Verify that3xyz =-(x +x-y) (y - z)+(z-x)

1.	Verify that3xyz =-(x +x-y) (y - z)+(z-x)
Answer» First We take R.H.S & use the Formula [( a-b)²= a²+b²-2ab] & simplify it then R.H.S becomes equal to L.H.S R.H.S ⇒ 1/2×(x + y + z) (x²+ y²-2xy +y²+ z²-2yz+x²+z²-2xz) [( a-b)²= a²+b²-2ab] ⇒ 1/2×(x + y + z) (2x²+ 2y²+2z²-2xy -2yz-2xz) ⇒ 1/2×(x + y + z) 2(x² + y²+ z² – xy – yz – xz) =(x + y + z) (x² + y²+ z² – xy – yz – xz) = x³+y³+z³-3xyz= L.H.S We know that, [x³+ y³ + z³– 3xyz = (x + y + z)(x²+ y² + z² – xy – yz – xz)] L.H.S = R.H.S [x³+ y³ + z³– 3xyz = (x + y + z)(x²+ y² + z² – xy – yz – xz)] Please let me know if you have any doubt and like my solution as well

Answer»

First We take R.H.S & use the Formula [( a-b)²= a²+b²-2ab] & simplify it then R.H.S becomes equal to L.H.S

R.H.S

⇒ 1/2×(x + y + z) (x²+ y²-2xy +y²+ z²-2yz+x²+z²-2xz)

[( a-b)²= a²+b²-2ab]

⇒ 1/2×(x + y + z) (2x²+ 2y²+2z²-2xy -2yz-2xz)

⇒ 1/2×(x + y + z) 2(x² + y²+ z² – xy – yz – xz)

=(x + y + z) (x² + y²+ z² – xy – yz – xz)

= x³+y³+z³-3xyz= L.H.S

We know that,

[x³+ y³ + z³– 3xyz = (x + y + z)(x²+ y² + z² – xy – yz – xz)]

L.H.S = R.H.S

[x³+ y³ + z³– 3xyz = (x + y + z)(x²+ y² + z² – xy – yz – xz)]

Please let me know if you have any doubt and like my solution as well

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