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已知:x+y+z=a,xy+yz+zx=b,xyz=c,试用含a.b.c的式子表示下列各式.1.x²+y²+z²2.(x-1)(y-1)(z-1)
题目详情
已知:x+y+z=a,xy+yz+zx=b,xyz=c,试用含a.b.c的式子表示下列各式.
1.x²+y²+z²
2.(x-1)(y-1)(z-1)
1.x²+y²+z²
2.(x-1)(y-1)(z-1)
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答案和解析
∵x+y+z=a
∴(x+y+z)²=a²
∴x²+xy+xz+xy+y²+zy+zx+zy+z²=a²
∴x²+y²+z²=a²-2(xy+xz+zy)
∵xy+yz+zx=b
∴x²+y²+z²=a²-2b
(x-1)(y-1)(z-1)
=[x(y-1)-(y-1)](z-1)
=xyz-zx-zy+z-xy+x+y-1
=xyz+(x+y+z)-( xy+yz+zx)
=c+a-b
∴(x+y+z)²=a²
∴x²+xy+xz+xy+y²+zy+zx+zy+z²=a²
∴x²+y²+z²=a²-2(xy+xz+zy)
∵xy+yz+zx=b
∴x²+y²+z²=a²-2b
(x-1)(y-1)(z-1)
=[x(y-1)-(y-1)](z-1)
=xyz-zx-zy+z-xy+x+y-1
=xyz+(x+y+z)-( xy+yz+zx)
=c+a-b
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