4PM1/01 Jun 2025, Question 5: Roots α and β of a Quadratic
Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 8 marks
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The question, in short
Prove an algebraic identity in α and β, then use the roots of 2x² − 6x − 7 = 0 to build a new quadratic with integer coefficients.
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Part (a) · 2 marks
Show that (α + β)³ − 3αβ(α + β) equals α³ + β³.
- 1(α + β)³ = α³ + 3α²β + 3αβ² + β³
- 23αβ(α + β) = 3α²β + 3αβ²
- 3Subtracting leaves α³ + β³
Answer(α + β)³ − 3αβ(α + β) = α³ + β³ shown
Part (b) · 6 marks
Without solving 2x² − 6x − 7 = 0, form an integer-coefficient quadratic whose roots are α²/β and β²/α.
- 1α + β = 3, αβ = −7/2
- 2Product = (α²/β)(β²/α) = αβ = −7/2
- 3Sum = (α³ + β³)/(αβ) = [(α + β)³ − 3αβ(α + β)]/(αβ)
- 4= [27 + 31.5]/(−7/2) = −117/7
- 5x² + (117/7)x − 7/2 = 0
- 614x² + 234x − 49 = 0
Answer14x² + 234x − 49 = 0
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