The exact problem I am having: $C(x)=1+ c_1x$ and $D(x)=1 + d_1 x$, where $c_1, d_1 \ll 1$ but $c_1 x$ or $d_1 x$ can be comparable to unity for large values of $x$, so can't use binomial expansion trivially. All I want to parametrize $D(x)/C(x)$ using a polynomial preferably. The whole purpose of requiring a polynomial is that we then can use orthonormal bases while fitting.
In the future I have to deal with $C(x)$ and $D(x)$ being higher order polynomials, so a solution which can be applied to a couple of generic $n$th order polynomials would be highly beneficial.