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WMA12/01 Jun 2023, Question 2: Cubic Roots after Division

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 6 marks

The question, in short

Given f(x) = 4x³ − 8x² + 5x + a with (2x − 3) a factor, use the factor theorem to find a, then show f(x) = 0 has only one real root.

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

Part (a) · 2 marks

Use the factor theorem with the factor (2x − 3) to show that the constant a equals −3.

  1. 1(2x − 3) is a factor, so f(3/2) = 0
  2. 24(3/2)³ − 8(3/2)² + 5(3/2) + a = 0
  3. 327/2 − 18 + 15/2 + a = 0
  4. 43 + a = 0, so a = −3

AnswerShown: f(3/2) = 27/2 − 18 + 15/2 + a = 0, so a = −3

Part (b) · 4 marks

Using a = −3, show that the cubic equation f(x) = 0 has exactly one real root.

  1. 1f(x) = 4x³ − 8x² + 5x − 3
  2. 2Divide by (2x − 3): f(x) = (2x − 3)(2x² − x + 1)
  3. 3Discriminant of 2x² − x + 1: (−1)² − 4(2)(1) = 1 − 8 = −7
  4. 4−7 < 0, so 2x² − x + 1 = 0 has no real roots
  5. 5So the only real root is x = 3/2

AnswerShown: f(x) = (2x − 3)(2x² − x + 1); 2x² − x + 1 has discriminant (−1)² − 4(2)(1) = −7 < 0, so x = 3/2 is the only real root

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