I am trying to understand that if $f(x)$ is any polynomial over a field $F$ such that $f(T) = 0$ then $m(x)\mid f(x).$ $T$ is a linear transformation from $V$ to $V$ and $m(x)$ is the minimal polynomial of $T$.
My book says apply the division algorithm to $f(x)$:
$f(x) = m(x)Q(x) + R(x)$ where either $R(x) = 0$ or $\deg R(x) < r = \deg m(x)$
Rewriting for $R(T)$ we have:
$R(T) = f(T) - m(T)Q(T) = 0$
Now my book claims that the above statement holds for $x$ and hence $R(x)=0$. I do not understand this implication. Why does this imply that $R(x)=0$?