Given that $f(z)=\dfrac{1}{2u^2(1-uz)}$ , show that \begin{equation*} \dfrac{f(z_1)}{(z_1-z_2)(z_1-z_3)}+\dfrac{f(z_2)}{(z_2-z_1)(z_2-z_3)}+\dfrac{f(z_3)}{(z_3-z_2)(z_3-z_1)}=\dfrac{2}{(1-au)(1-bu)(1-cu)} \end{equation*} where $a$,$b$,$c$ are any $z_i$
Given \begin{equation*} (z_2-z_3)(1-uz_3)(1-uz_2)-(z_1-z_3)(1-uz_3)(1-uz_1)+(z_1-z_2)(1-uz_2)(1-uz_1) , \end{equation*} is it possible to factorise to the form $u^2(z_1-z_2)(z_1-z_3)(z_2-z_3)$ ?
Is there any method to expand such polynomial efficiently without expanding everything?