若多項式 和 的 H.C.F. 及 L.C.M. 分別為 及 ，求 。
For Q.8, you may refer to
For 0.5-mark questions, (more…)
liszenga 留言＂why there aren’t answer of factorization Quiz and worksheet!!＂
教 sum and product of roots，設 為二次方程 的根（roots），則
梁：「但為何 (*) 是真的？」 (more…)
Let’s start with the following question.
By using the cross method, students may give the following two ‘possible answers’.
在中二的課，談到 cross method，我教同學用計算機。Casio 3950p (or 3650p) 較以往好，是它顯示的根（roots）是分數，不是點數，這樣可方便同學寫出答案。中五的同學，若你的 Casio 3950p 還未有 quadratic formula，快快看：
Is the topic “factorization" in junior form useful? Urm, apart from simplification, solving equation etc, will factorization play a critical role in advanced mathematics? No idea, but at least, I could tell you that we may generalize the idea of factorization in set theory. Take it easy. I just wanna say something in my daily teaching. This topic may have no application (?) in further study in mathematics, however, it may be useful as one of the tools of training of students’ mind. When letting my students play with some challenging (at least, not-straight-forward) problems, nealy all of them are willing to do and ask for help. Well, it is a good moment of promoting the beauty of mathematics!
The following are questions asked from one of my students, Lo, in class. Factorize
Questions above are not difficult, but at least, we could not just apply identities directly without doing some grouping beforehand. The art (art? yes, sometimes, it is more than ‘technique’, it may be an ‘art’) is: how?
Teachers can easily set up factorization problems by just expanding polynomials, i.e., starting from , we can come up with and ask students to factorize it back. However, if we change z (say) into number, the problem may not be easy to solve. That is, considering
Now, may be students find it difficult to factorize .
Further, if we consider the product of trinomials (three terms) with 3 variables , it may already be a nightmare, say, could you factorize the following
Following this idea, we can create many, like
It is extremely easy to set up questions above, while, it may not be easy to solve them immediately.
Just help you a bit, observe the following
Well, it is also an art to strike a balance: give some challenging questions to students without frightening them.
When I was in F.2, I found an old little mathematics book of only one single theme: factorization. I’d lost it long time ago. It taught me many techniques in factorization, as for example, I knew how to factorize something like (read my old post if you want to). But, it is quite demanding to ask a F.2 student to factorize the following
though it is just a piece of cake for F.6 or F.7 students to use the techniques in complex numbers and obtain
There was a funny problem called Tschebotarev problem concerning the factorization of , see if I can find more and share with you next time.
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Just wanna add something below.
When introducing the identity
in class, one student, Choi, puzzled that, ‘why can’t we write something like
Good question! When putting concrete numbers, it works fine, e.g.
But, what exactly is the question asking? Factorization of polynomials.
The mathematics object on the L.H.S., is a polynomial in ; however, the thing on the R.H.S., like is NOT a polynomial in . It is because the indices involved are NOT non-negative integers. Hence, the suggestion by the student should not be acceptable.
This is one of the rules. The other is the kind of coefficients.
The following is a question in the uniform test.
One student, Lai, gave the solution as
If you were a teacher, will you give marks?
Well, you may have the “correct answer" in your mind, it should be , right? But why the answer given by Lai was not an answer? More, if somebody gives the following, what is your comment?
Urm, let me further my discussion by giving two more examples.
Many students know how to factorize , that is
Urm, can I further my calculation in writing something like
Do you think the above is an answer to the factorization problem?
Some may say we cannot factorize , but after introducing the complex numbers, can we say
is a process of factorization?
All in all, the above may force us to think about what do we mean by factorization exactly?