已知m^2-5m+1=0,求m^3+1/m^3 和m-1/m的值

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已知m^2-5m+1=0,求m^3+1/m^3 和m-1/m的值
已知m^2-5m+1=0,求m^3+1/m^3 和m-1/m的值

已知m^2-5m+1=0,求m^3+1/m^3 和m-1/m的值
m²-5m+1=0
m²+1=5m,方程式两边同时除以m,得到
m+1/m=5
（m+1/m）^3=m^3+1/m^3+3(m+1/m),所以
m^3+1/m^3=(m+1/m)^3-3(m+1/m)=5^3-3×5=110
(m+1/m)^2=m^2+1/m^2+2,所以,m^2+1/m^2=(m+1/m)^2-2=5^2-2=23
(m-1/m)^2=m^2+1/m^2-2=23-2=21,
m-1/m=±√21
多谢!

m^2-5m=1
2m^2-5m+1/m^2
=m^2+m^2-5m+1/m^2
=m^2+1+1/m^2
m^2-1=5m
两边同时除以m得
m-1/m=5
m^2+1/m^2=(m-1/m)^2+2
=25+2
=27
∴原式=27+1=28

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