Why doesn't FullSimplify get rid of the common factor in this expression?

FullSimplify does not get rid of the z factor in the following expression:

FullSimplify[(E^(-I x) y z + (1 + E^(I y)) (x + y) z)/z, z != 0]

Upon removing a single piece of the expression, such as the first y factor or the 1+ in the brackets, the simplification suddenly succeeds. As far as I can tell it is only this special combination of variables that fails.

Is there any obvious problem with simplifying this?
Is there a way to tweak the FullSimplify function to make it succeed? (this is just a part of a longer expression - and I want to avoid looking for such problems manually).

I'm using Mathematica version 9.0.0.0 on Win7 64 bit.

• How about Simplify[(E^(-I x) y z + (1 + E^(I y)) (x + y) z)/z // Factor] ? – b.gates.you.know.what Jan 27 '13 at 16:04
• Maybe FullSimplify[Expand@(E^(-I x) y z + (1 + E^(I y)) (x + y) z)/z, z != 0]? – user1066 Jan 27 '13 at 16:37
• Or FullSimplify[Cancel[(E^(-I x) y z + (1 + E^(I y)) (x + y) z)/z], z != 0]. What I find interesting is that if you use e.g. (x+a) instead of (x+y) in your original expression, it seems to work directly... – Pinguin Dirk Jan 27 '13 at 16:42
• All these suggestions work. I'm not sure how to judge this, but the one involving Cancel seems like the most direct. As for an explanation for this strange behavior, I guess I shouldn't hold my breath. @PinguinDirk - do you want to turn your comment into an answer so I can accept it? – Joe Jan 28 '13 at 7:23