數學高一習題(１顆星 )
2013/09/09 17:55
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Q : 是否存在某數n使 n^4+2n^3+2n^2+2n+1 為完全平方數?
A :
(1) : 令 n^4+2n^3+2n^2+2n+1 = k
(n^4+n^3+n^2)+(n^3+n^2+n)+(n+1)
= n^2(n^2+n+1)+n(n^2+n+1)+(n+1)
= n(n+1)(n^2+n+1)+(n+1)
= (n^2+1)(n+1)^2
(2) :
(n^4+2n^2+1)+2n(n^2+1) < (n^2+n+1)^2
= 根據 (１) : k>(n^2+n)^2
A :
(1) : 令 n^4+2n^3+2n^2+2n+1 = k
(n^4+n^3+n^2)+(n^3+n^2+n)+(n+1)
= n^2(n^2+n+1)+n(n^2+n+1)+(n+1)
= n(n+1)(n^2+n+1)+(n+1)
= (n^2+1)(n+1)^2
(2) :
(n^4+2n^2+1)+2n(n^2+1) < (n^2+n+1)^2
= 根據 (１) : k>(n^2+n)^2
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