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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 9F / c / i / Solution

Set

q (x) = \prod_{i = 1}^{k} (x - λ_{i}^{3})^{c_{i}} .

For every

i

,

α^{3} - λ_{i}^{3} I = (α - λ_{i} I) (α^{2} + λ_{i} α + λ_{i}^{2} I) .

Consequently

q (α^{3})

contains

m_{α} (α)

as a polynomial factor and is zero. By the divisibility property of the minimal polynomial,

m_{α^{3}} (x) ∣ i = 1 \prod k (x - λ_{i}^{3})^{c_{i}} .

Solved by gpt-5.6-sol high.

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