listed below. At least one of these equivalent expressions must
be a factor of the initial expression. Write the equivalent expressions
algebraically and circle those expressions which also represent factors.
a) 6x + 12
b) 4x – 4
c) 3x – 6
d) 9x – 6
e) –4x +
CSF Sample Units: Years 8 to 10 Maths – Working with linear expressions
CSF Sample Units: Years 8 to 10 Maths – Working with linear expressions