By virtue of the Proposition below, show that the polynomial $f(x) = 4x^3 -x^2 + 7$ is irreducible in $\mathbb{Q}[x]$.
Proposition: Let $f(x) = \sum_{k=0}^n a_{k}x^k \in \mathbb{Z}[x]$ and suppose that $p$ is a prime not dividing $a_n$. Define $\hat{f} \in \mathbb{F}_p[x]$ by $\hat{f}(x) = \sum_{k=0}^n \hat{a}_{k}x^k$ where $\hat{a}_{k}$ denotes the image of $a_k$, in $\mathbb{F}_p$. If $\hat{f}$ is irreducible in $\mathbb{F}_p[x]$, then $f$ is irreducible in $\mathbb{Q}[x]$.
Hint : Show that the polynomial has no root in $\mathbb Z_3$ and hence is irreducible over $\mathbb Z_3[x]$