SCHWEITZER 2001 Problem 2

Let $\alpha \leq -2$ be an integer. Prove that for every pair $(\beta_0, \beta_1)$ of integers there exists a uniquely determined sequence $0\leq q_0, \ldots, q_k<\alpha ^ 2 - \alpha$ of integers, such that $q_k\neq 0$ if $(\beta_0, \beta 1)\neq (0,0)$ and $$\beta_i=\sum_{j=0}^k q_j(\alpha - i)^j,\text{ for }i=0,1$$