We're asked to determine which of the expressions COULD equal zero.

So, let's set each expression equal to zero and see if we can solve it.


I. $x ⊕ y$
We get: $x ⊕ y = 0$
Apply formula to get: $x^2 - xy = 0$
Factor: $x(x - y) = 0$
If x = 1 and y = 1, then we get: $1(1 - 1) = 0$
Works!!
Check the answer choices....eliminate A and D (since they say that $x ⊕ y$ CANNOT equal 0)



II. $xy ⊕ y$
We get: $xy ⊕ y = 0$
Apply formula to get: $(xy)^2 - (xy)(y) = 0$
Simplify to get: $x^2y^2 - xy^2 = 0$
Factor: $xy^2(x-1)= 0$
If x = 1 and y = 1, then we get: $(1)(1^2)(1-1)= 0$
Works!!
Check the answer choices....eliminate C (since is say that $xy ⊕ y$ CANNOT equal 0)



III. $x ⊕ (x + y)$
We get: $x ⊕ (x + y)=0$
Apply formula to get: $x^2 - x(x+y) = 0$
Expand to get: $x^2 - x^2 - xy = 0$
Simplify to get: $-xy = 0$
Since we're told that $xy ≠ 0$, it is IMPOSSIBLE for $x ⊕ (x + y)$ to equal 0

Answer: B

Cheers,
Brent