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4MA1/1HR Jun 2018, Question 21: Expand and Simplify

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1HR (Higher) · Calculator · 4 marks

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The question, in short

Show an algebraic identity for x(x − 1)(x + 1), then prove that a whole number minus its cube is always a multiple of 6.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 1 mark

Expand x(x − 1)(x + 1) to show it equals x³ − x.

  1. 1(x − 1)(x + 1) = x² − 1
  2. 2x(x² − 1) = x³ − x

Answerx³ − x

Part (b) · 3 marks

Prove that n − n³ is always a multiple of 6 for whole numbers n.

  1. 1Let the number be n: n³ − n = n(n − 1)(n + 1)
  2. 2(n − 1), n, (n + 1) are three consecutive whole numbers
  3. 3One of them is even and one is a multiple of 3
  4. 4So n³ − n is always a multiple of 2 × 3 = 6

Answerthe difference between a whole number and the cube of this number is always a multiple of 6

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