If (x+1/x)^2=3

Question

then x^72+x^66+x^54+x^36+x^6+1=??
Plz help so confuse

Answer 1

Just solve
(x+1/x)^2=3
x^2+2+1/x^2 = 3
x^2 = (1±√3 i)/2
A little playing around will show you that that is the same as
x = ±i^(1/6)
So, since the polynomial is just a bunch of powers of x^6 = i, we have

x^72+x^66+x^54+x^36+x^6+1
= i^12 + i^11 + i^9 + i^6 + i + 1
= 1-i+i-1+i+1
= 1+i

So, just plug that into the polynomial and you have