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4PM1/02 Nov 2023, Question 10: Roots α and β of a Quadratic

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 10 marks

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The question, in short

Given a quadratic's roots α and β with α + β = −5/2 and α³ + β³ = 115/8, prove αβ = 4, then build an integer-coefficient quadratic from two transformed roots.

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

Part (a) · 3 marks

Using the sum and sum of cubes of α and β, show that their product αβ equals 4.

  1. 1α³ + β³ = (α + β)³ − 3αβ(α + β)
  2. 2115/8 = (−5/2)³ − 3αβ(−5/2) = −125/8 + (15/2)αβ
  3. 3(15/2)αβ = 240/8 = 30
  4. 4αβ = 4

Answerαβ = 4 shown

Part (b) · 7 marks

Find an integer-coefficient quadratic equation whose roots are (α² + 1)/β and (β² + 1)/α.

  1. 1Sum = (α³ + β³ + α + β)/(αβ) = (115/8 − 5/2)/4 = 95/32
  2. 2α² + β² = (α + β)² − 2αβ = 25/4 − 8 = −7/4
  3. 3Product = (α²β² + α² + β² + 1)/(αβ) = (16 − 7/4 + 1)/4 = 61/16
  4. 4x² − (95/32)x + 61/16 = 0
  5. 532x² − 95x + 122 = 0

Answer32x² − 95x + 122 = 0

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