What does $K_1(R)$ tell us about $GL_n(R)/E_n(R)$? Let $D$ be a division ring, and $R=D[t_1,\ldots,t_n]$. If $GL_m(R)$ is the usual group of invertible matrices over $R$, then by $E_m(R)$ I mean the subgroup of $GL_m(R)$ generated by the elementary matrices of the form $I_m+r\epsilon(i,j)$, $r\in U(R)$. It is known that the whitehead group $K_1(R)$ of $R$ is isomorphic to the commutator quotient group of $U(D)$. Is there any way we can extract from this, anything about $GL_m(R)/E_m(R)$?
 A: In general what you are asking about is the problem of $K_1$-stabilization, that is the study of maps $GL(n, R)/E(n, R)\to GL(n+1, R)/E(n+1, R)$. For rings of finite dimension all such maps are bijective starting with some $n$ depending on the dimension.
The first relevant link is "On the stabilization of general linear group" by L. Vaserstein. I believe that any paper on the subject cites this one, so you can explore them by following the link.
For the particular problem about polynomials over a division ring, take a look at "The general linear group of polynomial rings over regular rings" by T. Vorst. Here's the abstract:

In this note we shall prove for two types of regular rings $A$ that every element of $GL_r(A[X_1,\ldots,X_n])$ is a product of an element of $E_r(A[X_1,\ldots,X_n])$ (the group of elementary matrices) and an element of $GL_r(A)$, for $r\geq3$ and $n$ arbitrary. This is a kind of $GL_r$-analogue of results of Lindel and Mohan-Kumar and is an extension of a result of Suslin.

