【独乐乐，不如众乐乐，解5次方程“降次成功！”渴求有团队力量的加入！】

风花飘飘

10次周转，才转了1次。………………………………………………

风花飘飘

第一转的结果记在这里：
(-31*u-30*r)*v^4+(-62*u^2-120*r*u-4*r*s-56*r^2)*v^3+(-62*u^3-180*r*u^2+(-12*r*s-168*r^2)*u+6*r*s^2-18*r^2*s-48*r^3)*v^2+(-31*u^4-120*r*u^3+(-12*r*s-168*r^2)*u^2+(12*r*s^2-36*r^2*s-96*r^3)*u-4*r*s^3+18*r^2*s^2-28*r^3*s-16*r^4)*v-30*r*u^4+(-4*r*s-56*r^2)*u^3+(6*r*s^2-18*r^2*s-48*r^3)*u^2+(-4*r*s^3+18*r^2*s^2-28*r^3*s-16*r^4)*u+r*s^4-6*r^2*s^3+14*r^3*s^2-15*r^4*s+30*r-38=0

风花飘飘

【共根判定解法】若x^3+y*x+m=0与x^2+n=0有共根，则n*y^2-2*n^2*y+m^2+n^3=0成立。
A=(x-x2)*(x-x3)=
(x+r-s+u+v)*(x+r+u+v) = x^2+(2*v+2*u-s+2*r)*x+v^2+(2*u-s+2*r)*v+u^2+(2*r-s)*u-r*s+r^2
B=(x-x3)*(x-x4)*(x-x5)=
(x+r+u+v)*(x-u)*(x-v) = x^3+r*x^2+(-v^2+(-u-r)*v-u^2-r*u)*x+u*v^2+(u^2+r*u)*v
则有A与B有共根。
M=u*v^2+(u^2+r*u)*v
N=(2*v+2*u-s+2*r)*x+v^2+(2*u-s+2*r)*v+u^2+(2*r-s)*u-r*s+r^2
Y=r*x-v^2+(-u-r)*v-u^2-r*u=(n^2+m*(-n)^(1/2))/n

则n*y^2-2*n^2*y+m^2+n^3=0成立，

Y^2= r^2*x^2+(-2*r*v^2+(-2*r*u-2*r^2)*v-2*r*u^2-2*r^2*u)*x+v^4+(2*u+2*r)*v^3+(3*u^2+4*r*u+r^2)*v^2+(2*u^3+4*r*u^2+2*r^2*u)*v+u^4+2*r*u^3+r^2*u^2

N^2=
(4*v^2+(8*u-4*s+8*r)*v+4*u^2+(8*r-4*s)*u+s^2-4*r*s+4*r^2)*x^2+(4*v^3+(12*u-6*s+12*r)*v^2+(12*u^2+(24*r-12*s)*u+2*s^2-12*r*s+12*r^2)*v+4*u^3+(12*r-6*s)*u^2+(2*s^2-12*r*s+12*r^2)*u+2*r*s^2-6*r^2*s+4*r^3)*x+v^4+(4*u-2*s+4*r)*v^3+(6*u^2+(12*r-6*s)*u+s^2-6*r*s+6*r^2)*v^2+(4*u^3+(12*r-6*s)*u^2+(2*s^2-12*r*s+12*r^2)*u+2*r*s^2-6*r^2*s+4*r^3)*v+u^4+(4*r-2*s)*u^3+(s^2-6*r*s+6*r^2)*u^2+(2*r*s^2-6*r^2*s+4*r^3)*u+r^2*s^2-2*r^3*s+r^4

M^2=
= u^2*v^4+(2*u^3+2*r*u^2)*v^3+(u^4+2*r*u^3+r^2*u^2)*v^2

N^3=
(8*v^3+(24*u-12*s+24*r)*v^2+(24*u^2+(48*r-24*s)*u+6*s^2-24*r*s+24*r^2)*v+8*u^3+(24*r-12*s)*u^2+(6*s^2-24*r*s+24*r^2)*u-s^3+6*r*s^2-12*r^2*s+8*r^3)*x^3+(12*v^4+(48*u-24*s+48*r)*v^3+(72*u^2+(144*r-72*s)*u+15*s^2-72*r*s+72*r^2)*v^2+(48*u^3+(144*r-72*s)*u^2+(30*s^2-144*r*s+144*r^2)*u-3*s^3+30*r*s^2-72*r^2*s+48*r^3)*v+12*u^4+(48*r-24*s)*u^3+(15*s^2-72*r*s+72*r^2)*u^2+(-3*s^3+30*r*s^2-72*r^2*s+48*r^3)*u-3*r*s^3+15*r^2*s^2-24*r^3*s+12*r^4)*x^2+(6*v^5+(30*u-15*s+30*r)*v^4+(60*u^2+(120*r-60*s)*u+12*s^2-60*r*s+60*r^2)*v^3+(60*u^3+(180*r-90*s)*u^2+(36*s^2-180*r*s+180*r^2)*u-3*s^3+36*r*s^2-90*r^2*s+60*r^3)*v^2+(30*u^4+(120*r-60*s)*u^3+(36*s^2-180*r*s+180*r^2)*u^2+(-6*s^3+72*r*s^2-180*r^2*s+120*r^3)*u-6*r*s^3+36*r^2*s^2-60*r^3*s+30*r^4)*v+6*u^5+(30*r-15*s)*u^4+(12*s^2-60*r*s+60*r^2)*u^3+(-3*s^3+36*r*s^2-90*r^2*s+60*r^3)*u^2+(-6*r*s^3+36*r^2*s^2-60*r^3*s+30*r^4)*u-3*r^2*s^3+12*r^3*s^2-15*r^4*s+6*r^5)*x+v^6+(6*u-3*s+6*r)*v^5+(15*u^2+(30*r-15*s)*u+3*s^2-15*r*s+15*r^2)*v^4+(20*u^3+(60*r-30*s)*u^2+(12*s^2-60*r*s+60*r^2)*u-s^3+12*r*s^2-30*r^2*s+20*r^3)*v^3+(15*u^4+(60*r-30*s)*u^3+(18*s^2-90*r*s+90*r^2)*u^2+(-3*s^3+36*r*s^2-90*r^2*s+60*r^3)*u-3*r*s^3+18*r^2*s^2-30*r^3*s+15*r^4)*v^2+(6*u^5+(30*r-15*s)*u^4+(12*s^2-60*r*s+60*r^2)*u^3+(-3*s^3+36*r*s^2-90*r^2*s+60*r^3)*u^2+(-6*r*s^3+36*r^2*s^2-60*r^3*s+30*r^4)*u-3*r^2*s^3+12*r^3*s^2-15*r^4*s+6*r^5)*v+u^6+(6*r-3*s)*u^5+(3*s^2-15*r*s+15*r^2)*u^4+(-s^3+12*r*s^2-30*r^2*s+20*r^3)*u^3+(-3*r*s^3+18*r^2*s^2-30*r^3*s+15*r^4)*u^2+(-3*r^2*s^3+12*r^3*s^2-15*r^4*s+6*r^5)*u-r^3*s^3+3*r^4*s^2-3*r^5*s+r^6


即：
n*y^2-2*n^2*y+m^2+n^3=
(8*v^3+(24*u-12*s+24*r)*v^2+(24*u^2+(48*r-24*s)*u+6*s^2-24*r*s+26*r^2)*v+8*u^3+(24*r-12*s)*u^2+(6*s^2-24*r*s+26*r^2)*u-s^3+6*r*s^2-13*r^2*s+10*r^3)*x^3+(12*v^4+(48*u-24*s+44*r)*v^3+(72*u^2+(136*r-72*s)*u+15*s^2-70*r*s+65*r^2-8)*v^2+(48*u^3+(136*r-72*s)*u^2+(30*s^2-142*r*s+134*r^2-16)*u-3*s^3+30*r*s^2+(8-71*r^2)*s+46*r^3-16*r)*v+12*u^4+(44*r-24*s)*u^3+(15*s^2-70*r*s+65*r^2-8)*u^2+(-3*s^3+30*r*s^2+(8-71*r^2)*s+46*r^3-16*r)*u-3*r*s^3+(15*r^2-2)*s^2+(8*r-25*r^3)*s+13*r^4-8*r^2)*x^2+(8*v^5+(36*u-16*s+34*r)*v^4+(70*u^2+(130*r-62*s)*u+12*s^2-60*r*s+60*r^2-8)*v^3+(70*u^3+(194*r-93*s)*u^2+(36*s^2-180*r*s+180*r^2-24)*u-3*s^3+36*r*s^2+(12-87*r^2)*s+56*r^3-24*r)*v^2+(36*u^4+(130*r-62*s)*u^3+(36*s^2-180*r*s+180*r^2-24)*u^2+(-6*s^3+72*r*s^2+(24-176*r^2)*s+114*r^3-48*r)*u-6*r*s^3+(36*r^2-4)*s^2+(24*r-58*r^3)*s+28*r^4-24*r^2)*v+8*u^5+(34*r-16*s)*u^4+(12*s^2-60*r*s+60*r^2-8)*u^3+(-3*s^3+36*r*s^2+(12-87*r^2)*s+56*r^3-24*r)*u^2+(-6*r*s^3+(36*r^2-4)*s^2+(24*r-58*r^3)*s+28*r^4-24*r^2)*u-3*r^2*s^3+(12*r^3-4*r)*s^2+(12*r^2-15*r^4)*s+6*r^5-8*r^3)*x+2*v^6+(10*u-4*s+10*r)*v^5+(24*u^2+(44*r-18*s)*u+3*s^2-18*r*s+21*r^2-2)*v^4+(32*u^3+(86*r-35*s)*u^2+(12*s^2-68*r*s+78*r^2-8)*u-s^3+12*r*s^2+(4-33*r^2)*s+24*r^3-8*r)*v^3+(24*u^4+(86*r-35*s)*u^3+(18*s^2-101*r*s+116*r^2-12)*u^2+(-3*s^3+36*r*s^2+(12-97*r^2)*s+70*r^3-24*r)*u-3*r*s^3+(18*r^2-2)*s^2+(12*r-31*r^3)*s+16*r^4-12*r^2)*v^2+(10*u^5+(44*r-18*s)*u^4+(12*s^2-68*r*s+78*r^2-8)*u^3+(-3*s^3+36*r*s^2+(12-97*r^2)*s+70*r^3-24*r)*u^2+(-6*r*s^3+(36*r^2-4)*s^2+(24*r-62*r^3)*s+32*r^4-24*r^2)*u-3*r^2*s^3+(12*r^3-4*r)*s^2+(12*r^2-15*r^4)*s+6*r^5-8*r^3)*v+2*u^6+(10*r-4*s)*u^5+(3*s^2-18*r*s+21*r^2-2)*u^4+(-s^3+12*r*s^2+(4-33*r^2)*s+24*r^3-8*r)*u^3+(-3*r*s^3+(18*r^2-2)*s^2+(12*r-31*r^3)*s+16*r^4-12*r^2)*u^2+(-3*r^2*s^3+(12*r^3-4*r)*s^2+(12*r^2-15*r^4)*s+6*r^5-8*r^3)*u-r^3*s^3+(3*r^4-2*r^2)*s^2+(4*r^3-3*r^5)*s+r^6-2*r^4=0

((n^2+m*(-n)^(1/2))/n)^2*n-2*n^2*((n^2+m*(-n)^(1/2))/n)+m^2+n^3=0

风花飘飘

每辗转一次最后得到一个以或r,或s,或u,或v视为未知数的4次方程。

继续降次……

风花飘飘

对于y^5-5*y-2=0来讲，我们不妨这样设：


X1= = (2*(-3*s^2+5*r*s-5*r^2-7*n^2)^(1/2)-3*s+6*r)/7

X2== -(2*(-3*s^2+5*r*s-5*r^2-7*n^2)^(1/2)-3*s-r)/7

X3=-(2*(-3*s^2+5*r*s-5*r^2-7*n^2)^(1/2)+4*s-r)/7

X4= ((-3*s^2+5*r*s-5*r^2-7*n^2)^(1/2)+2*s-4*r+7*n)/7

X5=((-3*s^2+5*r*s-5*r^2-7*n^2)^(1/2)+2*s-4*r-7*n)/7

用3个参数来演绎“根情”。

这样来设，我们可以直接去掉4次、3次项！！

表达式
(x-((2*(-3*s^2+5*r*s-5*r^2-7*n^2)^(1/2)-3*s+6*r)/7))*(x-(-(2*(-3*s^2+5*r*s-5*r^2-7*n^2)^(1/2)-3*s-r)/7))*(x-(-(2*(-3*s^2+5*r*s-5*r^2-7*n^2)^(1/2)+4*s-r)/7))*(x-(((-3*s^2+5*r*s-5*r^2-7*n^2)^(1/2)+2*s-4*r+7*n)/7))*(x-(((-3*s^2+5*r*s-5*r^2-7*n^2)^(1/2)+2*s-4*r-7*n)/7))

= (16807*x^5+sqrt(-3*s^2+5*r*s-5*r^2-7*n^2)*((882*s^2+5390*r*s-5390*r^2-5488*n^2)*x^2+(-504*s^3+378*r*s^2+(1890*r^2-4704*n^2)*s-1260*r^3+9408*n^2*r)*x-186*s^4+452*r*s^3+(-362*r^2-3192*n^2)*s^2+(3752*n^2*r-180*r^3)*s+90*r^4-3752*n^2*r^2-3136*n^4)+(-980*s^3+4165*r*s^2+(-6615*r^2-13720*n^2)*s+4410*r^3+27440*n^2*r)*x^2+(2961*s^4-8890*r*s^3+(13195*r^2+5880*n^2)*s^2+(3920*n^2*r-8610*r^3)*s+4305*r^4-3920*n^2*r^2-5488*n^4)*x-708*s^5+3937*r*s^4+(1120*n^2-9210*r^2)*s^3+(11630*r^3-840*n^2*r)*s^2+(-8235*r^4-4200*n^2*r^2+1568*n^4)*s+3294*r^5+2800*n^2*r^3-3136*n^4*r)/16807

风花飘飘

【设】：
X1=2*r-s+2*m
X2=-r+s-2*m
X3=-r-2*m
X4=m+n
X5=m-n
m = (((-3*s^2+5*r*s-5*r^2-7*n^2)^(1/2)+2*s-4*r)/7)

风花飘飘

【设】：
X1=2*r-s+2*m
X2=-r+s-2*m
X3=-r-2*m
X4=m+n
X5=m-n
m = (((-3*s^2+5*r*s-5*r^2-7*n^2)^(1/2)+2*s-4*r)/7)

风花飘飘

【设】：
X1=2*r-s+2*m
X2=-r+s-2*m
X3=-r-2*m
X4=m+n
X5=m-n
m = (((-3*s^2+5*r*s-5*r^2-7*n^2)^(1/2)+2*s-4*r)/7)

风花飘飘

表达式
(x-2*r+s-2*m)*(x+r-s+2*m)*(x+r+2*m)*(x-m-n)*(x-m+n)=0 = x^5+(-s^2+(3*r+4*m)*s-3*r^2-8*m*r-n^2-7*m^2)*x^3+(-r*s^2+(3*r^2+4*m*r)*s-2*r^3-4*m*r^2-2*m*n^2+2*m^3)*x^2+((2*m*r+n^2+3*m^2)*s^2+(-6*m*r^2+(-3*n^2-17*m^2)*r-4*m*n^2-12*m^3)*s+4*m*r^3+(3*n^2+17*m^2)*r^2+(8*m*n^2+24*m^3)*r+4*m^2*n^2+12*m^4)*x+((n^2-m^2)*r+2*m*n^2-2*m^3)*s^2+((3*m^2-3*n^2)*r^2+(10*m^3-10*m*n^2)*r-8*m^2*n^2+8*m^4)*s+(2*n^2-2*m^2)*r^3+(10*m*n^2-10*m^3)*r^2+(16*m^2*n^2-16*m^4)*r+8*m^3*n^2-8*m^5=0

表达式 (x-2*r+s-2*m)*(x+r-s+2*m)*(x+r+2*m) = x^3+2*m*x^2+(-s^2+(3*r+4*m)*s-3*r^2-8*m*r-4*m^2)*x+(-r-2*m)*s^2+(3*r^2+10*m*r+8*m^2)*s-2*r^3-10*m*r^2-16*m^2*r-8*m^3
表达式 (x-2*r+s-2*m)*(x+r-s+2*m)*(x-m-n) = x^3+(-r-n-m)*x^2+(-s^2+(3*r+4*m)*s-2*r^2+(n-5*m)*r-4*m^2)*x+(n+m)*s^2+((-3*n-3*m)*r-4*m*n-4*m^2)*s+(2*n+2*m)*r^2+(6*m*n+6*m^2)*r+4*m^2*n+4*m^3
表达式 (x-2*r+s-2*m)*(x+r-s+2*m)*(x-m+n) = x^3+(-r+n-m)*x^2+(-s^2+(3*r+4*m)*s-2*r^2+(-n-5*m)*r-4*m^2)*x+(m-n)*s^2+((3*n-3*m)*r+4*m*n-4*m^2)*s+(2*m-2*n)*r^2+(6*m^2-6*m*n)*r-4*m^2*n+4*m^3

任在深

简直是在受罪！？