KS3 Maths · Year 8 · Chapter 8: Algebra · Lesson 2

Expanding expressions with more than two brackets

Year 8About 75 minutesNo calculator55 marks

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Date

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Quick questions to start.

A1
Last lesson
Expand and simplify $(x - 7) (x + 8)$ .
[1 mark]
Answer
A2
Needed today
Expand $3 x (x^{2} - 2)$ .
[1 mark]
Answer
A3
Needed today
Simplify $5 x^{2} - 2 x + 3 x^{2} + 7 x$ .
[1 mark]
Answer
A4
Year 7
Work out $(- 3)^{3}$ .
[1 mark]
Answer

Example, then your turn

Year 8

Your teacher works through each example with you. Then try the one beside it.

Example 1

Expand and simplify $(x^{2} + 3 x - 5) (x + 2)$ .

Not drawn accurately

Hint Put the three terms along the top of a grid and

x

and

2

down the side. Fill all six boxes, then collect like terms.

B1
Expand and simplify $(x^{2} + 4 x - 3) (x - 5)$ .
[2 marks]
Answer

Example 2

Expand and simplify $(x + 1) (x + 4) (x - 3)$ .

Hint Expand two of the brackets first. You can choose any two. Then multiply that answer by the third bracket.

B2
Expand and simplify $(x + 2) (x - 5) (x + 6)$ .
[3 marks]
Answer

Example 3

Expand and simplify $(x + 3)^{3}$ . Explain why the answer is not $x^{3} + 27$ .

Hint

(x + 3)^{3}

means

(x + 3) (x + 3) (x + 3)

. Square the bracket first.

B3
Expand and simplify $(x - 4)^{3}$ .
[3 marks]
Answer

Example 4

Expand and simplify $(2 x + 1) (x - 3) (x + 2)$ .

Hint

(2 x + 1) (x - 3) = 2 x^{2} - 5 x - 3

. Then multiply each term by

x

and by

2

B4
Expand and simplify $(3 x - 2) (x + 1) (x + 4)$ .
[3 marks]
Answer

Example 5

Expand and simplify $(x + 5)^{2} (x - 2)$ .

Hint Expand

(x + 5)^{2}

first. Then multiply by

(x - 2)

B5
Expand and simplify $(x - 3)^{2} (x + 7)$ .
[3 marks]
Answer

Practice

Year 8

C1
Write an expression for the volume of this cuboid. Expand the brackets and simplify.
[3 marks]
Not drawn accurately
Answer
C2
$(x + 6) (x - 1) (x - 4)$ can be written as $x^{3} + a x^{2} + b x + c$ . Work out the values of $a$ , $b$ and $c$ .
[3 marks]
Answer $a = \dots\dots$ $b = \dots\dots$ $c = \dots\dots$

C3
Expand and simplify $(x + 2) (x - 2) (x + 7)$ .
[2 marks]
Answer
C4
Expand and simplify $(x + 2)^{3} - x^{3}$ .
[3 marks]
Answer
C5
Work out the missing number. $(x + 4) (x - 2) (x + □)$ is the same as $x^{3} + 5 x^{2} - 2 x - 24$ .
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Omar says: " $(x + 5)^{3} = x^{3} + 125$ ."
Omar is wrong. Show that he is wrong by expanding $(x + 5)^{3}$ correctly.
[3 marks]
Answer $(x + 5)^{3} = \dots\dots$

D2
A cuboid has a square base with sides $(x + 4)$ cm. Its height is $(x - 2)$ cm.
Not drawn accurately
(a)
Show that the volume of the cuboid is $(x^{3} + 6 x^{2} - 32)$ cm³.
[3 marks]
(b)
Work out the volume of the cuboid when $x = 6$ .
[2 marks]
Answer
Total for D2: 5 marks
D3
Solve $(x + 4) (x + 5) (x - 1) = x^{3} + 8 x^{2} + 7 x + 4$ .
[4 marks]
Answer
D4
A cube has edges of length $(x + 4)$ cm. A cuboid measures $x$ cm by $(x + 5)$ cm by $(x + 7)$ cm. Show that the volume of the cube is $(13 x + 64)$ cm³ more than the volume of the cuboid.
[5 marks]
Not drawn accurately
D5
Here is an expression: $(x + 2) (x + 3) (x + 4)$ .
(a)
Expand and simplify $(x + 2) (x + 3) (x + 4)$ .
[2 marks]
Answer
(b)
Use your answer with $x = 10$ to work out $12 \times 13 \times 14$ . Show your working.
[1 mark]
Answer
Total for D5: 3 marks

Extension

Stretch

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

E1
$(x + a) (x + b) (x + c) = x^{3} + 11 x^{2} + 34 x + 24$ , where $a$ , $b$ and $c$ are positive whole numbers. Work out $a$ , $b$ and $c$ .
[4 marks]
Hint 1 · What to try
Expand $(x + a) (x + b) (x + c)$ with letters. The number term is $ab c$ and the $x^{2}$ term is $(a + b + c) x^{2}$ .
Hint 2 · The first line
So $ab c = 24$ and $a + b + c = 11$ . List three whole numbers that multiply to $24$ .
Hint 3 · The full method
Check your three numbers by expanding the brackets. The $x$ term must be $34 x$ .
Answer $a = \dots\dots$ $b = \dots\dots$ $c = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can multiply a bracket with three terms by a bracket with two terms.
I can expand three brackets by expanding two and then multiplying by the third.
I can cube a bracket without leaving out the middle terms.
I can use expanding to compare volumes and solve equations.

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