已知x,y均为实数,且满足xy+x+y=17,x^2y+xy^2=66,求x^4+x^3y+x^2y^2+xy^3+y^4的值
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![已知x,y均为实数,且满足xy+x+y=17,x^2y+xy^2=66,求x^4+x^3y+x^2y^2+xy^3+y^4的值](/uploads/image/z/977102-62-2.jpg?t=%E5%B7%B2%E7%9F%A5x%2Cy%E5%9D%87%E4%B8%BA%E5%AE%9E%E6%95%B0%2C%E4%B8%94%E6%BB%A1%E8%B6%B3xy%2Bx%2By%3D17%2Cx%5E2y%2Bxy%5E2%3D66%2C%E6%B1%82x%5E4%2Bx%5E3y%2Bx%5E2y%5E2%2Bxy%5E3%2By%5E4%E7%9A%84%E5%80%BC)
已知x,y均为实数,且满足xy+x+y=17,x^2y+xy^2=66,求x^4+x^3y+x^2y^2+xy^3+y^4的值
已知x,y均为实数,且满足xy+x+y=17,x^2y+xy^2=66,求x^4+x^3y+x^2y^2+xy^3+y^4的值
已知x,y均为实数,且满足xy+x+y=17,x^2y+xy^2=66,求x^4+x^3y+x^2y^2+xy^3+y^4的值
xy+(x+y)=17
x^2y+xy^2=66
xy(x+y)=66
所以由韦达定理
xy和x+y是方程a^2-17a+66=0的根
(a-6)(a-11)=0
a=6,a=11
所以x+y=6,xy=11
或x+y=11,xy=6
若x+y=6,xy=11,则x和y是方程b^2-6b+11=0的根
判别式小于0,无解
若x+y=11,xy=6,则x和y是方程c^2-11c+6=0的根
判别式大于0,有解
所以x+y=11,xy=6
x^4+x^3y+x^2y^2+xy^3+y^4
=x^3(x+y)+x^2y^2+y^3(x+y)
=(x+y)(x^3+y^3)+x^2y^2
=(x+y)[(x+y)(x^2-xy+y^2)]+x^2y^2
=(x+y)^2[(x^2+2xy+y^2)-3xy]+x^2y^2
=(x+y)^2[(x+y)^2-3xy]+(xy)^2
=11^2*(11^2-3*6)+6^2
=12499
xy+(x+y)=17
x^2y+xy^2=66
xy(x+y)=66
所以由韦达定理
xy和x+y是方程a^2-17a+66=0的根
(a-6)(a-11)=0
a=6,a=11
所以x+y=6,xy=11
或x+y=11,xy=6
若x+y=6,xy=11,则x和y是方程b^2-6b+11=0的根
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xy+(x+y)=17
x^2y+xy^2=66
xy(x+y)=66
所以由韦达定理
xy和x+y是方程a^2-17a+66=0的根
(a-6)(a-11)=0
a=6,a=11
所以x+y=6,xy=11
或x+y=11,xy=6
若x+y=6,xy=11,则x和y是方程b^2-6b+11=0的根
判别式小于0,无解
若x+y=11,xy=6,则x和y是方程c^2-11c+6=0的根
判别式大于0,有解
所以x+y=11,xy=6
x^4+x^3y+x^2y^2+xy^3+y^4
=12499
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已知 xy+x+y = 17, xy(x+y) = 66,可以求出
xy = 6或11,x+y = 11或6
则:
x^4+x^3y+x^2y^2+xy^3+y^4
=(x^2+y^2)^2-2(xy)^2 +xy(x^2+y^2)+(xy)^2
=((x+y)^2-2xy)^2+xy((x+y)^2-2xy)-(xy...
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已知 xy+x+y = 17, xy(x+y) = 66,可以求出
xy = 6或11,x+y = 11或6
则:
x^4+x^3y+x^2y^2+xy^3+y^4
=(x^2+y^2)^2-2(xy)^2 +xy(x^2+y^2)+(xy)^2
=((x+y)^2-2xy)^2+xy((x+y)^2-2xy)-(xy)^2
当x+y=11,xy=6时,原表达式=(121-12)^2+6*(121-12)-36=12499;
当x+y=6,xy=11时,原表达式=(36-22)^2+11*(36-22)-121=229;
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