Let $f(x)=x^3+ax+2$ $\in \mathbb{Q}[x].$ If $f(x)$ is reducible over $\mathbb{Q}$ , find all the possible values of $a$.
Attempt
Suppose $\alpha\in \mathbb{Q}$ be zero of $f(x)$ now $\alpha^3+a\alpha+2=0 $ follows $ a=\dfrac{-2-\alpha^3}{\alpha}$.
therefore for any $ a=\dfrac{-2-\alpha^3}{\alpha}$ $\alpha\in \mathbb{Q}-\{0\}$ , $f(x)$is reducible over $\mathbb{Q}$
Is my attempt correct? Can anyone verify answer?