A-LevelFurther MathematicsRoots of polynomial equations9231/12 · Oct/Nov 2022 · Q29 Marks

The equation x⁴ + 3x² + 2x+6 = 0 has roots α, β, γ, δ.

(a)Find a quartic equation whose roots are `1/(α²)`, `1/(β²)`, `1/(γ²)`, `1/(δ²)` and state the value of `1/(α²) + 1/(β²) + 1/(γ²) + 1/(δ²)`.[4]
(b)Find the value of `β² γ² δ² + α² γ² δ² + α² β² δ² + α² β² γ²`.[3]
(c)Find the value of `1/(α⁴) + 1/(β⁴) + 1/(γ⁴) + 1/(δ⁴)`.[2]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

[B1] `y = x^{-2}` leading to `x = y^{-1/2}` (Correct substitution.)
[M1] `(y^{-1/2})^4 + 3(y^{-1/2})^2 + 2(y^{-1/2}) + 6 = 0` leading to `1 + 3y + 2y^{3/2} + 6y^2 = 0` (Obtains an equation not involving radicals.)
[A1] `36y^4 + 32y^3 + 21y^2 + 6y + 1 = 0`
[B1 FT] `1/(α²) + 1/(β²) + 1/(γ²) + 1/(δ²) = -32/36 = -8/9`
[B1] `αβγδ = 6` (SOI)
[M1] `(β² γ² δ² + α² γ² δ² + α² β² δ² + α² β² γ²)/(α² β² γ² δ²)` relates to sums of coefficients (Relates to coefficients.)
[A1] `-32`
[M1] `Σ 1/(α⁴) = (Σ 1/(α²))² - 2 Σ 1/(α²β²)` (Uses formula for sum of squares.)
[A1] `- 61/162`

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Topic: Roots of polynomial equations, A-Level Further Mathematics (syllabus 9231). Structured question, 9 marks.
Source: A-Level Further Mathematics, Oct/Nov 2022 Paper 1 Variant 2, question 2.

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