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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.
The expression in the question x^2(x - y) + y^2(y - x) is equivalent to (x - y)^2(x + y) for the following reason:
x^2(x - y) + y^2(y - x)
= x^2(x - y) - y^2(x - y) (taking out a common factor of -1 from the second bracket)
= (x - y)(x^2 - y^2) (taking out a common factor of (x – y))
= (x - y)(x - y)(x + y) (factoring (x^2 – y^2) using a difference of squares)
= (x - y)^2(x + y) (putting like terms together)
(x - y)^2
= x^2 - 2xy + y^2 (foiling (x - y)(x – y))
= x^2 + 2xy + y^2 – 4xy (since 2xy – 4xy = -2xy in the previous equation)
= (x + y)^2 – 4xy (factoring x^2 + 2xy + y^2 using trinomial factoring)
= 3^2- 4*1 = 9 – 4 = 5 when we have x + y = 3 and xy = 1 from both conditions 1) & 2).
Then we have (x - y)^2(x + y) = 5*3 = 15.
Therefore, C is the answer.
Answer: C
