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4PM1/01R Jan 2020, Question 4: The Factor and Remainder Theorems

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 11 marks

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The question, in short

A cubic f(x) = 2x³ + ax² + bx + 18 has unknown constants a and b; use the given remainder when f′(x) is divided by (x − 2) and the factor (x − 2) to work through parts a to c.

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

Part (a) · 6 marks

Use the two given conditions to set up equations and determine the constants a and b.

  1. 1f(2) = 16 + 4a + 2b + 18 = 0, so 4a + 2b = −34
  2. 2f′(x) = 6x² + 2ax + b
  3. 3f′(2) = 24 + 4a + b = 5, so 4a + b = −19
  4. 4Subtract: b = −15
  5. 54a = −19 + 15 = −4, so a = −1

Answera = −1, b = −15

Part (b) · 3 marks

Rewrite the cubic as a product of linear factors.

  1. 1f(x) = 2x³ − x² − 15x + 18
  2. 2Divide by (x − 2): 2x² + 3x − 9
  3. 32x² + 3x − 9 = (x + 3)(2x − 3)
  4. 4f(x) = (x − 2)(x + 3)(2x − 3)

Answerf(x) = (x − 2)(x + 3)(2x − 3)

Part (c) · 2 marks

Using your factorised form, solve f(x) = 0 algebraically.

  1. 1(x − 2)(x + 3)(2x − 3) = 0
  2. 2x = 2, x = −3, x = 3/2

Answerx = 2, x = −3, x = 3/2

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

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