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If x+y+z=0, show that x^3+y^3+z^3=3xyz.

Question

If $x + y + z = 0$ , show that $x^{3} + y^{3} + z^{3} = 3 x yz$ .

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Noah Discussed

x + y + z = 0

, show that

x^{3} + y^{3} + z^{3} = 3 x yz

6 mins ago

Discuss this question LIVE

6 mins ago

Text solutionVerified

We know that

x^{3} + y^{3} + z^{3} - 3 x yz = (x + y + z) (x^{2} + y^{2} + z^{2} - x y - yz - z x)

Given

x + y + z = 0

x^{3} + y^{3} + z^{3} - 3 x yz = (0) (x^{2} + y^{2} + z^{2} - x y - yz - z x)

\Rightarrow x^{3} + y^{3} + z^{3} - 3 x yz = 0

\Rightarrow x^{3} + y^{3} + z^{3} = 3 x yz

Hence proved.

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