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Expand More Than One Binomial

In this worksheet, students will learn how to expand more than one binomial.

'Expand More Than One Binomial' worksheet

Key stage:  KS 4

Curriculum topic:  

Curriculum subtopic:  

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

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

Expanding three (or more!) brackets if very similar, although it might look a little tricky at the start. Let's have a look at an example first.

 

(x + 3)(x - 5)(x + 4)

We are not going to do it all at once. Instead, we are just going to expand and simplify the first two brackets.

Use whatever method you are most comfortable with. 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 and multiply it out with the 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 here is to take time to expand each set of brackets.

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 first and then multiply the result by the third bracket

Expand and simplify

(r + 2)(r + 1)(r + 4)

r³ + 8

r³ + 7r² + 14r + 8

30r³

r² + 3r + 2

Write an expression for a volume of this cuboid.

Expand and simplify

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

x³ - 6x² + 3x + 10

30x³

x³ - 7x² + x + 10

2x³ - 7x² + x + 10

Which of the answers below shows the correct expansion of (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 a correct expanded and fully simplified answer.

Column A

Column B

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

Expand and simplify (2x - 5)³

Remember to include appropriate signs within your answer.

Column A

Column B

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

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

 

Remember to include appropriate signs with your answers.

Column A

Column B

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

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?

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

Expand and simplify

(r + 2)(r + 1)(r + 4)

CORRECT ANSWER
r³ + 7r² + 14r + 8
EDDIE SAYS
Did you follow the recipe for success? First multiply out (r + 2)(r + 1). This gives us r² + 3r + 2. Now multply this answer by the third bracket: (r² + 3r + 2)(r + 4) = r³ + 7r² + 14r + 8
  • Question 3

Write an expression for a volume of this cuboid.

CORRECT ANSWER
4 x³ - 8 x² - 11 x - 3
EDDIE SAYS
Do you remember how to find the volume of a cuboid? You need to multiply lenght × width × height. Here we don't have numbers for the dimensions of the cuboid, but rather algebraic expressions, but the method is the same! So our calculation would be: (2x + 1)(2x + 1)(x - 3) Now expand and simplify as you did for other exercises, and the answer will be 4 x³ - 8 x² - 11 x - 3.
  • Question 4

Expand and simplify

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

CORRECT ANSWER
2x³ - 7x² + x + 10
EDDIE SAYS
2x³ - 7x² + x + 10 is the correct answer. When you multiply out (2x - 5)(x + 1) correctly, you should get 2x&up2; - 3x - 5. Now multiply this by the third bracket. (2x&up2; - 3x - 5)(x - 2) = 2x&up2; - 3x - 5 If you made a mistake, double check the signs. If you have forgotten that negative × negative gives a positive and that a positive × negative gives a negative, your simplifying would be incorrect, causing mistakes further down in your working.
  • Question 5

Which of the answers below shows the correct expansion of (x - 1)³ ?

CORRECT ANSWER
x³ - 3x² + 3x - 1
EDDIE SAYS
This looks like a single bracket, but in fact, it's three brackets that you 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 you expand these triple brackets, you get the required answer of 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.

CORRECT ANSWER
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. The expression for the volume can be found by expanding triple brackets. (x - 4)(x - 4)(x - 4) = (x² - 8x + 16)(x - 4) = x³ - 12x² + 48x - 64
  • Question 7

Match each set of triple brackets with a correct expanded and fully simplified answer.

CORRECT ANSWER

Column A

Column B

(x+5)(x+1)(x+4)
x³ + 10x² + 29 x + 20
(x+3)(x+2)(x−1)
x³ + 4x² + x - 6
(x−4)(x+2)(x−3)
x³ - 5x² - 2 x + 24
(x−4)²(x−3)
x³ - 11x² + 40 x - 48
EDDIE SAYS
Always remember to multiply out the first two brackets first, and only then multiply out your answer by the third bracket. Another way of looking at this question is to look at the numbers (not x's) in the brackets. 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 answer. Similarly, in the second set of brackets, 3 × 2 × -1 = -6 and there is -6 at the end of the simplified expression.
  • Question 8

Expand and simplify (2x - 5)³

Remember to include appropriate signs within your answer.

CORRECT ANSWER
EDDIE SAYS
Write out this questions as three brackets (2x - 5)(2x - 5)(2x - 5) Now expand the first two brackets: (2x - 5)(2x - 5) = 4x² - 20x + 25 And multiply your result out by the third bracket: (4x² - 20x + 25)(2x - 5) = 8x³ - 60x² + 150x - 125
  • Question 9

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

 

Remember to include appropriate signs with your answers.

CORRECT ANSWER
EDDIE SAYS
The 3 outside of the brackets might have thrown you off, but you can ignore it for the majority of your workings and worry about it at the very end. Expand (z + 4)(z - 3)(z + 2) using the method you have done before: first two front brackets, then the bracket at the end. Did you get z³ + 3z² - 10z - 24 ? Now multiply every term by 3. 3z³ + 9 z² - 30z - 72
  • Question 10

Expand and simplify (2x – 1)(3x + 2)(5x – 6).

CORRECT ANSWER
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
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