待定系数法是一个非常重要的数学方法,其理论根据是下面要提到的“多项式恒等定理”。这个定理如果应用高等代数中的代数基本定理来证明是十分容易的。但由于代数基本定理要到大学才能学到,所以在中学教科书中对“多项式恒等定理”没有加以证明。本文对此提供一个中学生能接受的证法,供大家参考。引理若α_nx~n+α_(n-1)x~(n-1)+…+α_2x~2+α_1x+α_n≡0,则α_n=a_(n-1)=…=α_1=α_o=0。证明先证α_0=0。假若不然,设a_0≠0。由于|α_0|≠0,n是一个给定的自然数,因此一定可以取到充分小的正数ε,使得下面几个不等 The undetermined coefficient method is a very important mathematical method, and its theoretical basis is the “polynomial identity theorem” mentioned below. This theorem is very easy to prove if you apply the basic algebra theorem in higher algebra. However, since the fundamental theorem of algebra must be learned in universities, there is no proof of the “polynomial identity theorem” in the textbook of middle school. This article provides a proof method acceptable to middle school students for your reference. Lemma if α_nx~n+α_(n-1)x~(n-1)+...+α_2x~2+α_1x+α_n≡0, then α_n=a_(n-1)=...=α_1=α_o=0 . Prove that α_0=0. If not, set a_0≠0. Since |α_0|≠0,n is a given natural number, we must be able to get a sufficiently small positive number ε, so that the following range