Question

if a+b+c=0 .find the value of a÷bc^3+b÷ca^3+c÷ab^3​

Answer 1

Step-by-step explanation:

x3+y3+z3−3xyz=(x+y+z)(x2+y2+z2−xy−yz−zx)

If x + y + z = 0,

x3+y3+z3−3xyz=0=>x3+y3+z3=3xyz

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.

(a+b−c)3+(c+a−b)3+(b+c−a)3=x3+y3+z3=3xyz

3xyz=3∗(−2c)∗(−2b)∗(−2a)=−24abc

Hope this helped!

(a+b−c)3+(c+a−b)3+(b+c−a)3

=(−2c)3+(−2b)3+(−2a)3 As a+b+c=0

=−8(a3+b3+c3)−8(3abc−3abc)

=−8(a3+b3+c3−3abc)−24abc

=−4(a+b+c)((a−b)2+(b−c)2+(c−a)2)−24abc

=−24abc As a+b+c=0