If x + y + z = 0, then (x + y - z) 3+ (y + z - x) 3+ (z + x - y) 3= k(xyz), where k is equal to:
-24
The correct answer is -24.
If we set $a+b+c=0$, the right side becomes $0 \times (a^2+b^2+c^2-ab-bc-ca) = 0$. This leads directly to the simplified identity: if $a+b+c=0$, then $a^3+b^3+c^3 - 3abc = 0$, which means $a^3+b^3+c^3 = 3abc$. This identity is particularly useful in problems where the sum of variables is zero, as seen in this question. Understanding when and how to apply this identity can significantly simplify complex algebraic expressions and equations involving cubes.
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