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Question
If x + y + z = 0, show that x3 + y3 + z3 = 3xyz.
Solution
Since, x + y + z = 0
= x + y = −z(x + y)3 = (−z)3
= x3 + y3 + 3xy(x + y) = (−z)3
= x3 + y3 + 3xy(−z) = −z3 ...[∵ x + y = −z]
= x3 + y3 − 3xyz = (−z)3
= x3 + y3 + z3 = 3xyz
Hence, if x + y + z = 0, then
x3 + y3 + z3 = 3xyz
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