The discriminant of a monic, cubic polynomial
$$p(x) := x^3 + b x^2 + c x + d$$
is the invariant
$$\Delta := -4 b^3 d+b^2 c^2+18 bc d-4 c^3-27 d^2;$$
it has the useful property that $p$ has three distinct, real roots iff $\Delta > 0$. (If $p$ has a repeated real root, then $\Delta = 0$, but in this case the nonrepeated root is always rational.)
On the other hand, by the Rational Root Theorem, if a monic polynomial $x^n + \cdots + d$ has a rational root $r$, then $r$ is an integer that divides $d$. These facts together suggest a way of generating examples:
- Pick a triple $(b, c, d)$ of integers.
- Compute $\Delta$; if $\Delta \leq 0$, $p$ does not have three real roots, so start over and pick a new triple. Otherwise, $p$ has three real roots.
- For each of the factors $s$ of $d$. Computing $p(\pm s)$. If any of these is values is zero, then $p$ has a rational root, so start over and pick a new triple. Otherwise, if none of these is zero, none of the roots of $p$ are rational, that is, $p$ satisfies the condition.
A quick Maple script shows that $2922$ ($31.2\%$) of the $21^3 = 9261$ monic, cubic polynomials with integer coefficients in $-10, \ldots, 10$ satisfy the condition, so the above procedure is efficient in the sense that in practice, one needn't try too many triples $(b, c, d)$ to produce examples.