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Thread: Polynomial Division

  1. #1
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    Polynomial Division

    Hi,

    How do I go about the following polynomial division? Particularly with the fact that the coefficients of the numerator are smaller than the denominator.

    h(x) = 6x^4 + 3x^3 + 6x^2 + 3
    g(x) = 12x^3 - 30x^2 + 18x - 12

    What is h(x) / g(x)?

    I know there is a remainder.

    Thanks
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  2. #2
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    Re: Polynomial Division

    I don't know how to show long division in latex.. annoying.

    $6x^4 + 3x^3 + 6x^2 + 3 = \dfrac 1 2 x (12x^3-30x^2+18x-12) + (12x^3-3x^2+6x+3)$

    $ (12x^3-3x^2+6x+3) = 1 (12x^3-30x^2+18x-12) + (27x^2-12x+15)$

    so

    $\dfrac{6x^4 + 3x^3 + 6x^2 + 3}{12x^3-30x^2+18x-12} = \left(\dfrac 1 2 x + 1\right) + \dfrac{27x^2-12x+15}{12x^3-30x^2+18x-12}$
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  3. #3
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    Re: Polynomial Division

    Thank you.

    But your first substitution equates to

    6x^4 - 3x^3 + 6x^2 +3.

    Which is not the original expression.
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  4. #4
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    Re: Polynomial Division

    I get x/2 + 3/2 with remainder 14x^2 - 7x + 7
    Too lazy to check if correct!!
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  5. #5
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    Re: Polynomial Division

    Thanks, that's pretty much inline with what I managed.
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  6. #6
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    Re: Polynomial Division

    Quote Originally Posted by Stormin63 View Post
    h(x) = 6x^4 + 3x^3 + 6x^2 + 3
    g(x) = 12x^3 - 30x^2 + 18x - 12
    Dividing each by 3 makes process a little easier:
    (long division arrangement):
    4x^3 - 10x^2 + 6x - 4 / 2x^4 + x^3 + 2x^2 + 0x + 1
    Last edited by DenisB; Oct 1st 2017 at 07:53 AM.
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  7. #7
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    Re: Polynomial Division

    h(x) = 6x^4 + 3x^3 + 6x^2 + 3
    g(x) = 12x^3 - 30x^2 + 18x - 12

    What is h(x) / g(x)?

    The way I was taught to do this:
    12x^3 divides into 6x^4, (1/2) x times.

    Multiplying all of 12x^3- 30x^2+ 18x- 12 gives 6x^4- 15x^3+ 9x^2- 6x. Subtract that from 6x^4- 3x^3+ 6x^2+ 0x+ 3 (don't forget that "0x"!) leaves 12x^3- 3x^2+ 6x+ 3. Now, of course, 12x^3 divides into 12x^3 exactly once. Subtracting 1(12x^3- 30x^2+ 18x- 12) from 12x^3- 3x^2+ 6x+ 3 leaves 27x^2- 12x+ 15.

    h(x)/g(x) is (1/2)x+ 1 with remainder 27x^2- 12x+ 15.
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  8. #8
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    Re: Polynomial Division

    I did the equivalent of that you did, Hall;
    how cum we don't get same results?
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