已知x^4-5x^3+8x^2-5x+1=0 求x+1/x

来源：学生作业帮助网编辑：作业帮时间：2024/11/19 04:34:15

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已知x^4-5x^3+8x^2-5x+1=0 求x+1/x
已知x^4-5x^3+8x^2-5x+1=0 求x+1/x

已知x^4-5x^3+8x^2-5x+1=0 求x+1/x
x^4-5x^3+8x^2-5x+1=0
x^2(x^2-5x+8-5/x+1/x^2)=0
x^2[(x^2+1/x^2)-5(x+1/x)+8]=0
x^2[(x+1/x)^2-5(x+1/x)+6]=0
x^2(x+1/x-6)(x+1/x+1)=0
若x=0,则原式不成立
所以x≠0,x^2≠0
∴(x+1/x-6)(x+1/x+1)=0
∴x+1/x=6或-1

x^4-5x^3+8x^2-5x+1=0
x^4-2x^2+1-5x^3+10x^2-5x=0
(x^2-1)^2-5(x-1)^2=0
(x+1)^2(x-1)^2-5(x-1)^2=0
(x-1)^2(x^2+2x+1-5)=0
(x-1)^2(x^2+2x-4)=0
所以x=1,或x=-1±√5
所以x+1/x=2,x+1/x=(5√5+3)/4,x+1/x=(-5√5-3)/4

LS的:
x+1/x=-1恒不成立

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