Question

已知x+y=1，x3+y3=三分之一，求x的五次方+y的五次方的值，要有解题过程，最好通俗易懂

Answer 1

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

Answer 2

x3+y3
=(x+y)(x^2+y^2-xy)
=x^2+y^2-xy
=(x+y)^2-3xy
=1-3xy
=1/3
xy=2/9
x+y=1
(x+y)^2=1
x^2+y^2+2xy=1
x^2+y^2=5/9

x^5+y^5
=(x^5+y^5)(x+y)/(x+y)
=x^6+x^5y+xy^5+y^6
=(x^2+y^2)(x^4+y^4-x^2y^2)+xy(x^2+y^2)
=(x^2+y^2)*[(x^2+y^2)^2-3x^2y^2]+xy(x^2+y^2)
=5/9*(25/81-12/81)+2/9*5/9
=65/729+90/729
=155/729