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A-LevelFurther MathematicsRoots of polynomial equations9231/12 · Oct/Nov 2021 · Q410 Marks

The cubic equation x³ + 2x² + 3x + 3 = 0 has roots α, β, γ.

  1. (a)Find the value of α² + β² + γ².[2]
  2. (b)Show that α³ + β³ + γ³ = 1.[2]
  3. (c)Use standard results from the list of formulae (MF19) to show that Σ[from r=1 to n] ((α+r)³ + (β+r)³ + (γ+r)³) = n + ½n(n+1)(an²+bn+c), where a, b and c are constants to be determined.[6]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

  1. [M1] (a) Uses formula for sum of squares.
  2. [A1] (a) -2
  3. [M1] (b) Uses original equation or formula for sum of cubes.
  4. [A1] (b) AG
  5. [B1] (c) Expands (α+r)³ = α³+3α²r+3αr²+r³
  6. [M1 A1] (c) Collects like terms and uses results from parts (a) and (b).
  7. [M1] (c) Applies formulae from MF19.
  8. [M1 A1] (c) Simplifies.

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About this question

Topic
Roots of polynomial equations, A-Level Further Mathematics (syllabus 9231). Structured question, 10 marks.
Source
A-Level Further Mathematics, Oct/Nov 2021 Paper 1 Variant 2, question 4.

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