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4PM1/02R Jan 2022, Question 4: Roots α and β of a Quadratic

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

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The question, in short

For the quadratic 2x² + 4x + 3 = 0 with roots α and β, use root properties rather than solving to derive results about powers of the roots and build a new quadratic.

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

Part (a) · 4 marks

Using sum and product of roots, prove that α² + β² equals 1.

  1. 1α + β = −2, αβ = 3/2
  2. 2α² + β² = (α + β)² − 2αβ
  3. 3= (−2)² − 2 × 3/2
  4. 4= 4 − 3 = 1

Answerα² + β² = 1 shown

Part (b) · 3 marks

By relating α⁴ + β⁴ to α² + β² and αβ, determine its value.

  1. 1α⁴ + β⁴ = (α² + β²)² − 2α²β²
  2. 2= 1² − 2 × (3/2)²
  3. 3= 1 − 9/2 = −7/2

Answer−7/2

Part (c) · 3 marks

From earlier results, construct a quadratic with integer coefficients whose roots are α⁴ and β⁴.

  1. 1Sum of roots = −7/2
  2. 2Product of roots = (αβ)⁴ = (3/2)⁴ = 81/16
  3. 3x² + (7/2)x + 81/16 = 0
  4. 416x² + 56x + 81 = 0

Answer16x² + 56x + 81 = 0

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