positives and negatives

[squreshi]

If a, b, c, and d are integers and ab2c3d4 > 0, which of the following must be positive?

I. a2cd
II. bc4d
III. a3c3d2

I only
II only
III only
I and III
I, II, and III

I don't understand how do solve this problem. Please help

[RA]

Is the answer E?

Once you confirm the answer I will post the logic/theory I used to arrive at the answer.

[Paul]

I got (C ) III only

[RonPurewal]

ok, so first off, we know that none of the integers is zero (because the product would be zero if any of them were). so therefore, we can actually speak in terms of a dichotomy between "positive" and "negative", without having to worry about zero rearing its ugly head. that's good.

here's the story:
odd powers retain the sign of the original number.
even powers always come out positive.

THEREFORE

b^2 is guaranteed to be positive, so we don't know anything about b (because the outcome would be the same whether b were positive or negative).
d^4 is guaranteed to be positive, so we don't know anything about d (because the outcome would be the same whether d were positive or negative).

(a)(c^3) is positive, which means a and c^3 have the same sign.
because c^3 has the same sign as does c, this means that a and c are either both positive or both negative.

--

(I)
d could be either positive or negative, a fact that is itself enough to show that this whole expression can be either positive or negative.

(II)
b could be either positive or negative, a fact that is itself enough to show that this whole expression can be either positive or negative.

(III)
d^2 is positive, so you can ignore it (multiplication by a positive number has no effect on the sign of an expression).
therefore, we're really only dealing with the sign of the product (a^3)(c^3).
since odd powers retain signs, as mentioned above, we're really dealing only with the sign of the product ac.
this is positive.
therefore, III must be positive.

ans = c