  Let $k,a\in \mathbb{N}$ and let $p_1,\cdots,p_k\in \mathbb{Q}[n]$ with zero
constant term. We show that for any finite coloring of $\mathbb{Q}$, there are
non-zero $x,y\in \mathbb{Q}$ such that there exists a color which contains a
set of the form $$\Big\{x,\frac{x}{y^a},x+p_{1}(y),\cdots,x+p_{k}(y)\Big\}$$
and there are non-zero $v,u\in \mathbb{Q}$ such that there exists a color which
contains a set of the form $$\Big\{v,v\cdot
{u^a},v+p_{1}(u),\cdots,v+p_{k}(u)\Big\}.$$
