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: ,

Let’s start with the following question.

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

  4. Byran, where’s the link?

    迴響 由 johnmayhk — 2008/12/03 @ 9:41 下午 | 回覆

  5. http://byronxgalilee.wordpress.com/

    迴響 由 byronxgalilee — 2008/12/04 @ 7:18 下午 | 回覆


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