Question

prove : (x-y)^3+(y-z)^3+(z-x)^3=3(x-y)(y-z)(z-x)

Answer 1

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Answer 2

Answer:

Required to prove,

$(x-y)^3+(y-z)^3+(z-x)^3=3(x-y)(y-z)(z-x)$

Recall the formula

$a^3 + b^3+ c^3 -3abc = (a + b + c)(a^2 + b^2+ c^2- ab - bc - ca)$

Solution:

We have the formula

$a^3 + b^3+ c^3 -3abc = (a + b + c)(a^2 + b^2+ c^2- ab - bc - ca)$--------------(1)

Substituting the values of

a = x-y,   b = y-z and c = z-x we get

a+b+c = x-y+y-z+z-x = 0

Substituting the value of a+b+c in equation (1) we get

a³ + b³+c³ - 3abc = 0

a³ + b³+c³ = 3abc -------------------(2)

Substituting the value for a, b, and c in equation (2) we get

$(x-y)^3+(y-z)^3+(z-x)^3=3(x-y)(y-z)(z-x)$

Hence proved.

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