X的平方=X+1 Y的平方=Y+1 Y+X=1 X的五次方+Y的五次方=?

x^5+y^5
=x^2*x^2*x+y^2*y^2*y
=(x+1)^2*x+(y+1)^2*y
=(x^2+2x+1)*x+(y^2+2y+1)*y
=(3x+2)*x+(3y+2)*y
=3x^2+2x+3y^2+2y
=3x+3+2x+3y+3+2y
=5(x+y)+6
=11

x, y 是方程 t^2-t-1= 0 的两个根
x+y = 1
xy = -1
x^2+y^2 = (x+y)^2 -2xy = 3
x^3+y^3 = (x+y)(x^2+y^2) -xy(x+y)
= 3 +1 =4
x^5+y^5
= (x^2+y^2)(x^3+y^3) - x^2y^2(x+y)
= 3*4 -1 = 11

x²-x-1=0y²-y-1=0所以x和y是方程a²-a-1=0的根x+y=1,xy=-1(x+y)²=1²x²+y²+2xy=1x²+y²=1-2xy=3x³+y³=(x+y)(x²-xy+y²)=1×(3+1)=4x的5次方+y的5次方...

x^2=x+1
x^3=x^2 *x=(x+1)x=x^2+x=(x+1)+x=2x+1
x^5=(x^3 *x^2)=(2x+1)(x+1)=2x^2+3x+1=2(x+1)+3x+1=5x+3
同理y^5=5y+3
所以x^5+y^5
=5x+3+5y+3
=5(x+y)+6
=5*1+6
=11