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已知:x,y,z满足x+y+z=1,x^2+y^2+z^2=2,x^3+y^3+z^3=3,求x^4+y^4+z^4的值
题目详情
已知:x,y,z满足x+y+z=1,x^2+y^2+z^2=2,x^3+y^3+z^3=3,求x^
4+y^4+z^4的值
4+y^4+z^4的值
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答案和解析
令a[n]=x^n+y^n+z^n
则a[n+3]=(x+y+z)a[n+2]-xya[n+1]-yza[n+1]-zxa[n+1]+xyza[n]
a[4]=a[3]-(xy+yz+zx)a[2]+xyza[1]
=>a[4]=3-2(xy+yz+zx)+xyz
由于2(xy+yz+zx)=(x+y+z)^2-(x^2+y^2+z^2)=-1
xyz=[(x+y+z)^3-3(x+y+z)(x^2+y^2+z^2)+2(x^3+y^3+z^3)]/6=1/6
故x^4+y^4+z^4=25/6
则a[n+3]=(x+y+z)a[n+2]-xya[n+1]-yza[n+1]-zxa[n+1]+xyza[n]
a[4]=a[3]-(xy+yz+zx)a[2]+xyza[1]
=>a[4]=3-2(xy+yz+zx)+xyz
由于2(xy+yz+zx)=(x+y+z)^2-(x^2+y^2+z^2)=-1
xyz=[(x+y+z)^3-3(x+y+z)(x^2+y^2+z^2)+2(x^3+y^3+z^3)]/6=1/6
故x^4+y^4+z^4=25/6
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