Question:
If \[ x+y+z=0 \] find:
\[ (x+y)^3+(y+z)^3+(z+x)^3 \]
Solution:
Since \[ x+y+z=0 \]
\[ x+y=-z \]
\[ y+z=-x \]
\[ z+x=-y \]
Therefore,
\[ (x+y)^3+(y+z)^3+(z+x)^3 \]
\[ =(-z)^3+(-x)^3+(-y)^3 \]
\[ =-(x^3+y^3+z^3) \]
Now using identity:
\[ x^3+y^3+z^3-3xyz = (x+y+z)(x^2+y^2+z^2-xy-yz-zx) \]
Since \[ x+y+z=0 \]
\[ x^3+y^3+z^3=3xyz \]
Therefore,
\[ -(x^3+y^3+z^3) = -3xyz \]
Hence,
\[ (x+y)^3+(y+z)^3+(z+x)^3 = -3xyz \]