已知 x+y=1 x3+y3 = 1／3 求 x5+y5=?( x3 表示 x的 3次 )

x+y=1，x3+y3=三分之一
(x+y)(x²-xy+y²)=1/3
(x+y)³-3xy(x+y)=1/3
1-3xy=1/3
3xy=2/3
xy=2/9
x²+y²=1-2xy=1-4/9=5/9
所以
x的五次方+y的五次方
=(x²+y²)(x³+y³)-x²y³-x³y²
=5/9*1/3-x²y²(x+y)
=5/27-(2/9)²
=5/27-4/81
=11/81

x+y=1
(x+y)^2=x^2+2xy+y^2=1
(x+y)^3=x^3+y^3+3xy(x+y)=1
而x^3+y^3=1/3，代入得：
3xy=2/3
xy=2/9
(x+y)^5
=(x+y)^2*(x+y)^3
=[x^2+(4/9)+y^2]*[x^3+y^3+(2/3)]
=x^5+y^5+x^2y^2(y+x)+(2/3)(x^2+y^2)+(4/9)(x^3+y^3)+(8/27)
=x^5+y^5+(4/81)+(4/27)+(8/27)+(2/3)[(x+y)^2-2xy]
=x^5+y^5+(40/81)+(2/3)[1-(4/9)]
=x^5+y^5+(70/81)
=1
所以x^5+y^5=11/81

x+y=1 (x+y)^2=x^2+2xy+y^2=1 (x+y)^3=x^3+y^3+3xy(x+y)=1 而x^3+y^3=1/3,代入得：3xy=2/3 xy=2/9 由于x = 1 - y;故代入xy = 2/9;中得 y(1-y) = 2/9 = 1/3×(1-1/3);故y = 1/3;x = 1 - y =2/3;将x、y的值代入所求中...