Number of roots of $x^3+2x-1$ in $\mathbb Q$

Question

How many roots does $x^3+2x-1$ have in $\mathbb Q$?

I know that it has one real and two complex conjugate roots because the determinant is $-59$.

Answer 1

By the rational root theorem, if there is a rational root $p/q$ in lowest terms, then $p$ divides $a_{0}$ (i.e. $1$ is divisible by $p$, so what can $p$ be?). We can also say that $q$ divides $a_{3}=1$. What must $q$ be? Make sure you check whether the solution you get works!

Answer 2

the only real root is this here $x=\frac{\sqrt[3]{\frac{1}{2} \left(9+\sqrt{177}\right)}}{3^{2/3}}-2 \sqrt[3]{\frac{2}{3 \left(9+\sqrt{177}\right)}}$ and this root is not $\in \mathbb{Q}$ thus not such solution exists.