Question

已知a是方程x²+1=3x的根，试求a³-2a²-5a+1/a²+1的值

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Answer 1

解：因为a是方程x²+1=3x的根
所以a²+1=3a
所以1=3a-a²
所以
a³-2a²-5a+1/a²+1
=(a³-2a²-5a+3a-a²)/(a²+3a-a²)
=(a³-3a²-2a)/3a
=a(a²-3a-2)/3a
=a(-1-2)/3a
=-3a/3a
=-1

Answer 2

a是方程x²+1=3x的根
∴a²+1=3a
∴a²-3a+1=0
a³-2a²-5a+1/a²+1
=[(a³-3a²+a)+(a²-3a+1)-3a]/[(a²-3a+1)+3a]
=[a(a²-3a+1)+(a²-3a+1)-3a]/[(a²-3a+1)+3a]
=-3a/3a=-1

Answer 3

解：因为a是方程x²+1=3x的根，所以a²+1=3a，或a²-3a+1=0。
待求式分子部分：a³-2a²-5a+1=(a³-3a²-2a)+(a²-3a+1)
=a(a²-3a-2)
=a(a²-3a+1-3)
=-3a；
那么，(a³-2a²-5a+1)/(a²+1)=-3a/(a²+1)
=-3a/(3a)
=-1。

Answer 4

(a³-2a²-5a+1)/(a²+1)
= (a³-3a²+ a²+1-5a)/(a²+1)
=(a (a²-3a)+a²+1-5a)/(a²+1)
=(a*(-1)+3a-5a)/3a
=-3a/3a
=-1