已知x+y=1..x的平方+Y的平方=2.....那么X的7次方+Y的7次方=？

x+y=1
(x+y)^2=1
x^2+y^2+2xy=1
xy=[1-(x^2+y^2)]/2=-1/2
x^3+y^3=(x+y)(x^2-xy+y^2)=1*(2+1/2)=5/2

x^2+y^2=2
(x^2+y^2)^2=2^2
x^4+y^4+2(xy)^2=4
x^4+y^4=7/2

x^7+y^7
=(x^3+y^3)(x^4+y^4)-x^3y^4-x^4y^3
=(5/2)/(7/2)-x^3y^3(x+y)
=35/4-1*(xy)^3
=35/4+1/8
=71/8

x^7+y^7=x^7+(2-x^2)^3(1-x)=x^7-x^6-6x^5+6x^4+12x^3-12x^2-8x+8
=(x^2-2)^3(x-1)=y^4
又x+y=1 x^2+y^2=2
y=1/2+3^(1/2)/2或者y=1/2-3^(1/2)/2
x^7+y^7=y^4
y^4=7/4+3^(1/2)或者y^4=7/4-3^(1/2)