For a linear operator on a finite-dimensional vector space, if we have a monic polynomial $p_T(x)$ such that $\operatorname{deg}(p_T) = \dim V$ and $p_T(T) = 0$, $p_T$ can be factored into two monic polynomials $r(x), s(x)$ such that there are two vector spaces $V_1 = \{v \in V | r(T)v = 0\}$ and $V_2 = \{v \in V | s(T)v = 0\}$.
Apparently, $V_1 + V_2 = V$ is a direct sum. Why is this?
I understand that in all cases where we plug in $T$ we can form $p_T(T) = a_1r(T) + a_2s(T)$, which satisfies the constraints for a direct sum, but I don't see why the constraints for being a direct sum should be satisfied when we plug in other values into our decomposed polynomials.
This seems false.
Consider $T = \pmatrix{0 & 1 \\ 0 & 0}$ on $\mathbb{C}^2$. Then we can take $p_T(x) = x^2$ and $r(x) = s(x) = x$. We have $$V_1 = V_2 = \{v \in \mathbb{C}^2 : Tv = 0\} = \left\{\pmatrix{\lambda \\ 0} : \lambda \in \mathbb{C}\right\}$$
so $V_1 + V_2 \ne \mathbb{C}^2$ and $V_1 \cap V_2 \ne \{0\}$.