Question

x³ + y³ + z³ - 3xyz = ½( x + y + z ) [ ( x - y )² + ( y - z )² + ( z - y )² ]​

Answer 1

Answer:

= x³ + y³ + z³ - 3xyz = LHS

Step-by-step explanation:

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

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

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

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

= x[x² + y² + z² - xy - yz - zx] + y[x² + y² + z² - xy - yz - zx] + z[x² + y² + z² - xy - yz - zx]

= x³ + xy² + xz² - x²y - xyz - x²z + x²y + y³ + yz² - xy² - y²z - xyz + zx² + y²z + z³ - xyz - yz² - xz²

On simplifying,

= x³ + y³ + z³ - 3xyz = LHS

Answer 2

Answer:

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