Question

Solve This : Mods And Stars... Verify that:


\begin{gathered}x {}^{3} +y {}^{3} +z {}^{3} –3xyz \\ = ( \frac{1}{2} ) (x+y+z)[(x–y) {}^{2} +(y–z) {}^{2} +(z–x) {}^{2} ]\end{gathered} x 3 +y 3 +z 3 –3xyz =( 2 1 ​ )(x+y+z)[(x–y) 2 +(y–z) 2 +(z–x) 2 ] ​

Answer 1

$\: \huge\fbox\pink{Søluctiøn}$

$\: RHS = \frac{1}{2} (x + y + z)[ {(x - y)}^{2} + {(y - z)}^{2} + {(z - x)}^{2} ]$

$= \frac{1}{2} (x + y + z)$

$\: [ {x}^{2} + {y}^{2} - 2xy + {y}^{2} + {z}^{2} - 2yz + {z}^{2} + {x}^{2} - 2xz ]$

$= \frac{1}{2} (x + y + z)[ {2x}^{2} + {2y}^{2} + {2z}^{2} - 2xy - 2yz - 2zx ]$

$= \frac{1}{2} (x + y + z) \times 2 \times [ {x}^{2} + {y}^{2} + {z}^{2} - xy - yz - zx ]$

$= (x + y - z)( {x}^{2} + {y}^{2} + {z}^{2} - xy - yz - zx)$

$= {x}^{3} + {y}^{3} + {z}^{3} - 3xyz$

$\: = \: LHS$

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