TITLE: Why does the rational root theorem not include polynomials with rational coefficients?
QUESTION [9 upvotes]: Why does the rational root theorem only work when the polynomial has integer coefficients? Specifically, why would it not apply if the coefficients were rational?
Recall the rational root theorem (or rational root test, rational zero theorem, rational zero test or $p/q$ theorem) states a constraint on rational solutions of a polynomial equation
$$ a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}=0$$
with integer coefficients ${\displaystyle a_{i}\in \mathbb {Z}}$.
The theorem states that each rational solution $\,x = p⁄q\,$ written in lowest terms (so  $p$ and $q$ are relatively prime), satisfies:

$p$ is an integer factor of the constant term $a_0$, and
$q$ is an   integer factor of the leading coefficient $a_n$.

REPLY [5 votes]: If the coefficients are rational, you can multiply the polynomial by the least common denominator to get a second polynomial in integer coefficients that has the same zeroes. Rational root theorem applies.
If the coefficients are irrational numbers, all bets are off.