Quod Erat Demonstrandum

2007/12/02

[初中] Factorization

Filed under: Junior Form Mathematics — johnmayhk @ 3:50 下午
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The following is a question in the uniform test.
Factorize 12xy - 6x^2.
One student, Lai, gave the solution as
12xy - 6x^2
=12x(y - \frac{x}{2})
If you were a teacher, will you give marks?
Well, you may have the “correct answer" in your mind, it should be 6x(2y - x), right? But why the answer given by Lai was not an answer? More, if somebody gives the following, what is your comment?
12xy - 6x^2
= 24x^3y(\frac{1}{2x^2} - \frac{1}{3xy})
Urm, let me further my discussion by giving two more examples.
Many students know how to factorize x^2 - y^2, that is
x^2 - y^2
= (x + y)(x - y)
Urm, can I further my calculation in writing something like
x^2 - y^2
= (x + y)(x - y)
= (x + y)(\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y})
Do you think the above is an answer to the factorization problem?
Some may say we cannot factorize x^2 + 1, but after introducing the complex numbers, can we say
x^2 + 1
= (x + i)(x - i) where i^2 = -1
is a process of factorization?
All in all, the above may force us to think about what do we mean by factorization exactly?

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