I am currently assisting a course for future teachers at university level for a joint education in maths and physics.
We defined the annihilator for an element A of a K-algebra $\mathcal A$ as the set of polynomials over the field K, that return zero. So
$$Ann(A):=\{f\in K[x]: \tilde f(A)=0\}$$
The tilde indicates the evaluation-homomorphism. We proved that any such f can be decomposed into a unique normed polynomial h and some rest, such that every $f = g\cdot h$. This can be used to show that a nilpotent matrix of order k, has $M_A(x)=x^k$ as minimal polynomial.
My problem: it is an interesting tool to find the minimal polynomial of say a real valued matrix. But I do not see the relevance for future teachers as this seems to be "just" a technical result without any applications that one could explain to someone without extensive mathematical education.
So, are there any nice applications for this method in the realm of euclidean geometry? or physics?
I am asking to avoid the tunnel syndrom or the double discontinuity that Felix Klein complained about in 1908 for maths education. Maybe someone can give some inspiration?