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Math Help - Polynomial division vs synthetic division

  1. #1
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    Polynomial division vs synthetic division

    If I am simplifing an expression in which I cannot factor I can use polynomial division, I can also use synethic division to get the same answer. I know that I can only use synethic division when I am dividing by a linear equation.

    When I change the leading coefficient on the linear equation I found that I have to divide my numbers when I do synthetic division by that number, why does this have to happen

    example using poly division

    (42x^2-33)/(7x+7) = 6x-6+9/7x+7

    using synthetic I get 42x-42 with a remainder of 9

    so I can say that r= 9/x+1 Is the reason that I divide by 7 because I take that out a 7 and use -1 so I have to put the 7 back in so I have to divide everything by 7 to get the same answer when I use poly division?


    If I am using synthetic division to solve my answer will I always divide my two numbers by the leading cooefficient on my linear term.

    like if I want to divide 2x^2-17x-38 by 2x+3 if I use synthetic I will just divide by 2 when I am done
    50k^3+10k^2-35k-7 divided by 5k-4 when I do synthetic divide all terms by 5?
    Last edited by algebraisabeast; April 8th 2009 at 04:58 PM. Reason: clarify question
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  2. #2
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    For synthetic division, the divisor (the number that you're dividing by) the coefficient for x is always 1 in order for it to work. If the coefficient isn't one, you have to use polynomial long division to divide.
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  3. #3
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    Quote Originally Posted by algebraisabeast View Post
    If I am simplifing an expression in which I cannot factor I can use polynomial division, I can also use synethic division to get the same answer. I know that I can only use synethic division when I am dividing by a linear equation.

    When I change the leading coefficient on the linear equation I found that I have to divide my numbers when I do synthetic division by that number, why does this have to happen

    example using poly division

    (42x^2-33)/(7x+7) = 6x-6+9/7x+7

    using synthetic I get 42x-42 with a remainder of 9
    so I can say that r= 9/x+1 Is the reason that I divide by 7 because I take that out a 7 and use -1 so I have to put the 7 back in so I have to divide everything by 7 to get the same answer when I use poly division?


    If I am using synthetic division to solve my answer will I always divide my two numbers by the leading cooefficient on my linear term.
    Hi algebraisabeast,

    Your polynomial long division answer is correct:

    (42x^2-33) \div (7x+7) = 6x-6+\frac{9}{7x+7}

    Now, when you attempt to use synthetic division and the coefficient of x is not 1, then you must divide everything by that coefficient, the divisor and the dividend.

    Dividing everything by 7 you would have (6x^2-\frac{33}{7}) \div (x+1)

    Now set up your synthetic division process. Your divisor is now -1.

    Code:
    -1  |  6  0  -33/7
             -6   6
         -------------
           6 -6   9/7
    Above translates to 6x-6+\frac{\frac{9}{7}}{x+1} which further simplifies to

    6x-6+\frac{9}{7x+7}


    Quote Originally Posted by algebraisabeast View Post
    like if I want to divide 2x^2-17x-38 by 2x+3 if I use synthetic I will just divide by 2 when I am done
    50k^3+10k^2-35k-7 divided by 5k-4 when I do synthetic divide all terms by 5?
    This is correct. Now try one of those and see if you can do it.
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