Ta có: \(x+y+z=0\Rightarrow x+y=-z\)
\(x+y+z=0\Rightarrow\left(x+y+z\right)^3=0\)
\(\Leftrightarrow\left(x+y\right)^3+3\left(x+y\right)^2z+3\left(x+y\right)z^2+z^3=0\)\(\Leftrightarrow\left(x+y\right)^3+3\left(x+y\right)z\left(x+y+z\right)+c^3=0\)
\(\Leftrightarrow\left(x+y\right)^3+c^3=0\) ( vì x + y+z =0)
\(\Leftrightarrow x^3+3x^2y+3xy^2+y^3+z^3=0\)
\(\Leftrightarrow x^3+y^3+z^3+3xy\left(x+y\right)=0\)
\(\Leftrightarrow x^2+y^2+z^2+3xy\left(-z\right)=0\)
\(\Leftrightarrow x^3+y^3+z^3-3xyz=0\)
\(\Leftrightarrow x^3+y^3+z^3=3xyz\) ( đpcm)
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