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一、关于实系数一元n次方程虚根成对定理证明的教学。通用高中《数学》第三册103页给出了这个定理,课本上是这样叙述的:“还可以证明:如果虚数a+bi是实系数一元n次方程f(x)=0的根,那么它的共轭虚数a-bi也是这个方程的根。”但课本中却没有给出这这个定理的证明。是不是这个定理的证明学生无能力接受呢?回答是否定的。这个定理用学生学过的复数知识完全可以获得证明,而且学生还有能力来推导出这个定理的证明。据此,我认为应引导学生来证明这个定理,这样不但能使学生知其然,还能知其所以然,从而使学生把这个定理学得牢固,用得踏实。另外通过对定理的证明还可对前面学习过的复数知识进行复习和应用。实践证明、所达效果不出所料。
I. Teaching on the proof of the virtual root pair theorem for the real coefficient n-th order equation. This theorem is given on page 103 in Book 3 of General High School “Mathematics,” which is stated in the textbook: “It can also be proved that if the imaginary number a+bi is the root of the real coefficient one-dimensional n-order equation f(x)=0, then Its conjugate imaginary number a-bi is also the root of this equation." But the textbook does not give proof of this theorem. Isn’t the proof of this theorem that the student is incapable of accepting it? The answer is no. This theorem can be fully proved with the plural knowledge students have learned, and students have the ability to deduce the proof of this theorem. Based on this, I think that we should guide students to prove this theorem. This will not only make students aware of the facts, but also help them understand why, so that students can learn this theorem firmly and use it pragmatically. In addition, through the proof of the theorem, we can also review and apply the complex knowledge we have learned in the past. Practice has proved that the results achieved are not expected.