 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-3/9f/c/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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}} .

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