We begin our discussion with a brief summary of methods for determining the Galois groups of cubics & quartics. This discussion will lead us to the notion of resolvents which is a central tool in the computation of Galois groups. We then apply these tools to the family of irreducible $f(x)=x^9+ax^6+bx^3+c$ defined over $\mathbb{Q}[x]$. We conclude with a computation Gal$(f)$ in select cases.