# We havef=X^3-5X^2+a,ainRR.How to prove that f has at most a root in ZZ?

Jul 13, 2018

See below

#### Explanation:

The Rational root theorem states the following: given a polynomial with integer coefficients

$f \left(x\right) = {a}_{n} {x}^{n} + {a}_{n - 1} {x}^{n - 1} + \ldots + {a}_{1} x + {a}_{0}$

all the rational solutions of $f$ are in the form $\frac{p}{q}$, where $p$ divides the constant term ${a}_{0}$ and $q$ divides the leading term ${a}_{n}$.

Since, in your case, ${a}_{n} = {a}_{3} = 1$, you are looking for fractions like $\frac{p}{1} = p$, where $p$ divides $a$.

So, you can't have more than $a$ integer solutions: there are exactly $a$ numbers between $1$ and $a$, and even in the best case they all divide $a$ and are solutions of $f$.