# 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?