# Expand More Than One Binomial

In this worksheet, students will learn how to expand triple sets of brackets to reach the simplest possible expression.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra

Curriculum subtopic:   Notation, Vocabulary and Manipulation, Algebraic Expressions

Difficulty level:

### QUESTION 1 of 10

You may have already learnt how to expand single brackets such as 3a(2b + 7), and double brackets such as (x + 5)(x - 7).

If you are not feeling confident with either of these skills, you might want to revise these before trying this activity, as it builds on the same skills.

Expanding three (or more!) brackets is very similar, although it might look a little tricky at the start.

Let's have a look at an example now.

e.g. (x + 3)(x - 5)(x + 4)

We are not going to calculate this all at once.

Instead, we are just going to expand and simplify the first two brackets initially.

Use whatever method you are most comfortable with to do this, and then simplify your answer.

You might want to highlight the like terms to help you with simplifying:

(x + 3)(x - 5) = x2 + 3x - 5x - 15 = x2 - 2x - 15

We are now going to use this answer to multiply out with our third bracket:

(x2 - 2x - 15)(x + 4) = x3 + 4x2 - 2x2 - 8x -15x - 60 =  x3 + 2x2 - 23x - 60

Here we are then:

(x + 3)(x - 5)(x + 4) = x3 + 2x2 - 23x - 60

The key to success with these questions is to take our time to expand each set of brackets carefully.

In this activity, we are going to expand triple sets of brackets to reach the simplest possible expression using the method described above.

You may want to have a pen and paper handy to keep track of your working.

Don't rush it; it's easy to make a mistake with so many terms, different powers and signs to think about!

What is the correct method for expanding triple brackets?

Multiply everything by everything at once

Multiply out the first two brackets initially and then multiply your result by the third bracket

Expand and simplify:

(r + 2)(r + 1)(r + 4)

r³ + 8

r³ + 7r² + 14r + 8

30r³

r² + 3r + 2

Find an expression for a volume of this cuboid.

Record your answer by typing the correct numbers in the spaces below.

r³ + 8

r³ + 7r² + 14r + 8

30r³

r² + 3r + 2

Expand and simplify:

(2x - 5)(x + 1)(x - 2)

x³ - 6x² + 3x + 10

30x³

x³ - 7x² - 11x + 10

2x³ - 7x² - 11x + 10

Which of the options below shows the correct expansion of the expression below?

(x - 1)³

x³ - 1

x³ - 3x² + 3x - 1

2x³

x³ - 3x - 1

The cube below has a side length of (x - 4) cm.

Find an expression for the volume of this cube, giving your answer in a fully simplified form.

x³ - 64

4x² - 32x + 16

x² - 8x + 16

x³ - 12x² + 48x - 64

Match each set of triple brackets with its correctly expanded and fully simplified expression.

## Column B

(x + 5)(x + 1)(x + 4)
x³ + 10x² + 29x + 20
(x + 3)(x + 2)(x − 1)
x³ - 5x² - 2x + 24
(x − 4)(x + 2)(x − 3)
x³ + 4x² + x - 6
(x − 4)2(x − 3)
x³ - 11x² + 40x - 48

Expand and simplify:

(2x - 5)³

## Column B

(x + 5)(x + 1)(x + 4)
x³ + 10x² + 29x + 20
(x + 3)(x + 2)(x − 1)
x³ - 5x² - 2x + 24
(x − 4)(x + 2)(x − 3)
x³ + 4x² + x - 6
(x − 4)2(x − 3)
x³ - 11x² + 40x - 48

Expand:

3(z + 4)(z - 3)(z + 2)

## Column B

(x + 5)(x + 1)(x + 4)
x³ + 10x² + 29x + 20
(x + 3)(x + 2)(x − 1)
x³ - 5x² - 2x + 24
(x − 4)(x + 2)(x − 3)
x³ + 4x² + x - 6
(x − 4)2(x − 3)
x³ - 11x² + 40x - 48

Expand and simplify:

(2x – 1)(3x + 2)(5x – 6)

30x³ - 31x² - 16x + 12

-5³

30x³ + 71x² + 52x + 12

30x³ + 31x² + 16x + 12

• Question 1

What is the correct method for expanding triple brackets?

Multiply out the first two brackets initially and then multiply your result by the third bracket
EDDIE SAYS
This is so important to remember! Don't get yourself into a pickle by trying to do it all at once. Expand the first two brackets initially (this will give you marks in an exam already). Then use your result to multiply out the third bracket. Let's give this method a try now!
• Question 2

Expand and simplify:

(r + 2)(r + 1)(r + 4)

r³ + 7r² + 14r + 8
EDDIE SAYS
Did you follow our recipe for success? First multiply out the first two brackets: (r + 2)(r + 1) = r² + 3r + 2 Now multiply this answer by the third bracket: (r² + 3r + 2)(r + 4) = r³ + 7r² + 14r + 8
• Question 3

Find an expression for a volume of this cuboid.

Record your answer by typing the correct numbers in the spaces below.

EDDIE SAYS
Did you remember how to find the volume of a cuboid? We need to multiply: length × width × height. Here we don't have numbers for the dimensions of the cuboid, but rather algebraic expressions. Our method will be the same though. Our calculation would be: (2x + 1)(2x + 1)(x - 3) Now we need to expand and simplify as we did in the other exercises. If we do this, we reach an answer of: 4x³ - 8x² - 11x - 3 Did you fill those gaps accurately with the correct numbers and symbols?
• Question 4

Expand and simplify:

(2x - 5)(x + 1)(x - 2)

2x³ - 7x² - 11x + 10
EDDIE SAYS
Let's start with the first pair of brackets: (2x - 5)(x + 1) = 2x2 + 2x - 5x - 5 = 2x2 - 3x - 5 Now multiply this by the third bracket: (2x2 - 3x - 5)(x - 2) = 2x3 - 4x2 - 3x2 - 6x - 5x + 10 = 2x3 - 7x2 - 11x + 10 If you made a mistake, double check the signs. If you have forgot that: negative × negative = positive positive × negative = negative; your simplifying would be incorrect, causing mistakes further down your working.
• Question 5

Which of the options below shows the correct expansion of the expression below?

(x - 1)³

x³ - 3x² + 3x - 1
EDDIE SAYS
This may look like a single bracket, but in fact, it's three brackets that we need to expand! Can you see the power attached to the outside of the bracket? This expression is cubed, which means the question can be re-written as: (x - 1)(x - 1)(x - 1) When we expand the first two brackets, we reach: x2 - 2x + 1 If we multiply this by the final bracket, we get: (x2 - 2x + 1)(x - 1) = x3 - 3x2 +3x - 1
• Question 6

The cube below has a side length of (x - 4) cm.

Find an expression for the volume of this cube, giving your answer in a fully simplified form.

x³ - 12x² + 48x - 64
EDDIE SAYS
To find the volume of a cube, we need to cube the side length. This would give us: (x - 4)³ = (x - 4)(x - 4)(x - 4) If we expand the first pair of brackets, we reach: (x - 4)(x - 4) = x² - 8x + 16 Then let's multiply this by our final bracket: (x² - 8x + 16)(x - 4) = x³ - 12x² + 48x - 64
• Question 7

Match each set of triple brackets with its correctly expanded and fully simplified expression.

## Column B

(x + 5)(x + 1)(x + 4)
x³ + 10x² + 29x + 20
(x + 3)(x + 2)(x − 1)
x³ + 4x² + x - 6
(x − 4)(x + 2)(x − 3)
x³ - 5x² - 2x + 24
(x − 4)2(x − 3)
x³ - 11x² + 40x - 48
EDDIE SAYS
Always remember to multiply out the first two brackets first, and, only then, multiply your answer by the third bracket. A shortcut here is to consider the values of the numbers present (not x's) and to look for multiples which we know. For the first triple brackets, we have 5, 1 and 4. Multiplying these together gives us 20, which is the value of the constant in the matching expression. Similarly, in the second set of brackets: 3 × 2 × -1 = -6 There is -6 at the end of the simplified expression. Remember this shortcut in case it is helpful and saves time for you in the future.
• Question 8

Expand and simplify:

(2x - 5)³

EDDIE SAYS
Let's write out this question as three brackets: (2x - 5)(2x - 5)(2x - 5) Now expand the first two brackets: (2x - 5)(2x - 5) = 4x² - 20x + 25 And multiply our result by the remaining third bracket: (4x² - 20x + 25)(2x - 5) = 8x³ - 60x² + 150x - 125 Did you remember to write the missing numbers with the correct signs?
• Question 9

Expand:

3(z + 4)(z - 3)(z + 2)

EDDIE SAYS
The whole number '3' outside of the brackets might have thrown you off, but we can ignore it and just worry about it at the very end. First let's expand (z + 4)(z - 3)(z + 2) using the method we have used before: first two front brackets, then the bracket at the end. Did you get z³ + 3z² - 10z - 24? Now multiply every term by 3 to reach our final answer: 3z³ + 9z² - 30z - 72 Did you add the correct numbers into the gaps with the correct signs?
• Question 10

Expand and simplify:

(2x – 1)(3x + 2)(5x – 6)

30x³ - 31x² - 16x + 12
EDDIE SAYS
This is a tough question! Work through it very slowly to make sure you don't get signs confused. You need to be patient and careful. (2x – 1)(3x + 2) = 6x² + x - 2 (6x² + x - 2)(5x - 6) = 30x³ - 31x² - 16x + 12 Great work - you are becoming a brackets expert!
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