The reason all GMAT test-prep programs stress Testing Cases is that it
always works, even when you're totally stumped on what Number Properties they're testing. However, here, as MAAJ pointed out, it shouldn't be that hard to deduce the Number Properties themselves and speed through the choices:
xy>0
This means that
x and y are the same sign.
yz<0
This means that
y and z are different signs.
Learning to read these kinds of "<0" or ">0" statements in this way will really speed you up on test day. It's not a very difficult deduction to make, now, that if x and y are the same and y is different from z,
x and z are different signs.
Now, the answers:
A) xyz --> xy>0, we know that, so it all depends on z. We don't know the sign of z, so, we don't know.
Eliminate.
B) xy>0, we know that, and z^2>0 because anything^2 is always positive.
Eliminate.
C) y^2>0, by rule, and xz will be negative since they are different signs. This must be negative,
and is the answer.
D) y^2>0 and z^2>0 by rule, but we don't know the sign of x, so it depends.
Eliminate.
E) All three terms are positive by rule, so this must always be positive.
Eliminate.
On test day, we wouldn't go through all 5 choices; I just did it here to show you that a small amount of number theory can really make this type of problem very doable, and very efficient, regardless of what they ask us for or what they give us.
