# Minimal Polynomial

I have found the following characteristic polynomial: $$(x+2)(x-2)^2$$

I need to write down all the possible minimal polynomial, so I wrote: $${(x+2),(x-2),(x+2)(x-2),(x+2)(x-2)^2,(x-2)^2}$$

Why is it wrong?

So the options are $(x+2)(x-2)$ and $(x+2)(x-2)^2$.
• I meant the power must be positive. If the characteristic polynomial is $(x-r_1)^{m_1} \cdots (x-r_n)^{m_n}$, with $r_i$ distinct, then the minimal polynomial can be anything of the form $(x-r_1)^{k_1} \cdots (x-r_n)^{k_n}$ with $1 \leq k_i \leq m_i$ for $i=1 \cdots n$. – Matt B Aug 30 '15 at 11:09