KS3 Maths · Year 8 · Chapter 2: Equations and formulae · Lesson 2

Factorising algebraic expressions

Year 8About 70 minutesNo calculator49 marks

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Date

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Quick questions to start.

A1
Needed today
Work out the highest common factor of $18$ and $24$ .
[1 mark]
Answer
A2
Needed today
Simplify $x^{3} \div x$ .
[1 mark]
Answer
A3
Last lesson
Expand $4 y (3 y - 2)$ .
[1 mark]
Answer
A4
Year 7
Factorise $14 x + 21$ .
[1 mark]
Answer

Example, then your turn

Year 8

Factorising is expanding backwards. Find the highest common factor (HCF) of all the terms: the biggest number, and as many of the letter as every term has. Put it outside the bracket. Check by expanding. Your teacher works through each example with you. Then try the one beside it.

Example 1

Factorise $15 a^{2} + 20 a - 35$ as much as possible.

Hint The HCF of

15

20

and

35

5

. The last term has no

a

, so

a

is not common.

B1
Factorise as much as possible (i) $24 t - 16$ (ii) $18 y^{2} + 27 y - 36$ .
[2 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$

Example 2

Factorise $14 y^{2} - 21 y$ as much as possible.

Hint

7

is the HCF of

14

and

21

, and both terms have at least one

y

B2
Factorise as much as possible (i) $12 x^{2} + 18 x$ (ii) $20 p - 15 p^{2}$ .
[2 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$

Example 3

Factorise $8 m^{3} + 12 m^{2}$ as much as possible.

Hint Both terms have at least

m^{2}

m^{3} = m^{2} \times m

B3
Factorise as much as possible (i) $15 k^{3} - 10 k^{2}$ (ii) $6 d^{2} + 18 d^{3}$ .
[2 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$

Example 4

Factorise $10 y^{3} - 15 y^{2} + 25 y$ as much as possible.

Hint The HCF of the numbers is

5

, and every term has at least one

y

B4
Factorise $12 x^{3} + 8 x^{2} - 20 x$ as much as possible.
[2 marks]
Answer

Example 5

Expand $2 x^{2} (x + 3) + 2 x (x^{2} - 5)$ , collect like terms and factorise as much as possible.

Hint Expand both brackets first. Then collect the

x^{3}

terms.

B5
Expand $3 y (y^{2} + 2) + y^{2} (3 y - 9)$ , collect like terms and factorise as much as possible.
[3 marks]
Answer

Practice

Year 8

C1
Factorise as much as possible (i) $6 ab + 9 a$ (ii) $8 x y - 12 x^{2}$ .
[2 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$
C2
Fill in the boxes: $18 x^{2} - 30 x = 6 x (□ - □)$ .
[2 marks]
Answer $6 x (\dots\dots - \dots\dots)$

C3
Use factorising to work out $37 \times 14 + 37 \times 6$ . Show your working.
[2 marks]
Answer
C4
A rectangle has area $(6 x^{2} + 15 x)$ cm². One side is $3 x$ cm. Find an expression for the other side.
[2 marks]
Not drawn accurately
Answer
C5
Factorise $2 x^{4} - 10 x^{2}$ as much as possible.
[2 marks]
Answer
C6
Factorise $24 m^{2} n - 16 m n^{2}$ as much as possible.
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Leo factorises $18 x^{2} - 12 x$ and gets $3 x (6 x - 4)$ . He says, "That is factorised as much as possible."
Leo is wrong. Explain why, and factorise the expression as much as possible.
[3 marks]
AnswerCorrect answer $\dots\dots\dots$

D2
The sides of a triangle are $(4 x + 6)$ cm, $(2 x + 9)$ cm and $(6 x - 3)$ cm.
Not drawn accurately
(a)
Write an expression for the perimeter of the triangle. Factorise it as much as possible.
[3 marks]
Answer
(b)
The perimeter is $60$ cm. Work out the value of $x$ .
[2 marks]
Answer
Total for D2: 5 marks
D3
$2 n$ , $2 n + 2$ and $2 n + 4$ are three consecutive even numbers. Show that their sum is always a multiple of $6$ .
[3 marks]
D4
A rectangle has area $(8 x^{2} + 12 x)$ cm². One side is $4 x$ cm.
Not drawn accurately
(a)
Find an expression for the other side.
[2 marks]
Answer
(b)
Write an expression for the perimeter of the rectangle. Factorise it as much as possible.
[2 marks]
Answer
Total for D4: 4 marks
D5
Expand and simplify $4 x (x^{2} - 3) - 2 x^{2} (x - 5)$ . Factorise your answer as much as possible.
[3 marks]
Answer

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Ravi says, "If you add any five consecutive whole numbers, you always get a multiple of $10$ ." Show that Ravi is wrong. Then prove that the sum is always a multiple of $5$ , and say which of the five numbers the sum is $5$ times.
[4 marks]
Hint 1 · What to try
Try $1 + 2 + 3 + 4 + 5$ .
Hint 2 · The first line
Call the numbers $n$ , $n + 1$ , $n + 2$ , $n + 3$ and $n + 4$ . Add them and collect like terms.
Hint 3 · The full method
$5 n + 10 = 5 (n + 2)$ . Which of the five numbers is $n + 2$ ?

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the highest common factor of the terms in an expression.
I can factorise with a number, a letter or a power outside the bracket.
I can expand, collect and then factorise as much as possible.
I can use factorising to find a missing side and to show a number fact.

Answers and mark scheme

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