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4PM1/02R Jun 2020, Question 2: The Factor and Remainder Theorems

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 11 marks

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The question, in short

Given cubic f(x) = x³ + px + q with remainders −12 at x = 1 and 30 at x = 4, find p and q, then use them to factorise and solve f(x) = 0.

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

Part (a) · 6 marks

Use the two remainder conditions to form simultaneous equations and find the constants p and q.

  1. 1f(1) = 1 + p + q = −12 ⇒ p + q = −13
  2. 2f(4) = 64 + 4p + q = 30 ⇒ 4p + q = −34
  3. 3Subtract: 3p = −21
  4. 4p = −7, q = −6

Answerp = −7, q = −6

Part (b) · 1 mark

Substitute the found p and q into f(x) and verify that f(3) equals 0.

  1. 1f(3) = 3³ − 7 × 3 − 6
  2. 2= 27 − 21 − 6 = 0

Answerf(3) = 0 shown

Part (c) · 3 marks

Factorise the cubic into a product of three linear factors using the known root.

  1. 1(x − 3) is a factor
  2. 2x³ − 7x − 6 = (x − 3)(x² + 3x + 2)
  3. 3= (x − 3)(x + 2)(x + 1)

Answer(x − 3)(x + 2)(x + 1)

Part (d) · 1 mark

Use the factorised form to state all solutions of f(x) = 0.

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

Answerx = 3, x = −2, x = −1

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

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