Official Solution: \(x^6 - y^6 =\) A. \((x^3 + y^3)(x^2 + y^2)(x - y)\)
B. \((x^3 - y^3)(x^3 - y^3)\)
C. \((x^2 + y^2)(x^2 + y^2)(x + y)(x - y)\)
D. \((x^4 + y^4)(x + y)(x - y)\)
E. \((x^3 + y^3)(x^2 + xy + y^2)(x - y)\)
An efficient way to attack this problem is to rephrase the expression in the stem. Since a sixth power is a square, we can look at the difference of sixth powers as the difference of squares, and factor:
\(x^6 - y^6 = (x^3)2 - (y^3)2 = (x^3 + y^3)(x^3 - y^3)\)
This new expression is not among the answer choices, but as a search strategy, we should focus on these factors.
Choice (A) contains \((x^3 + y^3)\), so we should determine whether the rest of (A) works out too \((x^3 - y^3)\). The terms \((x^2 + y^2)(x - y)\) look promising, since they give us \(+x^3\) and \(-y^3\), but we get cross-terms that don't cancel: \((x^2 + y^2)(x - y) = x^3 - x^2y + xy^2 - y^3\).
Choice (B) is close but not right. \((x^3 - y^3)(x^3 - y^3) = (x^3 - y^3)^2\), which also gives us cross-terms that don't cancel: \((x3 - y3)^2 = x^6 - 2x^3y^3 + y^6\). (Moreover, the sign is wrong on the \(y^6\) term.)
Let's skip to choice (E), since we see \((x^3 + y^3)\). The remaining terms multiply out as follows:
\((x^2 + xy + y^2)(x - y) = x^3 + x^2y + xy^2 - x^2y - xy^2 - y^3 = x^3 - y^3\). This is what we were looking for. We can verify that the other answer choices do not multiply out to \(x^6 - y^6\) exactly.
The correct answer is (E).
Of course, you can also plug simple numbers. Don't pick 0 for either \(x\) or \(y\), because too many terms will cancel. Also, don't pick the same number for \(x\) and \(y\), because then the result is 0 (and every answer choice gives you 0 as well). But if you pick 2 and 1, you can keep the quantities relatively small and still eliminate wrong answers.
\(2^6 - 1^6 = 64 - 1 = 63\). Our target is 63.
\((8 + 1)(4 + 1)(2 - 1) = 45\)
\((8 - 1)(8 - 1) = 49\)
\((4 + 1)(4 + 1)(2 + 1)(2 - 1) = 75\)
\((16 + 1)(2 + 1)(2 - 1) = 51\)
\((8 + 1)(4 + 2 + 1)(2 - 1) = 63\)
Answer: E
_________________