Quick way to determine existence of integral root of a polynomial in one variable

Question

Suppose $p(x) \in \mathbb{Z}[x]$ and if there exist $b \in \mathbb{Z}$ s.t. $p(b)=0$, then $x-b|p(x)$. The other technique can be to put all $b \in \mathbb{Z}$.

But this require to check every $b \in \mathbb{Z}$ where $\mathbb{Z}$ is an infinite set.

So is there a quick way of determining whether or not there exist such $b \in \mathbb{Z}$.

Answer 1

Suppose you have $p(x)=(x-b)f(x)$ for some polynomial $f(x)$...

Product of constant term of $f$ and $-b$ should be same as constant term of $p$.

Thus, $b$ should divide constant term...

So, it is enough to check for factors of $p(0)$..

Answer 2

But this require to check every b∈ℤ where ℤ is an infinite set.

False. Only is required check the divisors ($+$ and $-$) of $p(0)$.