范德蒙方法解三次方程之 U+V 代数运算笔记
作者:互联网
设 x3 + px + q = 0 的根为 x1、x2、x3。即 (x - x1)(x - x2)(x - x3) = x3 + px + q = 0,展开可得:
x1 + x2 + x3 = 0 ①
x1x2 + x1x3 + x2x3 = p ②
x1x2x3 = -q ③
x3 = 1 的三个根 1、ω、ω2 满足 ω3 = 1 和 1 + ω + ω2 = 0
于是:
3x1 = (x1 + x2 + x3) + (x1 + ωx2 + ω2x3) + (x1 + ω2x2 + ωx3)
3x2 = (x1 + x2 + x3) + ω2(x1 + ωx2 + ω2x3) + ω(x1 + ω2x2 + ωx3)
3x3 = (x1 + x2 + x3) + ω(x1 + ωx2 + ω2x3) + ω2(x1 + ω2x2 + ωx3)
记 u = x1 + ωx2 + ω2x3,v = x1 + ω2x2 + ωx3,则
3x1 = u + v
3x2 = ω2u + ωv
3x2 = ωu + ω2v
标签:代数,2x2,笔记,3x2,x2,x3,2x3,x1,德蒙 来源: https://www.cnblogs.com/readalps/p/15413125.html