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Basic question on spivak's calculus

  1. Dec 23, 2013 #1
    http://postimg.org/image/lh7ga876t/ [Broken]

    Hi, I have a basic question concerning definition of the word 'factorization'. Does Spivak consider factorization as development of factors ? He goes from saying the "factorization" x2−3x+2=(x−1)(x−2) is really a triple use of P9 and goes on showing development.

    P9 says : If a,b, and c are any numbers, then : a⋅(b+c)=a⋅b+a⋅c
    Also, when Spivak does the following : (x−1)(x−2)=x(x−2)+(−1)(x−2) does he use any property or just assumes it as like this ? I know whats happening, just curious if there's any justification to it.

    Thank you !
     
    Last edited by a moderator: May 6, 2017
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  3. Dec 23, 2013 #2

    SammyS

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    Note: Use the X2 icon for exponents (superscripts).

    Here's the image you posted:
    attachment.php?attachmentid=65052&stc=1&d=1387849695.jpg
    I suppose Spivak does assume that x-1 is the same as x + (-1) .

    Then of course, ##\displaystyle\ (x-1)(a)\ ## is equivalent to ##\displaystyle\ x(a)+(-1)(a)\ ## . Correct? (Assuming we can distribute from the left as well as from the right.)

    Then just let ##\displaystyle\ a = (x-2) \ ## .
     

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    Last edited by a moderator: May 6, 2017
  4. Dec 23, 2013 #3
    Yeah, that I understood. The other thing which I don't understand is why does he talk about using P9 to factorize if he's showing the development of factors. How does it make any sense ?thank you!
     
  5. Dec 23, 2013 #4

    SammyS

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    It looks like he's using P9 to expand (multiply out) the factorized form, (x-1)(x-2), verifying that it is the correct factorization for x2 - 3x + 2 .
     
  6. Dec 24, 2013 #5
    Yeah, I noticed that. I just was expecting the inverse, the factorization. Anyway, thank you for the help!
     
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