I am reading the following lemma about polynomials:
Suppose that $x$ is a root of a polynomial $P$, $\pmod p$. Then the irreducible polynomial $(T - x)$ is a factor of $P$
I am not sure I understand this. Let's say we have the polynomial $P = 3T - 5, \pmod 7$.
The root of this polynomial is $x \equiv 4 \pmod 7$.
But now how can I combine this with the lemma above to figure out the irreducible polynomial that is a factor of $P$? I guess it means $(T - 4)$ is a factor of $P$ does that just mean that $P = 3(T - 4)$? If yes, how is this representation useful?