How do I use partial fraction decomposition to split up the following:
$$\frac{1}{(1-q_1z)(1-q_2z)}$$
where both $q_1$ and $q_2$ are probabilities, and $q_1 = 1 - p_1$ and $q_2 = 1 - p_2$
The answer is supposed to be
$$\frac{q_1}{1-q_1z} - \frac{q_2}{1-q_2z}$$
but I only got as far as
$$\frac{1}{(1-q_1z)(1-q_2z)} = \frac{A}{1-q_1z} + \frac{B}{1-q_2z}$$ $$1 = A(1-q_2z) + B(1-q_1z)$$ $$A + B = 1$$ $$-Aq_2 -Bq_1 = 0$$
Solving the last 2 equations, I got
$$A = \frac{q_1}{q_1-q_2}$$ $$B = -\frac{q_2}{q_1-q_2}$$