Consider a polynomial $$p(t)=\sum_{i=1}^n a_it^{b_i}+a_0$$ of degree $b_n$, show that it admits at most $n+1$ nonnegative roots and at least one complex root. Assume all $b_i$ are positive and $a_i$ are real numbers.
Attempt: Using successively Rolle's Theorem we can prove that it admits at most $2^{n-1}$ nonnegative roots.