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A-LevelFurther MathematicsRoots of polynomial equations9231/12 · May/June 2019 · Q69 Marks

The equation x³ − x + 1 = 0 has roots α, β, γ.

  1. (i)Use the relation x = y⅓ to show that the equation y³ + 3y² + 2y + 1 = 0 has roots α³, β³, γ³. Hence write down the value of α³ + β³ + γ³.[3]
  2. (ii)Find the value of S₋₃.[2]
  3. (iii)Show that S₆ = 5 and find the value of S₉.[4]

Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

  1. [1] Obtains an equation in y not involving radicals.
  2. [1] AG
  3. [1] S₃ = -3
  4. [2] S₋₃ = (α³β³ + β³γ³ +α³γ³) / (α³β³γ³) = 2 / (-1) = -2
  5. [2] Uses (Σα)² = Σα² + 2Σαβ. AG.
  6. [2] Sums y³ - 3y² + 2y−1=0 for y=α³,β³,γ³.

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About this question

Topic
Roots of polynomial equations, A-Level Further Mathematics (syllabus 9231). Structured question, 9 marks.
Source
A-Level Further Mathematics, May/June 2019 Paper 1 Variant 2, question 6.

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