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Given x+y+z=0
⟹x+y=−z
Cubing on both sides
(x+y)
3
=(−z)
3
⟹x
3
+y
3
+3x
2
y+3xy
2
=−z
3
⟹x
3
+y
3
+3xy(x+y)=−z
3
⟹x
3
+y
3
+3xy(−z)=−z
3
⟹x
3
+y
3
−3xyz=−z
3
⟹x
3
+y
3
+z
3
=3xyz
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