Solution
= Solution
Set
$$
q(x)=\prod_{i=1}^k(x-\lambda_i^3)^{c_i}.
$$
For every $i$,
$$
\alpha^3-\lambda_i^3I
=(\alpha-\lambda_iI)
(\alpha^2+\lambda_i\alpha+\lambda_i^2I).
$$
Consequently $q(\alpha^3)$ contains $m_\alpha(\alpha)$ as a polynomial factor and is zero. By the divisibility property of the <minimal polynomial>,
$$
\boxed{m_{\alpha^3}(x)\mid
\prod_{i=1}^k(x-\lambda_i^3)^{c_i}}.
$$