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Synthetic Division?

  1. Feb 15, 2006 #1
    I'm always curious as to how this works? I mean, it's amazing that, even though dividing by binomials, that this sort of mechanism works but I don't understand quite the qwerks behind this.

    Can someone fulfill my curiousity?
     
  2. jcsd
  3. Feb 15, 2006 #2

    0rthodontist

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    It's just like regular long division except in regular long division the unknown variable always equals 10 (if you're working in base 10).

    What kind of explanation are you looking for, a proof of the algorithm?
     
  4. Feb 15, 2006 #3
    Yes. A proof of the algorithm would better help me understand the workings behind this algorithm.

    EDIT: Whats the difference between regular long division using base 10 and synthetic division? Don't they both use a decimal based system?
     
  5. Feb 17, 2006 #4
    By synthetic division, I take it you mean what I'd call polynomial division?

    When you do a long division, you can write

    341 into 212784 = 624

    as

    (3.100 + 4.10 + 1) into (2.100000 + 1.10000 + 2.1000 + 7.100 + 8.10 + 4) equals 6.100 + 2.20 + 4.

    Alternatively, you could write it as

    (3x2 + 4x + 1) into (2x5 + 1x4 + 2x3 + 7x2 + 8x + 4) = 6x2 + 2x + 4

    and then write x=10.

    Do you see what Orthodontist means when he says that synthetic division is just like long division, except that the unknown variable is always 10?
     
  6. Feb 18, 2006 #5
    Maybe I'm getting the wrong idea from this, but you're saying that if you divide ANY polynomial expression through synthetic division, I would always get 10 as one of my roots?
     
  7. Feb 18, 2006 #6

    HallsofIvy

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    No, he didn't say anything like that- he said that if you put x= 10 the polynomials you get the numbers in the example.
     
  8. Feb 20, 2006 #7
    synthetic division is for lazy people.

    Long division always wins
     
  9. Feb 20, 2006 #8
    Not true at all.

    Intelligent people always go for the more efficient method of solving problems.

    As for putting x = 10 into the polynomials, I don't understand this step:

    Can you please tell me how you derive that?
     
  10. Feb 20, 2006 #9

    HallsofIvy

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    " 3.100+ 4.10+ 1" should be 3*100+ 4*10+ 1 which is, of course, 341.
     
  11. Feb 20, 2006 #10

    Tide

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    Synthetic division IS long division - only done more efficiently.
     
  12. Feb 20, 2006 #11
    Sorry, yeah, I always use a '.' to mean multiplication now. Force of habit etc...
     
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