Question

Verify that x³ + y³ + z³ - 3xy = 1/2 ( x+y+z) [ ( x-y)² + ( y - z )² ( z - x)²​

Answer 1

Step-by-step explanation:

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Answer 2

Solution :-

We have to verify :-

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

Solving RHS :-

= 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 ]

= 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 -zx² + yx² + y³ + yz² -xy² -y²z - xyz + zx² + zy² + z³ - xyz - yz² - z²x

= x³ + y³ + z³ -xyz - xyz - xyz

= x³ + y³ + z³ -2xyz - xyz

= x³ + y³ + z³ - 3xyz

Thus,

LHS = RHS

Hence, Proved $.$