若实数x,y满足 x+y=3,x^2+y^2=7,求x^5+y^5的值

题目

若实数x,y满足 x+y=3,x^2+y^2=7,求x^5+y^5的值

答案

x^2+y^2=(x+y)^2-2xy即7=3^2-2xy则2xy=9-7=2xy=1x^3+y^3=(x+y)(x^2-xy+y^2)=3×(7-1)=18(x^2+y^2)(x^3+y^3)=x^5+x^2y^3+y^2x^3+y^5将xy=1代入上式得x^5+y^5+y+x=x^5+y^5+3又(x^2+y^2)(x^3+y^3)=7×18=126即x^5+y^5+3...