已知x、y均为实数,且满足xy+x+y=17,x2y+xy2=66,则x4+x3y+x2y2+xy3+y4=_.
问题描述:
已知x、y均为实数,且满足xy+x+y=17,x2y+xy2=66,则x4+x3y+x2y2+xy3+y4=______.
答
x2y+xy2=xy(x+y)=66,
设xy=m,x+y=n,
由xy+x+y=17,得到m+n=17,由xy(x+y)=66,得到mn=66,
∴m=6,n=11或m=11,n=6(舍去),
∴xy=m=6,x+y=n=11,
x2+y2=112-2×6=109,x2y2=36
x4+y4=1092-36×2=11809
x4+x3y+x2y2+xy3+y4
=11809+6×109+36
=12499.
故答案为:12499