设m,n,p为任意非负整数,证明x^2+x+1|x^3m+x^(3n+1)+x^(3p+2)
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解决时间 2021-11-11 08:31
- 提问者网友：太高姿态
- 2021-11-11 00:44
设m,n,p为任意非负整数,证明x^2+x+1|x^3m+x^(3n+1)+x^(3p+2)
最佳答案
- 五星知识达人网友：英雄的欲望
- 2021-11-11 01:43
x^3m+x^(3n+1)+x^(3p+2)
=x^3m-1+x^(3n+1)-x+x^(3p+2)-x^2+1+x+x^2
=(x^3-1)(1+x^3+...+x^(3m-3)) + x(x^3-1)(1+x^3+...+x^(3n-3)) +x^2(x^3-1)(1+x^3+...+x^(3p-3))+1+x+x^2
=(x-1)(1+x+x^2)(1+x^3+...+x^(3m-3)) + x(x-1)(1+x+x^2)(1+x^3+...+x^(3n-3)) +x^2(x-1)(1+x+x^2)(1+x^3+...+x^(3p-3))+1+x+x^2
每一项都有因式1+x+x^2
因此，x^2+x+1|x^3m+x^(3n+1)+x^(3p+2)
=x^3m-1+x^(3n+1)-x+x^(3p+2)-x^2+1+x+x^2
=(x^3-1)(1+x^3+...+x^(3m-3)) + x(x^3-1)(1+x^3+...+x^(3n-3)) +x^2(x^3-1)(1+x^3+...+x^(3p-3))+1+x+x^2
=(x-1)(1+x+x^2)(1+x^3+...+x^(3m-3)) + x(x-1)(1+x+x^2)(1+x^3+...+x^(3n-3)) +x^2(x-1)(1+x+x^2)(1+x^3+...+x^(3p-3))+1+x+x^2
每一项都有因式1+x+x^2
因此，x^2+x+1|x^3m+x^(3n+1)+x^(3p+2)
全部回答
- 1楼网友：掌灯师
- 2021-11-11 02:35
证明什么，你题目是不是错了
- 2楼网友：迟山
- 2021-11-11 02:21
证明：原式=x^3m+x^(3n+1)+x^(3p+2)=[x^3m-1]+[x^(3n+1)-x]+[x^(3p+2)-x^2]+(1+x+x^2)
=[x^3m-1]+x[x^3n-1]+x^2[x^3p-1]+(1+x+x^2)
由于：x^2+x+1|x^3m-1；x^2+x+1|x^3n-1；x^2+x+1|x^3p-1
所以：x^2+x+1|x^3m+x^(3n+1)+x^(3p+2)
=[x^3m-1]+x[x^3n-1]+x^2[x^3p-1]+(1+x+x^2)
由于：x^2+x+1|x^3m-1；x^2+x+1|x^3n-1；x^2+x+1|x^3p-1
所以：x^2+x+1|x^3m+x^(3n+1)+x^(3p+2)
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