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4PM1/02R Jan 2023, Question 8: Roots α and β of a Quadratic

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 14 marks

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The question, in short

For 3x² − kx − 1 = 0 with positive integer k and roots α, β, prove a sum of squares identity, find k from a fourth-power sum, then build an integer-coefficient equation from two combined-root expressions.

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

Part (a) · 3 marks

Show that α² + β² equals (k² + 6)/9 using the roots of 3x² − kx − 1 = 0.

  1. 1α + β = k/3, αβ = −1/3
  2. 2α² + β² = (α + β)² − 2αβ
  3. 3= k²/9 + 2/3 = (k² + 6)/9

Answerα² + β² = (k² + 6)/9 shown

Part (b) · 5 marks

Given α⁴ + β⁴ = 466/81, determine the positive integer value of k.

  1. 1α⁴ + β⁴ = (α² + β²)² − 2(αβ)²
  2. 2466/81 = ((k² + 6)/9)² − 2/9
  3. 3k⁴ + 12k² − 448 = 0
  4. 4(k² − 16)(k² + 28) = 0
  5. 5k² = 16, k positive so k = 4

Answerk = 4

Part (c) · 6 marks

Hence form an equation with integer coefficients whose roots are (α³ + β)/β and (β³ + α)/α.

  1. 1Sum = (α⁴ + β⁴ + 2αβ)/(αβ) = (466/81 − 2/3)/(−1/3) = −412/27
  2. 2Product = (α³β³ + α⁴ + β⁴ + αβ)/(αβ) = (−1/27 + 466/81 − 1/3)/(−1/3) = −436/27
  3. 3x² + (412/27)x − 436/27 = 0
  4. 427x² + 412x − 436 = 0

Answer27x² + 412x − 436 = 0

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