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?

See also: Finding an irreducible polynomial of degree $n$ in $\mathbb Q[X]$ with real roots. — Martin Sleziak, May 23 '18 at 06:59

score 3 · Answer 1 · answered Jan 17 '15 at 01:22

3

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.

answered Jan 17 '15 at 01:22

Timbuc

1 Answers1