已知z=cos(2π/5)+isin(2π/5),则(1+z)(1+z^2)(1+z^3)(1+z^4)=

Question

已知z=cos(2π/5)+isin(2π/5) ,则(1+z )(1+z^2)(1+z^3)(1+z^4)=

Answer 1

z=cos(2π/5)+isin(2π/5)
∴ z^5=cos(2π)+isin(2π)=1
∵ (1+z)(1+z^2)(1+z^3)(1+z^4)
=(1-z)(1+z)(1+z^2)(1+z^4)(1+z^3)/(1-z)
=(1-z^8)*(1+z^3)/(1-z) 平方差公式
=(1-z^3)*(1+z^3)/(1-z)
=(1-z^6)/(1-z) 平方差公式
=(1-z)/(1-z)
=1