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A-LevelFurther MathematicsRoots of polynomial equations9231/12 · May/June 2025 · Q27 Marks

The cubic equation x³ +2x+1 = 0 has roots α, β, γ.

  1. (a)Find a cubic equation whose roots are α³ – 1, β³ – 1, γ³ – 1.[3]
  2. (b)Find the value of (α³ − 1)² + (β³ − 1)² + (γ³ − 1)². [2]
  3. (c)Find the value of (α³ − 1)³ + (β³ − 1)³ + (γ³ – 1)³.[2]

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

  1. [B1] Uses correct substitution.
  2. [M1] Substitutes and obtains an equation not involving radicals.
  3. [A1] y³ + 6y² + 20y + 16 = 0
  4. [M1] Uses Σα² = (Σα)² - 2Σαβ.
  5. [A1] -4
  6. [M1] Uses their equation from (a) or a valid formula for the sum of cubes.
  7. [A1] 96

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About this question

Topic
Roots of polynomial equations, A-Level Further Mathematics (syllabus 9231). Structured question, 7 marks.
Source
A-Level Further Mathematics, May/June 2025 Paper 1 Variant 2, question 2.

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