一元多项式在复数域内分解成一次因式的乘积(1)x^n-C(2n,2)x^(n-1)+C(2n,4)x^(n-2)+……+(-1)^nC(2n,2n)(2)x^(2n)+C(2n,2)x^(2n-2)(x^2-1)+C(2n,4)x^(2n-4)(x^2-1)^2+……+(x^2-1)^n(3)x^(2n+1)+C(2n+!,2)x^(2n-1)(x^2-1)+C(2n+1,4)x^(2n-3)(x^2

Question

一元多项式在复数域内分解成一次因式的乘积
(1)x^n-C(2n,2)x^(n-1)+C(2n,4)x^(n-2)+……+(-1)^nC(2n,2n)
(2)x^(2n)+C(2n,2)x^(2n-2)(x^2-1)+C(2n,4)x^(2n-4)(x^2-1)^2+……+(x^2-1)^n
(3)x^(2n+1)+C(2n+!,2)x^(2n-1)(x^2-1)+C(2n+1,4)x^(2n-3)(x^2-1)^2+……+x(x^2+1)^n
其中C(2n,i)和C(2n+1,j)等表示组合数

Answer 1

这三问都是一个来源:棣莫弗(de Moivre)公式,即(cos(θ)+i·sin(θ))^m = cos(mθ)+i·sin(mθ).(第3问最后一项少了系数).1) 对m = 2n,两边取实部得:cos(θ)^(2n)-C(2n,2)·cos(θ)^(2n-2)·sin(θ)^2+...+(-1)^n·C(...