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4PM1/01R Jun 2022, Question 8: Roots α and β of a Quadratic

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 10 marks

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The question, in short

Expand (x + y)³, then for 2x² + 3x + 4 = 0 with roots α and β, find α³ + β³ without solving, and build an integer-coefficient quadratic with roots α/β² and β/α².

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

Part (a) · 1 mark

Write out the full expansion of (x + y)³.

  1. 1(x + y)³ = x³ + 3x²y + 3xy² + y³

Answerx³ + 3x²y + 3xy² + y³

Part (b) · 4 marks

Given 2x² + 3x + 4 = 0 has roots α and β, work out α³ + β³ without finding α or β.

  1. 1α + β = −3/2, αβ = 4/2 = 2
  2. 2α³ + β³ = (α + β)³ − 3αβ(α + β)
  3. 3= (−3/2)³ − 3(2)(−3/2) = −27/8 + 9
  4. 4= 45/8

Answer45/8

Part (c) · 5 marks

Using earlier results, construct a quadratic with integer coefficients whose roots are α/β² and β/α².

  1. 1Sum = α/β² + β/α² = (α³ + β³)/(αβ)² = (45/8)/4 = 45/32
  2. 2Product = (α/β²)(β/α²) = 1/(αβ) = 1/2
  3. 3x² − (45/32)x + 1/2 = 0
  4. 432x² − 45x + 16 = 0

Answer32x² − 45x + 16 = 0

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