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Grade 6 · Higher · free GCSE maths worksheet

Expanding three brackets

Edexcel, AQA, OCR and Edexcel IGCSE

Multiplying two brackets first, then the third.

10 original questions, 31 marks. Type your answers below for an instant mark and a score, or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

How to do it

  1. Expand the first two brackets and simplify: (x + 3)(x - 2) = x² + x - 6
  2. Multiply every term of that quadratic by each term of the third bracket
  3. Write out all six terms before you collect anything
  4. Collect like terms and write the answer in descending powers: x³, x², x, number

Watch out: Each of the three terms in the quadratic must multiply both terms of the last bracket, so you should have six terms before collecting, not four.

Worked example

Expand and simplify (x + 3)(x − 2)(x + 5)

Expand the first two brackets

(x + 3)(x - 2)=x2 - 2x + 3x - 6
=x2 + x - 6

Multiply every term by (x + 5)

(x2 + x - 6)(x + 5)=x3 + 5x2 + x2 + 5x - 6x - 30

Collect like terms

=x3 + 6x2 - x - 30

x3 + 6x2 - x - 30

The questions

1

Expand and simplify (x + 1)(x + 2)(x + 4)

2 marks
Show answer
x3 + 7x2 + 14x + 8
2

Expand and simplify (x + 2)(x − 3)(x + 5)

3 marks
Show answer
x3 + 4x2 - 11x - 30
3

Expand and simplify (x − 1)(x − 4)(x + 2)

3 marks
Show answer
x3 - 3x2 - 6x + 8
4

Expand and simplify (x + 1)3

3 marks
Show answer
x3 + 3x2 + 3x + 1
5

Expand and simplify (x − 3)2(x + 4)

3 marks
Show answer
x3 - 2x2 - 15x + 36
6

Expand and simplify (3x − 1)(x + 2)(x − 2)

3 marks
Show answer
3x3 - x2 - 12x + 4
7

Expand and simplify (2x − 3)(x + 1)(x + 5)

3 marks
Show answer
2x3 + 9x2 - 8x - 15
8

A cuboid has length (x + 3) cm, width (x + 1) cm and height (x − 2) cm, where x > 2.

The volume of the cuboid is V cm3.

Show that V = x3 + 2x2 − 5x − 6

3 marks
Show answer
V = (x + 3)(x + 1)(x - 2) = (x2 + 4x + 3)(x - 2) = x3 - 2x2 + 4x2 - 8x + 3x - 6 = x3 + 2x2 - 5x - 6
9

(x + 4)(x − 2)(x − 5) = x3 + ax2 + bx + c, where a, b and c are integers.

  1. Find the value of a.
  2. Find the value of b.
  3. Find the value of c.
4 marks
Show answer
a) a = -3 b) b = -18 c) c = 40
10
  1. Expand and simplify (2x − 1)(3x + 1)
  2. Hence expand and simplify (x + 2)(2x − 1)(3x + 1)
4 marks
Show answer
a) 6x2 - x - 1 b) 6x3 + 11x2 - 3x - 2

Your score

Answer the questions above and your score appears here.

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper. It is also on the Edexcel International GCSE (4MA1).

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