Verify that x3 + y3 + z3 – 3xyz = 1/2 (x + y + z) [(x – y)2 + (y

1.	Verify that x3 + y3 + z3 – 3xyz = 1/2 (x + y + z) [(x – y)2 + (y – z)2 + (z – x)2 ] (OR) Verify that p3 + q3 + r3 – 3pqr = 1/2 (p + q + r) [(p – q)2 + (q – r)2 + (r – p)2 ]
Answer» Given x³ + y³ + z³ – 3xyz = 1/2 (x + y + z) [(x – y)² + (y – z)² + (z – x)²] R-H.S = 1/2 (x + y + z) [(x – y)² + (y – z)² + (z – x)² ] = 1/2 (x + y + z) [x² + y² – 2xy + y² + z² – 2yz + z² + x² – 2xz] = 1/2 (x + y + z) [2x² + 2y² + 2z² – 2xy – 2yz – 2zx] = 1/2 (x + y + z) (2) [x² + y² + z² – xy – yz – zx] = (x + y + z) (x² + y² + z² – xy – yz – zx) = L.H.S Hence proved.

Verify that x3 + y3 + z3 – 3xyz = 1/2 (x + y + z) [(x – y)2 + (y – z)2 + (z – x)2 ] (OR) Verify that p3 + q3 + r3 – 3pqr = 1/2 (p + q + r) [(p – q)2 + (q – r)2 + (r – p)2 ]

Answer»

Given x³ + y³ + z³ – 3xyz = 1/2 (x + y + z) [(x – y)² + (y – z)² + (z – x)²]

R-H.S = 1/2 (x + y + z) [(x – y)² + (y – z)² + (z – x)² ]

= 1/2 (x + y + z) [x² + y² – 2xy + y² + z² – 2yz + z² + x² – 2xz]

= 1/2 (x + y + z) [2x² + 2y² + 2z² – 2xy – 2yz – 2zx]

= 1/2 (x + y + z) (2) [x² + y² + z² – xy – yz – zx]

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

= L.H.S

Hence proved.