Video solution:

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Written solution:

By Vieta's formulas, we get that the product of the roots is $16$ and that their sum is $10.$

Given that all the roots are positive integers, we can see that the roots are $$1, 1, 2, 2, 2, 2.$$

The polynomial is therefore $$(z - 1)^2(z - 2)^4 = (z^2 - 2z + 1)$$ $$(z^4 - 8z^3 + 24z^2 - 32z + 16).$$

Calculating just the $z^3$ term, we get $$-32z^3 - 48z^3 - 8z^3 = -88z^3.$$

Thus, A is the correct answer.