I expect you are over thinking it.
if a+b+c=0 and none are 0 (and all are reak) then one must be neg, one pos and the other either one.
So when you multiply them together the answer could be positive or negative
a/|a|= plus or minus 1 etc
so maybe you are looking for all the possible combinations of
Or maybe you can discount some of those answers off. Think about it a bit.
Anyway, I know someone else is also answering this right now.
There are 2 cases for this (negative, negative, positive) or (positive, positive, negative).
For this problem, order doesn't matter.
negative, negative, positive
a and b are negative, and c is positive.
a/|a| would be -1 since a is negative.
b/|b| would be -1 since b is negative.
c/|c| would be 1 since c is positive.
abc/|abc| would be 1, since negative * negative * positive is positive.
-1 - 1 + 1 + 1 = 0
positive, positive, negative
a and b are positive, and c is negative.
a/|a| would be 1 since a is positive.
b/|b| would be 1 since b is positive.
c/|c| would be -1 since c is negative.
abc/|abc| would be -1, since positive * positive * negative is negative.
1 + 1 - 1 - 1 = 0
So I think there is only one value, 0.