Are there any characteristics of polynomials with rational roots?

In a few days iīm having a presentation in number theory.

Are there any characteristics of polynomials with rational roots? Both the expression and graphical.

How do I do this?

Re: Are there any characteristics of polynomials with rational roots?

Like what do you mean.

Like this one? Assume $\displaystyle deg(f(x)) \geq 2$

if $\displaystyle f(x) \in \mathbb{Q}[X] \text{ such that f(x) is irreducible in } \mathbb{Q}[X] \text{ then f has no roots in } \mathbb{Q} $ ?

or

$\displaystyle f(x) \in \mathbb{Z}[X] \text{ if f(x) is reducible in } \mathbb{Q}[X] \text{ then f(x) is reducible in also} \mathbb{Z}[X] $

Re: Are there any characteristics of polynomials with rational roots?

Sorry, didīnt get that :P

But the teacher gave me a tip

x^2+px+q=0

x=r1 V x=r2

(x-r1)(x-r2)=x2-x(r1+r2)+r2*r1

p=-(r2+r1)

q=r2*r1

Re: Are there any characteristics of polynomials with rational roots?