You can reduce the complexity to 1 variable by letting y=x*t and z=x*s. Then x+y+z=0 becomes x*(1+t+s)=0. I assume x is not 0 so s=-(1+t) and the expression to simplify becomes (1+ t^2 +(t+1)^2)*(1+t^5 -(1+t)^5) / (1+t^7 -(1+t)^7). (If x=0 then y+z=0 and z=-y and the denominator becomes y^7 - y^7 = zero). After applying the binomial theorem and collecting terms the numerator and denominator are 10*(t^6 + 3*t^5 +5*t^4 + 5*t^3 + 3t^2 +t) / 7*(t^6 + 3*t^5 +5*t^4 + 5*t^3 + 3t^2 +t)=10/7. So the expression always reduces to 10/7.
Fantastic!!! These type of problems are always my favorite in seeing just how they get solved. Pure genius!!!
You can reduce the complexity to 1 variable by letting y=x*t and z=x*s. Then x+y+z=0 becomes x*(1+t+s)=0. I assume
x is not 0 so s=-(1+t) and the expression to simplify becomes (1+ t^2 +(t+1)^2)*(1+t^5 -(1+t)^5) / (1+t^7 -(1+t)^7). (If x=0 then
y+z=0 and z=-y and the denominator becomes y^7 - y^7 = zero). After applying the binomial theorem and collecting terms
the numerator and denominator are 10*(t^6 + 3*t^5 +5*t^4 + 5*t^3 + 3t^2 +t) / 7*(t^6 + 3*t^5 +5*t^4 + 5*t^3 + 3t^2 +t)=10/7.
So the expression always reduces to 10/7.
Nice!
The Newton-Girard identities provide a systematic way to do this. Put
P1 = x + y + z
P2 = x^2 + y^2 + z^2
etc
E0 = 1
E1 = x + y + z All products of 1 variable
E2 = xy + yz + zx All products of 2 variables
E3 = xyz All products of 3 variables
E4 = E5 = ... = 0
The relevant identities are
P1*E0 = 1*E1
P2*E0 - P1*E1 = -2*E2
P3*E0 - P2*E1 + P1*E2 = 3*E3
P4*E0 - P3*E1 + P2*E2 - P1*E3 = -4*E4 = 0
P5*E0 - P4*E1 + P3*E2 - P2*E3 = 0
P6*E0 - P5*E1 + P4*E2 - P3*E3 = 0
P7*E0 - P6*E1 + P5*E2 - P4*E3 = 0
Here
P1=E1=0, E0=1
P2 = -2*E2
P3 = 3*E3
P4 + P2*E2 = 0
P5 + P3*E2 - P2*E3 = 0
P6 + P4*E2 - P3*E3 = 0
P7 + P5*E2 - P4*E3 = 0
Substituting for P2 and P3
P4 + (-2*E2)*E2 = 0, P4 = 2*E2^2
P5 + (3*E3)*E2 - (-2*E2)*E3 = 0, P5 = -5*E2*E3
P6 + (2*E2^2)E2 - 3*E3^2 = 0, P6 = 3*E3^2 - 2*E2^3
P7 - 5*E2^2^E3 - 2*E2^2*E3, P7 = 7*E2^2*E3
P2*P5/P7 = -2*E2*(-5*E2*E3)/(7*E2^2*E3)
= 10/7
Pretty good!
10/7
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