Given that xy > 0, which of the following must also be greater than ze

Given that xy > 0, which of the following must also be greater than zero?

(A) \(x + y\)
(B) \(x^2 + y\)
(C) \((x + y)^2\)
(D) \(x^2 - y^2 + 100\)
(E) \((x - y)^2\)

Notice two things:
1. The question asks which if the following MUST be greater than zero, not COULD be greater than zero.
2. xy > 0 means that x and y have the same sign, they are either both positive or both negative (and neither of them could be zero).

Right away we can notice that \((x + y)^2\) (option C) is either \((positive + positive)^2=positive^2=positive\) or \((negative + negative )^2=negative ^2=positive\). So, \((x + y)^2\) is always greater than zero.

As for other options:

(A) \(x + y\). If both x and y are negative, then x + y = negative.

(B) \(x^2 + y\). If say x = -1 and y = -100,000, then x^2 + y = negative.

(D) \(x^2 - y^2 + 100\). If say x = 1 and y = 100,000, then x^2 - y^2 + 100 = negative.

(E) \((x - y)^2\). If x = y, then (x - y)^2 = (x - x)^2 = 0^2 = 0. Not positive.

So, only C MUST be greater than 0.

Answer: C.

Hope it helps.