# Quod Erat Demonstrandum

## 2008/11/26

### F.2 Mathematics : a minor problem in factorization

Filed under: Junior Form Mathematics — johnmayhk @ 6:11 下午
Tags: ,

Factorize

$1 - x^2(1 - 2x)^2$.

Urm, it’s a piece of cake.

$1 - x^2(1 - 2x)^2$
$\equiv 1 - [x(1 - 2x)]^2$
$\equiv [1 + x(1 - 2x)][1 - x(1 - 2x)]$
$\equiv (1 + x - 2x^2)(1 - x + 2x^2)$
$\equiv (1 + 2x)(1 - x)(1 - x + 2x^2)$

Urm, if we expand the original expression a bit, a second approach is presented as follows.

$1 - x^2(1 - 2x)^2$
$\equiv 1 - x^2(1 - 4x + 4x^2)$
$\equiv 1 - x^2 + 4x^3 - 4x^4$
$\equiv (1 - x^2) + 4x^3(1 - x)$
$\equiv (1 - x)(1 + x) + 4x^3(1 - x)$
$\equiv (1 - x)(1 + x + 4x^3)$

The problem is here, if a F.2 student solves the question by following the second approach, he or she may not obtain the ‘correct’ answer. (Of course, a F.4 student knows the fact that $1 + x + 4x^3$ can be further factorized as $(1 + 2x)(1 - x + 2x^2)$.) Similar ‘problem’ appears in questions like factorize $x^6 - y^6$. As a teacher, how to resolve the ‘puzzle’ in front of a F.2 student?

## 5 則迴響 »

1. F.2V Uniform Test Results (Setter: Mr. Wong)

Full mark = 100
Max. = 100
Min. = 18
Mean = 65.65
S.D. = 22.13
No. of passing = 30

Mark　　No. of students
100　　　1
98　　　 1
94　　　 3
90　　　 1
88　　　 3

I have no data of other 4 classes, but it seems that the results in general may not be very good. Anyway, knowing students getting full marks or having great improvement is happy, but I am worry about students who were performing badly.

迴響 由 johnmayhk — 2008/11/26 @ 8:16 下午 | 回覆

2. F.4E General Mathematics Uniform Test (2008-11-27) Results (Setter: Mr. John Ng)

Full mark = 40
Max. = 40
Min. = 13
Mean = 31.94
S.D. = 5.89
No. of passing = 41 (out of 42)
No. of absence = 2

Mark　　No. of students
40　　　　2
39　　　　2
38.5　　　2
38　　　　1
37　　　　2
36　　　　2

迴響 由 johnmayhk — 2008/11/27 @ 6:18 下午 | 回覆

3. Mr.Ng!I open a new blog.Welcome to visit my blog!

迴響 由 byronxgalilee — 2008/12/03 @ 9:18 下午 | 回覆