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Mathematics

Verify that x3 + y3 +z3 - 3xyz = 12\dfrac{1}{2}(x + y + z)[(x - y)2 + (y - z)2 + (z - x)2]

Polynomials

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Answer

R.H.S

= 12\dfrac{1}{2}(x + y + z)[(x - y)2 + (y - z)2 + (z - x)2]

= 12\dfrac{1}{2}(x + y + z)[(x2 + y2 - 2xy) + (y2 + z2 - 2yz) + (z2 + x2 - 2zx)] [∵ (a - b)2 = (a)2 + (b)2 - 2ab]

= 12\dfrac{1}{2}(x + y + z)[2x2 + 2y2 + 2z2 - 2xy - 2yz - 2zx]

= 12\dfrac{1}{2}(x + y + z) 2[x2 + y2 + z2 - xy - yz - zx]

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

= x3 + y3 +z3 - 3xyz [∵ x3 + y3 +z3 - 3xyz = (x + y + z)(x2 + y2 + z2 - xy - yz - zx)]

Hence, L.H.S = R.H.S

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