运用同解关系式分解一元五次方程

风花飘飘

运用同解关系式分解一元五次方程：





X^5+A*X^4+B*X^3+C*X^2+D*X+E=0--------------（0）

x^3+a*x^2+b*x+c=0----------------(1)
x^2+m*x+n=0------------------------(2)

（0）= (1)*(2)

=(x^3+a*x^2+b*x+c)*(x^2+(((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2)*x^3)-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/(54*x^3))^(1/3)+(a^2*x^2+2*a*b*x+b^2)/(9*x^2*((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2)*x^3)-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/(54*x^3))^(1/3))+(2*a*x-3*n+2*b)/(3*x))*x+n)
= x^5+(a+m)*x^4+(b+a*m+n)*x^3+(c+b*m+a*n)*x^2+(c*m+b*n)*x+c*n=0

由同解关系式推导：
(m*x+n)*(a*x+b)^2-2*(m*x+n)^2*(a*x+b)+c^2+(m*x+n)^3=0
= m^3*x^3-2*a*m^2*x^3+a^2*m*x^3+3*m^2*n*x^2-4*a*m*n*x^2+a^2*n*x^2-2*b*m^2*x^2+2*a*b*m*x^2+3*m*n^2*x-2*a*n^2*x-4*b*m*n*x+2*a*b*n*x+b^2*m*x+n^3-2*b*n^2+b^2*n+c^2=0 （这个是一元三次方程）
令
m=
((((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2)*x^3)-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/(54*x^3))^(1/3)+(a^2*x^2+2*a*b*x+b^2)/(9*x^2*((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2)*x^3)-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/(54*x^3))^(1/3))+(2*a*x-3*n+2*b)/(3*x)))


n = (((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3)+(a^2*x^2+2*a*b*x+b^2)/(9*((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3))-(3*m*x-2*a*x-2*b)/3)


（省略……）


X^5-5*x-2=0的实数系数分解式共有4种方法。
下面是其中一种分解法（系数未用根式，非不能也！实在太繁琐是繁琐啊，还是用数值简洁啊。）：
X^5-5*x-2
=(x^3-1.773984198645598*x^2+2.595382746051033*x-3.625573102394295)*(x^2+1.773984198645598*x+0.5516369256709288)




方程组：(共4组实数解）
m^2-b-n=0-------------------------------------（A）
m*n*b-m*n^2-2=0----------------------------（B）
b*n^2-2*m+5*n=0=0--------------------------（C）
c*n+2=0---------------------------------------------（D）

方程组：(共4组实数解，代入n值）
m^2-b-(((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3)+(a^2*x^2+2*a*b*x+b^2)/(9*((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3))-(3*m*x-2*a*x-2*b)/3)=0-------------------------------------（A）
m*(((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3)+(a^2*x^2+2*a*b*x+b^2)/(9*((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3))-(3*m*x-2*a*x-2*b)/3)*b-m*(((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3)+(a^2*x^2+2*a*b*x+b^2)/(9*((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3))-(3*m*x-2*a*x-2*b)/3)^2-2=0----------------------------（B）
b*(((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3)+(a^2*x^2+2*a*b*x+b^2)/(9*((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3))-(3*m*x-2*a*x-2*b)/3)^2-2*m+5*(((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3)+(a^2*x^2+2*a*b*x+b^2)/(9*((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3))-(3*m*x-2*a*x-2*b)/3)=0--------------------------（C）
c*(((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3)+(a^2*x^2+2*a*b*x+b^2)/(9*((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3))-(3*m*x-2*a*x-2*b)/3)+2=0---------------------------------------------（D）






下楼将贴出另外3种分解式。

风花飘飘

x^5+A*X^4+B*X^3+C*X^2+D*X+E=0--------------（0）
分解为如下两个多项式的乘积
x^3+a*x^2+b*x+c=0----------------(1)
x^2+m*x+n=0------------------------(2)

（0）= (1)*(2)=(x^3+a*x^2+b*x+c)*(x^2+m*x+n)
= x^5+(a+m)*x^4+(b+a*m+n)*x^3+(c+b*m+a*n)*x^2+(c*m+b*n)*x+c*n=0

对于x^5-5*x-2=0来讲：
x^5-5*x-2=(x^3-m*x^2+b*x-2/n)  *  (x^2+m*x+n)

风花飘飘

风花飘飘

风花飘飘

x = ((c*sqrt(-(4*a^3*n^3-12*a^2*b*m*n^2+12*a*b^2*m^2*n+((-4*b^3)-27*c^2)*m^3+27*a*c^2*m^2)/(m-a)))/(2*3^(3/2)*m^2*(m-a)^2)-(2*a^3*n^3-6*a^2*b*m*n^2+a*(27*c^2*m^2+6*b^2*m^2*n)-27*c^2*m^3-2*b^3*m^3)/(54*a^3*m^3-162*a^2*m^4+162*a*m^5-54*m^6))^(1/3)+(a^2*n^2-2*a*b*m*n+b^2*m^2)/((9*a^2*m^2-18*a*m^3+9*m^4)*((c*sqrt(-(4*a^3*n^3-12*a^2*b*m*n^2+12*a*b^2*m^2*n+((-4*b^3)-27*c^2)*m^3+27*a*c^2*m^2)/(m-a)))/(2*3^(3/2)*m^2*(m-a)^2)-(2*a^3*n^3-6*a^2*b*m*n^2+a*(27*c^2*m^2+6*b^2*m^2*n)-27*c^2*m^3-2*b^3*m^3)/(54*a^3*m^3-162*a^2*m^4+162*a*m^5-54*m^6))^(1/3))-((-3*m*n)+a*n+2*b*m)/(3*(a*m-m^2))

风花飘飘

((45*sqrt(2203)*sqrt(859913)-298792*sqrt(3)*sqrt(5))^(2/3)-28*3^(1/6)*5^(1/6)*(45*sqrt(2203)*sqrt(859913)-298792*sqrt(3)*sqrt(5))^(1/3)-5501*3^(1/3)*5^(1/3))/(2*3^(7/6)*5^(7/6)*(45*sqrt(2203)*sqrt(859913)-298792*sqrt(3)*sqrt(5))^(1/3))