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A-LevelFurther MathematicsRoots of polynomial equations9231/12 · May/June 2020 · Q28 Marks

The cubic equation 6x³ +px² -3x-5 = 0, where p is a constant, has roots α, β, γ.

  1. (a)Find a cubic equation whose roots are α², β², γ².[3]
  2. (b)(i)It is given that α² + β² + γ² = 2(α+β+γ). Find the value of p.[3]
  3. (b)(ii)Find the value of α³ + β³ + γ³.[2]

Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

  1. [B1] y = x²
  2. [M1] 6y^1.5 + py - 3y^0.5 - 5 = 0 => y^0.5(6y-3) = -py+5 OR (6x-3)x = -px^2+5
  3. [A1] 36y³ - (p²+36)y² + (10p+9)y - 25 = 0
  4. [B1] α²+β²+γ² = p²/36 + 1
  5. [M1] p²/36 + 1 = 2(-p/6) => p²/36 + 1 = -p/3
  6. [A1] p = -6
  7. [M1] 6(α³ + β³ + γ³) = 6(α² + β² + γ²) + 3(α+β+γ) + 15
  8. [A1] α³ + β³ + γ³ = 5

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About this question

Topic
Roots of polynomial equations, A-Level Further Mathematics (syllabus 9231). Structured question, 8 marks.
Source
A-Level Further Mathematics, May/June 2020 Paper 1 Variant 2, question 2.

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