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)如果会的话加悬赏啊……
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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)如果会的话加悬赏啊……
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)
如果会的话加悬赏啊……
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)如果会的话加悬赏啊……
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)
证明什么,你题目是不是错了
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)