Below is a question given in exercises of book of Galois theory by Patrick Morandi which is used in the proof Lindemann-Weierstrauss theorem. Let $$ \sum_{i=1}^r a_i x^{\alpha_i} \text { and } \sum_{i=1}^s b_i x^{\beta_i} $$ be polynomial functions with $a_i, b_i$ nonzero rational numbers and $\alpha_i, \beta_i$ algebraic numbers. Assume that the $\alpha_i$ are distinct and that the $\beta_i$ are distinct. Writing $\sum_j a_j x^{\alpha_j} \cdot \sum_j b_j x^{\beta_j}$ in the form $\sum_k c_k x^{\gamma_j}$ with the $\gamma_j$ distinct, show that at least one of the $c_k$ is nonzero.
Assuming all $c_k =0$ isn' t helping me.Can anyone help me out?