<s>[INST] Long polynomial expansion with 34 roots 
This is a very tricky problem, I just need a few hints. I think the $(-x^{17})$ is also there for a specific trick. In the end if it is $ax^{17}$, I see that $a = 17 - 1 + 1 = 17$. 
Also, another possible approach is:
$$(1 + x + \cdots + x^{17})^2 = x^{17}$$
$$1 + x + \cdots + x^{17} = x^{17/2}$$
But that doesn't do much. Only hints please!
UPDATE:
$$P(x)=0\implies x\ne 1.$$
By the geometric series formula this changes to:
$$\left(\sum_{n=0}^{17}  x^n \right)^2 = x^{17} \text{ where } |x| < 1.$$
$$ \left( \frac{1 - x^{18}}{1-x}  \right)^2 = x^{17}.$$
$$(1 - x^{18})^2 = (1-x)^2(x^{17}) = x^{19} - 2x^{18} + x^{17}.$$
$$x^{36} - 2x^{18} + 1 = x^{19} - 2x^{18} + x^{17}.$$
$$x^{36} - x^{19} - x^{17} + 1 = 0.$$
$$x^{19}(x^{17} - 1) - (x^{17} - 1) = 0.$$
$$(x^{19} - 1)(x^{17} - 1) = 0.$$
With zero prod. property, we have to use roots of unity. 
$$x^{19} = 1 = e^{2\pi i*k}.$$
$$1\ne x = e^{2\pi i \cdot k/19}.$$
$$1\ne x = e^{2\pi i \cdot k/17} \space \text{for the other case}.$$
the smallest root obviously is $a_1 = 1/19, a_2 = 1/17, a_3 = 2/19, a_4 = 2/17, a_5 = 3/19$.
$$\sum a_k = \frac{6}{19} + \frac{3}{17} = \frac{102 + 57}{323} = \frac{159}{323} = \frac{m}{n}$$
$m + n = 482$.
 [/INST] From the update to the question:

The value of $|z_k| < 1$, because $(1 + x + ... + x^{17})^2 > x^{17}$.
By the geometric series formula this changes to:
$$\left(\sum_{n=0}^{17}  x^n \right)^2 = x^{17} \space \text{where} \space |x| < 1$$
$$ \left( \frac{1 - x^{18}}{1-x}  \right)^2 = x^{17}$$
$$(1 - x^{18})^2 = (1-x)^2(x^{17}) = x^{19} - 2x^{18} + x^{17}$$
$$x^{36} - 2x^{18} + 1 = x^{19} - 2x^{18} + x^{17}$$
$$x^{36} - x^{19} - x^{17} + 1 = 0$$
$$x^{19}(x^{17} - 1) - (x^{17} - 1) = 0$$
$$(x^{19} - 1)(x^{17} - 1) = 0$$
With zero prod. property, we have to use roots of unity. 
$$x^{19} = 1 = e^{2\pi i*k}$$
$$x = e^{2\pi i *k/19}$$
$$x = e^{2\pi i *k/17} \space \text{for the other case}$$
the smallest root obviously is $a_1 = 1/19, a_2 = 1/17, a_3 = 2/19, a_4 = 2/17, a_5 = 3/19$.
$$\sum a_k = \frac{6}{19} + \frac{3}{17} = \frac{102 + 57}{323} = \frac{159}{323} = \frac{m}{n}$$
$m + n = 482$.
</s>