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4PM1/02R Jan 2020, Question 5: Roots α and β of a Quadratic

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 12 marks

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The question, in short

Given roots α and β of 2x² + 3x + 6 = 0, prove two algebraic identities and use them to find α³ + β³ and α⁴ + β⁴, then form a quadratic with integer coefficients.

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Part (a) · 2 marks

Show that (α + β)³ − 3αβ(α + β) equals α³ + β³.

  1. 1(α + β)³ = α³ + 3α²β + 3αβ² + β³
  2. 23αβ(α + β) = 3α²β + 3αβ²
  3. 3Subtract: α³ + β³

Answer(α + β)³ − 3αβ(α + β) = α³ + β³ shown

Part (b) · 2 marks

Without solving 2x² + 3x + 6 = 0, find the value of α³ + β³.

  1. 1α + β = −3/2, αβ = 3
  2. 2α³ + β³ = (−3/2)³ − 3(3)(−3/2) = −27/8 + 27/2 = 81/8

Answer81/8

Part (c) · 2 marks

Show that (α² + β²)² − 2(αβ)² equals α⁴ + β⁴.

  1. 1(α² + β²)² = α⁴ + 2α²β² + β⁴
  2. 2Subtract 2(αβ)² = 2α²β²: α⁴ + β⁴

Answer(α² + β²)² − 2(αβ)² = α⁴ + β⁴ shown

Part (d) · 6 marks

Form a quadratic equation with integer coefficients whose roots are (α³ − β) and (β³ − α).

  1. 1α² + β² = (α + β)² − 2αβ = 9/4 − 6 = −15/4
  2. 2Sum = α³ + β³ − (α + β) = 81/8 + 3/2 = 93/8
  3. 3α⁴ + β⁴ = (−15/4)² − 2(9) = −63/16
  4. 4Product = (αβ)³ − (α⁴ + β⁴) + αβ = 27 + 63/16 + 3 = 543/16
  5. 5x² − (93/8)x + 543/16 = 0
  6. 616x² − 186x + 543 = 0

Answer16x² − 186x + 543 = 0

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