Irreducible polynomial with three real zeros

Question

Is there an irreducible (cubic) polynomial with rational coefficients with three real zeros?

(When I speak of irreducibility I mean over rational numbers.)

How about an irreducible polynomial of degree $n$ with rational coefficients with $n$ real zeros?

Answer 1

For example

$$f(x)=x^3-2x^2-x+1$$

It has no rational roots (why?) so it is irreducible, and

$$f(-1)<0<f(0)\;,\;\;f(1)<0$$

So you already have two real roots and thus also the third one must be real.