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If x=a+b+c then what is the value of (x-a)^3 + (x-b)^3 + (x-c)^3 - 3(x-a)(x-b)(x-c) = ?

+1 vote
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If x=a+b+c then what is the value of (x-a)^3 + (x-b)^3 + (x-c)^3 - 3(x-a)(x-b)(x-c) = ?
posted Sep 12, 2018 by anonymous

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1 Answer

+1 vote

Zero


If a+b+c=0 then a^3+b^3+c^3=3abc -----> (1)
Proof
a+b+c=0 => a+b=−c => (a+b)^3=(−c)^3 => a^3+b^3−3ab(a+b)=−c^3 => a^3+b^3−3ab(−c)=−c^3 => a^3+b^3+c^3=3abc


We have
x=a+b+c ---> (2)
(x-a)^3 + (x-b)^3 + (x-c)^3 - 3(x-a)(x-b)(x-c) ----> (3) =?


If you call x-a=a, x-b=b and x-c=c then (1)=(3) => x-a+x-b+x-c=0 => 3x=a+b+c ----> (4)
Comparing (2) and (4) x=3x => x=0 =>a+b+c=0 then as per (1) the answer in zero

answer Sep 13, 2018 by Hanifa Mammadov
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