Question

设ω=cos(2π/5) + i×sin(2π/5),请写出以ω,ω^3,ω^7,ω^9为根的方程

Answer 1

ω=cos(2π/5) + i×sin(2π/5),

说明1,w,w^2,w^3,w^4是x^5-1的五个根

所以w^5=1,且(x-1)(x-w)(x-w^3)(x-w^2)(x-w^4)=x^5-1

=（x-1）(x4-x3+x2-x+1)

所以 (x-w)(x-w^3)(x-w^2)(x-w^4)

=(x^5-1)/(x-1)

=x4-x3+x2-x+1

w^7=w^2,w^9=w^4

所以(x-w)(x-w^3)(x-w^7)(x-w^9)

=(x-w)(x-w^3)(x-w^2)(x-w^4)

=x4-x3+x2-x+1