# 七次方程

${\displaystyle ax^{7}+bx^{6}+cx^{5}+dx^{4}+ex^{3}+fx^{2}+gx+h=0,\,}$

${\displaystyle y(x)=ax^{7}+bx^{6}+cx^{5}+dx^{4}+ex^{3}+fx^{2}+gx+h\,}$

## 七次方程求根

${\displaystyle x^{7}+7ax^{5}+14a^{2}x^{3}+7a^{3}x+b=0\ }$

${\displaystyle x=u+v}$${\displaystyle uv+a=0}$，則以上方程化簡為 ${\displaystyle u^{7}+v^{7}+b=0}$。故 ${\displaystyle u^{7}}$${\displaystyle v^{7}}$ 皆為輔助方程的根。

${\displaystyle x_{k}=\omega _{k}{\sqrt[{7}]{y_{1}}}+\omega _{k}^{6}{\sqrt[{7}]{y_{2}}}}$     ${\displaystyle (k=1,2,\dots ,7)}$

${\displaystyle x^{7}-2x^{6}+(a+1)x^{5}+(a-1)x^{4}-ax^{3}-(a+5)x^{2}-6x-4=0}$

${\displaystyle {d=-4^{4}(4a^{3}+99a^{2}-34a+467)^{3}}}$

## 文獻

1. ^ Vasco Brattka, Kolmogorov's Superposition Theorem, Kolmogorov's heritage in mathematics, Springer
2. ^ V.I. Arnold, From Hilberts Superposition Problem to Dynamical Systems: 4
3. ^ Weisstein, Eric W. "Cyclic Pentagon." From MathWorld--A Wolfram Web Resource. [1]
4. ^ Weisstein, Eric W. "Cyclic Hexagon." From MathWorld--A Wolfram Web Resource. [2]