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

1 Answer
Jul 13, 2018

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.