已知a是方程x^2 -3x+1=0的根,试求a^3-2a^2 -5a+1/a^2 +1的值.

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已知a是方程x^2 -3x+1=0的根,试求a^3-2a^2 -5a+1/a^2 +1的值.
已知a是方程x^2 -3x+1=0的根,试求a^3-2a^2 -5a+1/a^2 +1的值.

已知a是方程x^2 -3x+1=0的根,试求a^3-2a^2 -5a+1/a^2 +1的值.
由题意得:a^2-3a+1=0移项得a^2=3a-1
(a^3-2a^2-5a+1)/(a^2+1)
=[a^2(a-2)-5a+1]/(3a-1+1)
=[(3a-1)(a-2)-5a+1]/(3a)
=(3a^2-6a-a+2-5a+1)/(3a)
=(3a^2-12a+3)/(3a)
=(a^2-4a+1)/(a)
=(3a-1-4a+1)/a
=-a/a
=-1

a^2-3a+1=0,a^2-3a=-1
a^3-2a^2-5a+1/a^2+1
=a(a^2-3a)+a^2-3a-2a+1/a^2+1
=-a-1-2a+1/a^2+1
=-3a+1/a^2

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