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A-LevelFurther MathematicsRoots of polynomial equations9231/12 · May/June 2015 · Q48 Marks

The roots of the cubic equation x³ – 7x² + 2x − 3 = 0 are α, β and γ. Find the values of Deduce a cubic equation, with integer coefficients, having roots 1/(αβ), 1/(βγ) and 1/(γα).

  1. (i)1 / (αβ)(βγ)(γα)᾿[2]
  2. (ii)1/(αβ) + 1/(βγ) + 1/(γα)[2]
  3. (iii)1/(αβ) + 1/(βγ) + 1/(γα)[2]

Mark scheme

Marking points from the official mark scheme (May/June 2015). The mark or mark type is shown in brackets.

  1. [B1] α+β+γ=7, αβ+βγ+γα=2, αβγ=3 (Stated or implied by working.)
  2. [B1] (i) 1 / (αβ)(βγ)(γα) = 1 / (αβγ)² = 1/9
  3. [M1A1] (ii) 1/(αβ) + 1/(βγ) + 1/(γα) = (α+β+γ)/(αβγ) = 7/3
  4. [M1A1] (iii) 1/(αβ)⋅1/(βγ) + 1/(βγ)⋅1/(γα) + 1/(γα)⋅1/(αβ) = (αβ+βγ+γα)/(αβγ)² = 2/9
  5. [M1A1V] ⇒ x³ - (7/3)x² + (2/9)x - (1/9) = 0 → 9x³−21x² + 2x−1 = 0

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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 2015 Paper 1 Variant 2, question 4.

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