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 $a_n$.
2 Answers
$\begingroup$The Rational Root Test proof depends on the polynomial coefs being integers. Let's recall it.
Theorem $ $ If $\,f(x) = f_n x^n + \cdots + f_0\,$ is a polynomial with $\,\color{#c00}{{\rm integer\ coefs}\ \,f_i\in \Bbb Z}\,$ and $\,f(x)\,$ has a rational root $\,x = a/b,\ \color{#0a0}{\gcd(a,b)=1},\,$ then $\,a\mid f_0\,$ and $\,b\mid f_n$
Proof $\ \ 0 = f(a/b)\ \Rightarrow\ 0 = b^n f(a/b)\ =\, f_n\, a^n\! + f_{n-1}\, a^{n-1}b+\cdots+f_1\, ab^{n-1}\! + f_0\, b^n$
Thus $\,\ (\overbrace{f_{n}\, a^{n-1}+f_{n-1}\,a^{n-2}b+\cdots+f_1\, b^{n-1}}^{\large{\rm an\ integer,\ since}\,\ \color{#c00}{f_i\ {\rm are\ integers}}})\,a\,=\, -f_0\, b^n,\ $ hence $\ a\mid b^n f_0\,\color{#0a0}{\Rightarrow}\, a\mid f_0,\, $ since $\,\color{#0a0}{\gcd(a,b)=1},\,\ a\mid bc\,\Rightarrow\,a\mid c,\,$ by Euclid's Lemma, so, by induction, $\,a\mid b^nc\,\color{#0a0}{\Rightarrow}\,a\mid c.$
Notice how the above proof depends crucially on the polynomial coefficients $\,\color{#c00}{f_i\,\ \rm being\ integers},\,$ which implies that the overbraced term is an integer and, hence, that $\,a\mid b^n f_0.\,$ Exactly the same applies to the reversed case, which deduces, symmetrically that $\, b\mid a^n f_n\,\Rightarrow\,b\mid f_n\ $ [or use $\ b^n f(a/b) = f_n\,a^n + ab (\ldots) + f_0\, b^n\,$ for $\,(\ldots) \in \Bbb Z\,$]
Besides identifying where the proof breaks down, there are obvious counterexamples, e.g. $\,x-a/b\,$ has a root $\,a/b\,$ that need not be an integer. Less trivial are quadratic examples
$\quad (x-a/b)\,(x-b/a)\, =\, x^2-(a/b+b/a)\,x + 1\,$ has a root $\,a/b\,$ that need not be $\,\pm1$.
Note also the hypothesis that the rational root is $\color{#0a0}{\rm reduced}$ (in lowest or least terms) is necessary, else e.g. $\, x = 4/6\, [= 2/3]\,$ is a root of $\,3\,x-2\,$ but $\, 6\nmid 3,\, 4\nmid 2.$
The above proof requires only that gcds exist, so it works over any GCD domain, e.g. any UFD (see here for more on this general case).
The Rational Root Test can also be viewed as a special case of Gauss's Lemma for polynomials.
$\endgroup$ 2 $\begingroup$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.
$\endgroup$
