real analysis – $∀x∈mathbb Q$, $f(x)≧0$ implies $∀x∈mathbb R$, $f(x)≧0$

Let $f(x)∈mathbb R(x)$ be polynomial over real number.

I think the following claim holds.

Claim

$∀x∈mathbb Q$, $f(x)≧0$ implies $∀x∈mathbb R$, $f(x)≧0$

I think this holds because of density of $mathbb Q$ in $mathbb R$.

My question

  1. Does this claim correct?
  2. Can we extend this statement to some general results?

Thank you in advance.