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4PM1/01 Jan 2023, Question 4: The Factor and Remainder Theorems

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 9 marks

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The question, in short

Given cubic f(x) = x³ + px² + qx + 6 with (x − 1) a factor and remainder 8 on division by (x + 1), find p and q, then solve f(x) = 0.

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

Part (a) · 6 marks

(i) Use the factor condition to show p = −2. (ii) Use the remainder condition to find q.

  1. 1f(1) = 0: 1 + p + q + 6 = 0, so p + q = −7
  2. 2f(−1) = 8: −1 + p − q + 6 = 8, so p − q = 3
  3. 3Add: 2p = −4, p = −2
  4. 4q = −7 − (−2) = −5

Answer(i) p = −2 shown (ii) q = −5

Part (b) · 3 marks

Using your p and q, solve the cubic equation f(x) = 0.

  1. 1f(x) = x³ − 2x² − 5x + 6 = (x − 1)(x² − x − 6)
  2. 2= (x − 1)(x − 3)(x + 2)
  3. 3x = 1, 3, −2

Answerx = 1, x = 3, x = −2

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

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