Let $f(t, x)$ be an irreducible polynomial in $\mathbb C[t, x]$ that is monic and cubic when regarded as a polynomial in $x$. Assume that for some $t_0$ the polynomial $f (t_0 , x)$ has one simple root and one double root. Prove that the splitting field $K$ of $f(x)$ over $\mathbb C(t)$ has degree 6.
I saw a counterexample show this exercise is wrong $$ f(x, t)=x^3-3t^2x-(t^4+t^2) $$ This polynomial meet the requirement irreducible and has the root as required when $t=1$ but it discriminant $D \in \mathbb C(t)$ So its splitting field just 3 order over $\mathbb C(t)$.
I wonder if this exercise is wrong or this couterexample have problem.