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4PM1/02R Jun 2023, Question 6: 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 f(x) = 2x² + 5x − p with roots α and β and α³ + β³ = −215/8, find p, then form an integer-coefficient quadratic with two given root expressions.

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

Part (a) · 5 marks

Using the sum and product of roots of 2x² + 5x − p = 0 together with α³ + β³ = −215/8, find the value of p.

  1. 1α + β = −5/2, αβ = −p/2
  2. 2α³ + β³ = (α + β)³ − 3αβ(α + β)
  3. 3= −125/8 − 3(−p/2)(−5/2) = −125/8 − 15p/4
  4. 4−125/8 − 15p/4 = −215/8, so 15p/4 = 45/4
  5. 5p = 3

Answerp = 3

Part (b) · 5 marks

Without solving f(x) = 0, form a quadratic equation with integer coefficients whose roots are (α + β)/α² and (α + β)/β².

  1. 1With p = 3: α + β = −5/2, αβ = −3/2
  2. 2Sum = (α + β)(α² + β²)/(αβ)² = [α³ + β³ + αβ(α + β)]/(αβ)²
  3. 3= (−215/8 + 15/4)/(9/4) = −185/18
  4. 4Product = (α + β)²/(αβ)² = (25/4)/(9/4) = 25/9
  5. 5x² + (185/18)x + 25/9 = 0, so 18x² + 185x + 50 = 0

Answer18x² + 185x + 50 = 0

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