In 1985, Bressoud and Goulden derived the formula for the constant term in $\prod_{(i,j)\in T} \frac{x_j}{x_i}\\\prod_{0\le i<j \le n}(\frac{x_i}{x_j})_{a_i}(\frac{qx_j}{x_i})_{a_j-1}$, where $T \subseteq \{(i,j)\mid 0\le i<j \le n\}$. This result implies the Andrews' $q$-Dyson identity. In 2006, Gessel and Xin proved the $q$-Dyson identity by considering both sides of the equality as polynomials in $q^{a_0}$. We use this approach to determine the coefficients of $x_0/x_1$ and $x_0/x_2$ in Laurent polynomials studied by Bressoud and Goulden.