verify that x^3 + y^3 + z^3 - 3xyz = 1/2(x + y + z)[(x - y)^2 + (y - z)^2 + (z - x)^2]
Answers
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)]
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Hope this will help you...
Solution:-
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)]