Answers

2015-04-25T07:10:46+05:30

This Is a Certified Answer

×
Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
xyz=1\\xy=\frac{1}{z}=z^{-1},\ \ yz=\frac{1}{x}=x^{-1},\ \ \ zx=\frac{1}{y}=y^{-1}

A=\frac{1}{1+x+y^{-1}} + \frac{1}{1+y+z^{-1}} + \frac{1}{1+z+x^{-1}}\\\\=\frac{y}{y+xy+1} + \frac{z}{z+yz+1} + \frac{x}{x+zx+1}\\\\=\frac{y(z+yz+1)+z(y+xy+1)}{(y+xy+1)(z+yz+1)}+\frac{x}{x+zx+1}\\\\=\frac{yz+y^2z+y+yz+xyz+z}{yz+y^2z+y+xyz+xy^2z+xy+z+yz+1}+\frac{x}{x+zx+1}\\\\=\frac{2yz+y^2z+y+1+z}{2yz+y^2z+2y+xy+z+2}+\frac{x}{x+zx+1}\\\\=1-\frac{xy+y+1}{2yz+y^2z+2y+xy+z+2}+\frac{x}{x+zx+1}

\\\\=1-\frac{x(2yz+y^2z+2y+xy+z+2)-(x+xz+1)(xy+y+1)}{(2yz+y^2z+2y+xy+z+2)(x+xz+1)}\\\\=1-\frac{2xyz+xy^2z +2xy+x^2y+xz+2x-x^2y-xy-x-x^2yz-xyz-xz-xy-y-1}{(2yz+y^2z+2y+xy+z+2)(x+xz+1)}\\\\=1-0

= 1
========================
you could try some quick calculations like:

let  x = 1 ,  y = 1  and z = 1         =>    A = 1/3 + 1/3 + 1/3 = 1
let  x = 1  , y = -1  and  z = -1      =>    A = 1/1 + 1/-1  + 1/1  = 1
let x = 2 , y = 1/2  and z = 1        =>    A = 1/5  + 1/2.5  + 1/2.5  = 1

So we can safely assume that the given complicated polynomial sum = 1

2 5 2
click on thanks button (azure blue) above