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4PM1/01 Jun 2023, Question 8: Solving Cubic Equations

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

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The question, in short

Using the derivative 18x² − 2x + 13 and the fact that 2x − 1 divides f(x), show that y = f(x) crosses the x-axis exactly once.

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

Solution · 9 marks

  1. 1f(x) = ∫ (18x² − 2x + 13) dx = 6x³ − x² + 13x + c
  2. 2(2x − 1) is a factor so f(1/2) = 0: 6/8 − 1/4 + 13/2 + c = 0, c = −7
  3. 3f(x) = 6x³ − x² + 13x − 7
  4. 4Divide by (2x − 1): f(x) = (2x − 1)(3x² + x + 7)
  5. 5Discriminant: 1² − 4 × 3 × 7 = −83 < 0
  6. 6So 3x² + x + 7 = 0 has no real roots and the curve meets the x-axis only at (1/2, 0)

Answerf(x) = 6x³ − x² + 13x − 7 = (2x − 1)(3x² + x + 7); discriminant of 3x² + x + 7 is −83 < 0, so only one intersection with the x-axis, at (1/2, 0) — shown

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