已知x-y不等于0,x^2-x=7,y^2-y=7,求x^3+y^3+x^2y+xy^2的值
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已知x-y不等于0,x^2-x=7,y^2-y=7,求x^3+y^3+x^2y+xy^2的值
![已知x-y不等于0,x^2-x=7,y^2-y=7,求x^3+y^3+x^2y+xy^2的值](/uploads/image/z/2354750-62-0.jpg?t=%E5%B7%B2%E7%9F%A5x-y%E4%B8%8D%E7%AD%89%E4%BA%8E0%2Cx%5E2-x%3D7%2Cy%5E2-y%3D7%2C%E6%B1%82x%5E3%2By%5E3%2Bx%5E2y%2Bxy%5E2%E7%9A%84%E5%80%BC)
已知x-y≠0,x²-x=7,y²-y=7,求x³+y³+x²y+xy²的值.
经观察可知,x、y相当于是方程t²-t=7的两个不等实根,方程整理为:
t²-t-7=0
由韦达定理,得:
x+y=1
xy=-7
所以:
x³+y³+x²y+xy²
=(x³+y³)+(x²y+xy²)
=(x+y)(x²+y²-xy)+xy(x+y)
=(x+y)(x²+y²)
=x²+y²
=(x+y)²-2xy
=1-2*(-7)
=1+14
=15
经观察可知,x、y相当于是方程t²-t=7的两个不等实根,方程整理为:
t²-t-7=0
由韦达定理,得:
x+y=1
xy=-7
所以:
x³+y³+x²y+xy²
=(x³+y³)+(x²y+xy²)
=(x+y)(x²+y²-xy)+xy(x+y)
=(x+y)(x²+y²)
=x²+y²
=(x+y)²-2xy
=1-2*(-7)
=1+14
=15
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