T be a linear operator on a finite dimensional vector space.

p(x) - Minimal Polynomial of T

q(x) - Characteristic Polynomial of T

I want to prove the p(x)|q(x)

I know that every root of q(x) is a root of p(x) and vice a versa. I guess we can use this fact to prove the above.

But I do not want to use this - because we might not be able to factorize the polynomials. Is there a better way please? Thanks