Char vs min polynomial

Let T be a linear operator on ${\mathbb{R}}^3$. Let $f(X)\in \mathbb{R}[X]$ denote its characteristic polynomial. Consider the following statements. (a) Suppose T is non-zero and 0 is an eigenvalue of T. If we write f(X) = X·g(X) in $\mathbb{R}[X]$, then the linear operator g(T) is zero. (b) Suppose 0 is an eigenvalue of T with at least two linearly independent eigenvectors. If we write f(X) = X·g(X) in $\mathbb{R}[X]$, then the linear operator g(T) is zero. Which of the following is true?