Let $P(x)$ be a polynomial with integer coefficients. In what conditions that $P(x)$ doesn't have a rational root?
From https://en.wikipedia.org/wiki/Rational_root_theorem, if $P(x)=a_nx^n+\cdots +a_0$ with integer coefficients, where $a_0\ne 0$, $a_n\ne 0$, then the only conceivable rational roots of $P(x)$ are of the form $\dfrac{a}{b}$, where $a|a_0$, $b|a_n$. However there are some cases that for $a|a_0$, $b|a_n$, $\dfrac{a}{b}$ migh not be a root of $P(x)$.
Are there any general criterion so that $P(x)$ doesn't have a rational root?
In general, if we have a polynomial $P(x)$ with integer coefficients, where $P(x)=a_0x^n+\cdots +a_n$, where $a_0\ne 0$, $a_n\ne 0$, then the only conceivable rational roots of $P(x)$ are of the form $\dfrac{a}{b}$, where $a$ is a divisor (possibly negative) of $a_n$ and $b$ is a positive divisor of $a_0$.