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A-LevelFurther MathematicsRoots of polynomial equations9231/12 · Oct/Nov 2011 · Q15 Marks

The equation x³ + px + q = 0 has a repeated root. Prove that 4p³ + 27q² = 0.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2011). The mark or mark type is shown in brackets.

  1. [M1] (N.B. Not α, β, γ) Let roots be a, a, and β. Writes down sum of roots, (1) 2α + β = 0
  2. [A1] sum of products in pairs (2) 2αβ + α² = p
  3. [A1] and product of roots. (3) α²β = -q From (1) β = −2α
  4. [M1] Eliminates β (or α). (2) ⇒ -4a² + a² = p ⇒ p = −3a² (3) ⇒ −2a³ = -q ⇒ q = 2a³
  5. [A1] Equates power of a (or β) α = (-p/3)³ = (q/2)² ⇒ 4p³ + 27q² = 0 (AG)

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About this question

Topic
Roots of polynomial equations, A-Level Further Mathematics (syllabus 9231). Structured question, 5 marks.
Source
A-Level Further Mathematics, Oct/Nov 2011 Paper 1 Variant 2, question 1.

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