已知xy=1,x+y=3,求x5+y5

问题描述:

已知xy=1,x+y=3,求x5+y5

x²+y²=(x+y)²-2xy=9-2=7
(x²+y²)(x+y)=21=x³+x²y+xy²+y³=x³+y³+xy(x+y)=x³+y³+3 所以 x³+y³=18
(x³+y³)(x²+y²)=7X18=126=x^5+y^5+x³y²+x²y³=x^5+y^5+x²y²(x+y)=x^5+y^5+3
所以x^5+y^5=123