Complex Numbers Simplified?

Since $x+y+z = 2$ , then $z = 2 - x -y$ .

$\Rightarrow xy + z - 1 = xy + 2 - x -y -1 = xy - x - y + 1 = (x-1)(y-1)$

$\Rightarrow \dfrac {1}{xy+z-1} + \dfrac {1}{yz+x-1} + \dfrac {1}{zx+y-1}$

$\quad = \dfrac {1}{(x-1)(y-1)} + \dfrac {1}{(y-1)(z-1)} + \dfrac {1}{(z-1)(x-1)}$

$\quad = \dfrac {x-1+y-1+z-1}{(x-1)(y-1)(z-1)} = \dfrac {-1}{(x-1)(y-1)(z-1)}$

Now $(x-1)(y-1)(z-1) = xyz - (xy+yz+zx)+(x+y+z)-1$ .

And $xy+yz+zx = \frac {1}{2} [(x+y+z)^2 - (x^2+y^2+z^2)] = \frac {1}{2} (4 - 3) = \frac {1}{2}$

Therefore,

$\dfrac {1}{xy+z-1} + \dfrac {1}{yz+x-1} + \dfrac {1}{zx+y-1}$

$= \dfrac {-1}{xyz - (xy+yz+zx)+(x+y+z)-1}$

$= \dfrac {-1}{4 - \frac {1}{2} +2-1} = \dfrac {-1}{\frac {9}{2}} = -\dfrac {2}{9} \approx \boxed {-0.222}$

$xy=\frac { 4 }{ z }$

Hence first term becomes :

$\large \frac { 1 }{ \frac { 4 }{ z } +z-1 } =\frac { z }{ { z }^{ 2 }-z+4 }$

Hence our desired sum becomes :

$S=\displaystyle \sum _{ cyc }^{ }{ \frac { z }{ { z }^{ 2 }-z+4 } }$

Now $2(xy+yz+xz)={(x+y+z)}^{2}-({x}^{2}+{y}^{2}+{z}^{2})$

$\Rightarrow xy+yz+xz=\frac{1}{2}$

We assume x,y,z to be roots of a cubic polynomial hence the polynomial is given by :

${ p }^{ 3 }-(\displaystyle \sum _{ cyc }^{ }{ x } ){ p }^{ 2 }+(\displaystyle \sum _{ cyc }^{ }{ xy } )p-xyz=0$

Putting the given values we get :

$2{p}^{3}-4{p}^{2}+p-8=0$

Since z is a root hence we have :

$2{z}^{3}-4{z}^{2}+z-8=0$

Factorising we get :

$2(z-1)({z}^{2}-z+4)=9z$

Hence we have :

$\frac { 2 }{ 9 } (z-1)=\frac { z }{ { z }^{ 2 }-z+4 }$

Our desired sum becomes :

$S=\frac { 2 }{ 9 } \displaystyle \sum _{ cyc }^{ }{ (z-1) }$

Which it turns out to be :

$S=\frac { 2 }{ 9 } (x+y+z-3)=\frac{-2}{9}$

This is my problem, Complicated Complex Fractions !!!

This answer is checked to have no cyclic order differences.

x = -0.229220113606684-1.25479322434906i

y = -0.229220113606684+1.25479322434906i

z = 2.45844022721337

2

3.00000000000003

3.99999999999999

All checked correct.

0.324097828269638

-0.27316002524593-0.278842938744237i

-0.27316002524593+0.278842938744237i

Sum = -0.222222222222222 such that -2/ 9