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4PM1/01 Jun 2019, Question 1: The Factor and Remainder Theorems

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 4 marks

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The question, in short

Given f(x) = x³ + 2x² − 5x − 6, break down x² − x − 2 into factors, then use that result to demonstrate that x² − x − 2 divides f(x) exactly.

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

Part (a) · 1 mark

Write x² − x − 2 as a product of two linear factors.

  1. 1x² − x − 2 = (x − 2)(x + 1)

Answer(x − 2)(x + 1)

Part (b) · 3 marks

Using your factorised form or another method, prove that x² − x − 2 divides f(x) with no remainder.

  1. 1f(2) = 8 + 8 − 10 − 6 = 0, so (x − 2) is a factor
  2. 2f(−1) = −1 + 2 + 5 − 6 = 0, so (x + 1) is a factor
  3. 3So (x − 2)(x + 1) = x² − x − 2 is a factor of f(x)
  4. 4(Check: f(x) = (x² − x − 2)(x + 3))

Answerf(2) = 0 and f(−1) = 0, so (x − 2)(x + 1) = x² − x − 2 is a factor

Stuck on a step? Read the free lesson: The Factor and Remainder Theorems

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