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1978年全国数学竞赛有这样一道题:对多项式x~(12)+x~9+x~6+x~3+1进行因式分解。具结果是x~(12)+x~9+x~6+x~3+1=(x~4+x~3+x~2+x+1)(x~8-x~7+x~5-x~4+x~3-x+1)。这个结果恰好是将等式左边x~(12)+x~9+x~6+x~3+1中的x~3换为x就是等式右边的第一个因式x~4+x~3+x~2+x+1,我们知道,x~4+x~2+1=(x~2+x+1)(x~2-x+1),这道题的结果也是将等式左边的x~2换为x就是等式右边的第一个因式x~2+x+1。由这两道题的结果使人想到:上述两例是否具有普遍性?对于这个问题的回答,我们有如下定理:
In 1978 the national mathematics competition has such a problem: the polynomial x ~ (12) + x ~ 9 + x ~ 6 + x ~ 3 +1 for factorization. The result is that x ~ (12) + x ~ 9 + x ~ 6 + x ~ 3 + 1 = (x ~ 4 + x ~ 3 + x ~ 2 + x + 1) ~ 5-x ~ 4 + x ~ 3-x + 1). This result happens to be that x ~ 3 in x ~ (12) + x ~ 9 + x ~ 6 + x ~ 3 + 1 on the left side of the equation is replaced by x as the first factor x ~ 4 + x ~ 3 + x ~ 2 + x + 1, we know that x ~ 4 + x ~ 2 + 1 = (x ~ 2 + x + 1) X ~ 2 to the left of the equation x is the first factor x ~ 2 + x + 1 to the right of the equation. From the results of these two questions, one can think of the following: Are the above two examples universal? In response to this question, we have the following theorem: