I see what I was missing now with problem 43 : the ability to read the question carefully!
I hate it when that happens.
If we look at the equation y(x)=x^n - ( the sum )
At x=0, y(0) = -a_0
At x=1, y(1) = 1 - ( the sum of a_i )
So for I: a_0 > 0 implies y(0) < 0 and the sum < 1 implies
y(1) > 0 . We can then apply the intermediate value theorem.
For II: a_0 > implies y(0) < 0 and the sum >1 implies y(1) < 0.
So there is no guarantee here.
For III: a_0 < 0 implies y(0) > 0 and the sum > 1 implies y(1) < 0.
Apply the intermediate value theorem.
So choice (E) is correct.