Rational Number Proof

I'll prove b):
$ \begin{aligned} (-1) f(z) =(-1) (c_0 + c_1 z+ c_2 z^2 + \ldots + & c_k z^k + \ldots) \\ a_1 z f(z) = a_1 (c_0 z + c_1 z^2 + \ldots + & c_{k-1} z^k + \ldots) \\ \vdots \\ a_k z^k f(z) = a_k & (c_0 z^k + \ldots) \end{aligned} $

If we add all of above equations we'll obtain:
$ \begin{aligned} & (-1+a_1 z + a_2 z^2 + \ldots + a_k z^k) f(z) = \\ & (- c_0) + (-c_1 + a_1 c_0) z +(-c_2 + a_1 c_1 + a_2 c_0) z^2 + \ldots \\ & +(-c_{k-1}+ a_1 c_{k-2} + \ldots + a_{k-1} c_0) z^{k-1} \end{aligned} $