A-LevelFurther MathematicsRoots of polynomial equations9231/12 · Oct/Nov 2015 · Q58 Marks

The cubic equation x³ + px² + qx + r = 0, where p, q and r are integers, has roots α, β and γ, such that α + β + γ = 15, α² + β² + γ² = 83. Write down the value of p and find the value of q. Given that α, β and γ are all real and that αβ + αγ = 36, find α and hence find the value of r.

(a)Write down the value of p and find the value of q.[3]
(b)Given that α, β and γ are all real and that αβ + αγ = 36, find α and hence find the value of r.[5]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2015). The mark or mark type is shown in brackets.

[B1] (a) α + β + γ = −p = 15 ⇒ p = −15
[M1] (a) 2(αβ + βγ + γα) = (α + β + γ)² − (α² + β² + γ²)
[A1] (a) ⇒ q = ¹⁄₂(225 − 83) = 71
[M1] (b) αβ + αγ = α(β + γ) = 36
[M1A1] (b) ⇒ α² − 15α + 36 = 0 ⇒ α = 3, α ≠ 12, e.g. since 12² > 83 or other reason
[B1] (b) βγ = 71 − 36 = 35
[A1] (b) ⇒ r = −αβγ = −3 × 35 = −105 (extra answer penalised)

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Topic: Roots of polynomial equations, A-Level Further Mathematics (syllabus 9231). Structured question, 8 marks.
Source: A-Level Further Mathematics, Oct/Nov 2015 Paper 1 Variant 2, question 5.

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