我这样来解 x^5-5*x-2=0

风花飘飘

x^5-5*x-2=0与(5*x^3+5*x+y)*(y*x^2+20*x-y+10)=0是同解方程，我们这样来做：

把 (5*x^3+5*x+y)*(y*x^2+20*x-y+10)除以5*y后分解因式为：

(y+5*x^3+5*x)*(x^2*y-y+20*x+10)/(5*y)=

=(x-((sqrt(y^2-10*y+100)-10)/y))
*(x-((sqrt(27*y^2+100)/(10*3^(3/2))-y/10)^(1/3)-1/(3*(sqrt(27*y^2+100)/(10*3^(3/2))-y/10)^(1/3))))
*(x-(-(sqrt(y^2-10*y+100)+10)/y))
*(x-((-sqrt(3)*%i/2-1/2)*(sqrt(27*y^2+100)/(10*3^(3/2))-y/10)^(1/3)-(sqrt(3)*%i/2-1/2)/(3*(sqrt(27*y^2+100)/(10*3^(3/2))-y/10)^(1/3))))
*(x-((sqrt(3)*%i/2-1/2)*(sqrt(27*y^2+100)/(10*3^(3/2))-y/10)^(1/3)-(-sqrt(3)*%i/2-1/2)/(3*(sqrt(27*y^2+100)/(10*3^(3/2))-y/10)^(1/3))))
=0
从而实现以y为待定参数的根式解。

风花飘飘

(y+5*x^3+5*x)*(x^2*y-y+20*x+10)=x^5-5*x-2

y = ((sqrt(25*x^10+4*x^7-250*x^6-104*x^5-20*x^3+617*x^2+520*x+108)-5*x^5-15*x-10)/(2*x^2-2))

x^2*((sqrt(25*x^10+4*x^7-250*x^6-104*x^5-20*x^3+617*x^2+520*x+108)-5*x^5-15*x-10)/(2*x^2-2))
-((sqrt(25*x^10+4*x^7-250*x^6-104*x^5-20*x^3+617*x^2+520*x+108)-5*x^5-15*x-10)/(2*x^2-2))
+20*x+10=0= (sqrt(x^5-5*x-2)*sqrt(25*x^5+4*x^2-125*x-54)-5*x^5+25*x+10)/2

风花飘飘

(x^2-m*x-2^(2/5))*(x^3+m*x+n*x+2^(3/5))=x^5-5*x-2=0

n*x^3-m^2*x^3-4^(1/5)*x^3-m*n*x^2-4^(1/5)*m*x^2+2^(3/5)*x^2-4^(1/5)*n*x-8^(1/5)*m*x+5*x=0

x = (sqrt((m^2+4^(6/5))*n^2+(-2*4^(1/5)*m^2+(4*8^(1/5)-2^(8/5))*m-4^(7/5)-20)*n-4*8^(1/5)*m^3+(4^(2/5)+20)*m^2+(-4^(6/5)*8^(1/5)-2^(8/5)*4^(1/5))*m+5*4^(6/5)+2^(6/5))+m*n+4^(1/5)*m-2^(3/5))/(2*n-2*m^2-2*4^(1/5))

m=((sqrt(x^2*(n^2-2^(7/5)*n+2^(4/5)+20)+x*(2^(8/5)*n+4)+x^4*(4*n-2^(12/5))+2^(13/5)*x^3+2^(6/5))+x*(-n-2^(2/5))-2^(3/5))/(2*x^2))
n=((x^2*(m^2+2^(2/5))+x*(2^(2/5)*m-2^(3/5))+2^(3/5)*m-5)/(-x*m+x^2-2^(2/5)))

风花飘飘

(x^2-m*x-2^(2/5))*(x^3+m*x+n*x+2^(3/5))=x^5-5*x-2=0

n*x^3-m^2*x^3-4^(1/5)*x^3-m*n*x^2-4^(1/5)*m*x^2+2^(3/5)*x^2-4^(1/5)*n*x-8^(1/5)*m*x+5*x=0
5*x-m*x^4+n*x^3+m*x^3-4^(1/5)*x^3-m*n*x^2-m^2*x^2+2^(3/5)*x^2-4^(1/5)*n*x-8^(1/5)*m*x-4^(1/5)*m*x=0

x = (sqrt((m^2+4^(6/5))*n^2+(-2*4^(1/5)*m^2+(4*8^(1/5)-2^(8/5))*m-4^(7/5)-20)*n-4*8^(1/5)*m^3+(4^(2/5)+20)*m^2+(-4^(6/5)*8^(1/5)-2^(8/5)*4^(1/5))*m+5*4^(6/5)+2^(6/5))+m*n+4^(1/5)*m-2^(3/5))/(2*n-2*m^2-2*4^(1/5))

m=((sqrt(x^2*(n^2-2^(7/5)*n+2^(4/5)+20)+x*(2^(8/5)*n+4)+x^4*(4*n-2^(12/5))+2^(13/5)*x^3+2^(6/5))+x*(-n-2^(2/5))-2^(3/5))/(2*x^2))
n=((x^2*(m^2+2^(2/5))+x*(2^(2/5)*m-2^(3/5))+2^(3/5)*m-5)/(-x*m+x^2-2^(2/5)))

风花飘飘

(n-m^2-2^(2/5))*x^2+(-m*n-2^(2/5)*m+2^(3/5))*x-2^(2/5)*n-2^(3/5)*m+5=0