If a=b+c, then find the value of a┬│-b┬│-c┬│-3abc.
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If a+b+c=0 then what is the value of (a+b-c) ^3+ (c+a-b) ^3+ (b+c-a) ^3?
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Punith Rajendran, I love Math!
Answered May 23, 2016 · Author has 135answers and 134k answer views
[math]x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)[/math]
If x + y + z = 0,
[math]x^3+y^3+z^3-3xyz=0 => x^3+y^3+z^3=3xyz[/math]
Assume:
x = a + b - c = -2c
y = a + c - b = -2b
z = b + c - a = -2a
(Since a+b+c=0, a+b=−c, b+c=−a, a+c=−b)
We see x + y + z = a + b + c =0.
[math](a+b−c)^3+(c+a−b)^3+(b+c−a)^3=x^3+y^3+z^3=3xyz[/math]
[math]3xyz=3*(-2c)*(-2b)*(-2a)= -24abc[/math]
Hope this helped!
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3 ANSWERS

Punith Rajendran, I love Math!
Answered May 23, 2016 · Author has 135answers and 134k answer views
[math]x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)[/math]
If x + y + z = 0,
[math]x^3+y^3+z^3-3xyz=0 => x^3+y^3+z^3=3xyz[/math]
Assume:
x = a + b - c = -2c
y = a + c - b = -2b
z = b + c - a = -2a
(Since a+b+c=0, a+b=−c, b+c=−a, a+c=−b)
We see x + y + z = a + b + c =0.
[math](a+b−c)^3+(c+a−b)^3+(b+c−a)^3=x^3+y^3+z^3=3xyz[/math]
[math]3xyz=3*(-2c)*(-2b)*(-2a)= -24abc[/math]
Hope this helped!
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