Let $(c_0,...,c_n)$ be a cash flow. Suppose $c_0<0$ and $c_k \geq 0$ for all $k \geq 1$. Then, $\exists$ unique positive root $\alpha$ to the equation
$$ 0 = c_0 + c_1 z + c_2 z^2 + c_3 z^3 + ... + c_n z^n $$
and if $\sum_{k=0}^n c_k > 0 $, then show the corresponding IRR (internal rate of return) is $r = \frac{1}{\alpha}-1$
Attempt
Notice since $c_0 < 0$, then $c_0 = - |c_0|$. Therefore, our equation is equivalent to (after moving $|c_0|$ to the other side and dividing the entire equation by it which is positive since $c_0$ is never zero),
$$ 1 = \frac{c_1}{|c_0|} z + \frac{c_2}{|c_0|} z^2 + ... + \frac{c_n}{|c_0|} z^n $$
Here is where I get stuck since I dont know or I cant see how the equation above has a root that is unique. Am I on the right track?
As for the second part, since $z = \frac{1}{1+r}$ is given then once we find the unique root $\alpha$, then rearranging the equation gives $r= \frac{1}{\alpha}-1$, but that part is easy. Any help with the polynomial part?
The polynomial has a unique real positive root by Descartes' rule of signs since there is just one sign change in the sequence of coefficients, between $\,c_0\,$ and the previous non-zero $\,c_j\,$.
Additionally, if $\,\sum_{k=0}^n c_k > 0\,$ then $\,p(0)=c_0 \lt 0\,$ and $\,p(1)=\sum_{k=0}^n c_k > 0\,$, so the unique positive root must lie in the interval $\,(0,1)\,$.
[ EDIT ] Alternatively, using OP's approach, it can be noted that $\, \frac{c_1}{|c_0|} z + \frac{c_2}{|c_0|} z^2 + ... + \frac{c_n}{|c_0|} z^n\,$ is strictly increasing on $\,\mathbb{R}^+\,$, since each $\,\frac{c_k}{|c_0|} z^k\,$ is increasing. It is therefore injective, so it can take the value $\,1\,$ at most once. But the expression is $\,0\,$ at $\,z=0\,$ and tends to $\,+\infty\,$ for $\,z \to \infty\,$, so it must take all values in the range $\,(0, \infty)\,$ by continuity. In particular, it must $\,=1\,$ for some $\,z \gt 0\,$.