Prove that in the polynomial $a_1 x^{n_1} + a_2 x^{n_2} + \dots + a_k x^{n_k}$, where $n_1 < n_2 < \dots < n_k$, each non-zero root has multiplicity not above $k-1$.
As I understand it, this can be proven by induction, taking $k=2$ as a base and assuming that $x^{n−1} \ne 0$, we get the expression $\dfrac{a_1}{a_2}+x^{n_2−n_1}=0$, since the number of roots of a polynomial of degree $n_2−n_1$ does not exceed $n_2−n_1$ even if multiple roots are counted taking into account the multiplicity. But I couldn’t figure out how to prove the induction transition.