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A-LevelFurther MathematicsRoots of polynomial equations9231/12 · Oct/Nov 2020 · Q18 Marks

The cubic equation x³ + bx² + cx + d = 0, where b, c and d are constants, has roots α, β, γ. It is given that αβγ =-1.

  1. (a)State the value of d.[1]
  2. (b)Find a cubic equation, with coefficients in terms of b and c, whose roots are α+1, β+1, γ+1.[3]
  3. (c)Given also that γ+1 =-α-1, deduce that (c-2b+3)(b-3) = b-c.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

  1. [B1] d=1
  2. [B1] y=x+1⇒ x = y −1
  3. [M1 A1] Substitutes and expands. y³ + (b−3)y² + (c - 2b + 3)y + b − c = 0
  4. [B1] β+1=-(b-3)
  5. [B1] −(α+1)(α+1) = c − 2b + 3
  6. [M1] Applies product of roots. −(α+1)(β+1)(α+1)= -(b−c)
  7. [A1] AG ⇒(c-2b+3)(b-3)=b-c

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About this question

Topic
Roots of polynomial equations, A-Level Further Mathematics (syllabus 9231). Structured question, 8 marks.
Source
A-Level Further Mathematics, Oct/Nov 2020 Paper 1 Variant 2, question 1.

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