Question

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

Answer 1

See below

Explanation:

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

`f(x) = a_n x^n + a_{n-1}x^{n-1}+...+a_1x+a_0`

all the rational solutions of `f` are in the form `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 `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`.