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A-LevelFurther MathematicsRoots of polynomial equations9231/12 · Oct/Nov 2016 · Q26 Marks

Find the cubic equation with roots α, β and γ such that α + β + γ = 3, α² + β² + γ² = 1, α³ + β³ + γ³ = −30, giving your answer in the form x³ + px² + qx + r = 0, where p, q and r are integers to be found.

  1. Find the cubic equation with roots α, β and γ such that α + β + γ = 3, α² + β² + γ² = 1, α³ + β³ + γ³ = −30, giving your answer in the form x³ + px² + qx + r = 0, where p, q and r are integers to be found.[6]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2016). The mark or mark type is shown in brackets.

  1. [M1A1] Σαβ=9-1⇒Σαβ=4
  2. [M1] Use of, e.g.: Σα³ – 3αβγ = Σα(Σα² -Σαβ) or (Σα)³ = Σα³ +3ΣαΣαβ-3αβγ
  3. [A1] Correct substitution in formula
  4. [A1] ⇒αβγ =-7
  5. [A1] Required cubic equation is x³-3x² +4x+7=0 must see final equation
  6. ALT METHOD: [M1] S3 - 3S2+4S₁ + 3r = 0
  7. [A1] 3r = 30 +3 × 1 – 4 × 3
  8. [A1] r=7

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About this question

Topic
Roots of polynomial equations, A-Level Further Mathematics (syllabus 9231). Structured question, 6 marks.
Source
A-Level Further Mathematics, Oct/Nov 2016 Paper 1 Variant 2, question 2.

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