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Math Help - Dividing polynomials

  1. #1
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    Dividing polynomials

    I'm learning how to divide polynomials by expressions like a + 1 and I'm stuck on a particular question:

    4x^4 - 3x^2 + x + 2 divided by 2x + 3

    The answer must be expressed in the form (2x+3)*Quotient+Remainder.

    Here's how I've been going about it.

    4x^4 - 3x^2 + x + 2 = (2x + 3)(ax^3 + bx + c) + R

    Then after expanding and collecting like terms, I have:

    2ax^4 + 3ax^3 + 2bx^2 + (2b+2c)x + 3c + R


    And it's here that I get stuck. The problem here is that I don't know how to go on. What for example do I do with the x^3 bit?

    Many thanks
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  2. #2
    Grand Panjandrum
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    Re: Dividing polynomials

    Quote Originally Posted by bubbletea999 View Post
    I'm learning how to divide polynomials by expressions like a + 1 and I'm stuck on a particular question:

    4x^4 - 3x^2 + x + 2 divided by 2x + 3

    The answer must be expressed in the form (2x+3)*Quotient+Remainder.

    Here's how I've been going about it.

    4x^4 - 3x^2 + x + 2 = (2x + 3)(ax^3 + bx + c) + R

    Then after expanding and collecting like terms, I have:

    2ax^4 + 3ax^3 + 2bx^2 + (2b+2c)x + 3c + R


    And it's here that I get stuck. The problem here is that I don't know how to go on. What for example do I do with the x^3 bit?

    Many thanks
    Have you done synthetic division? If so use it.

    If not try equating coeficients of like powers in each of your expressions, to get you started (assuming you have no algebraic mistakes) for the fourth power term:

    4=2a

    CB
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  3. #3
    Member sbhatnagar's Avatar
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    Re: Dividing polynomials

    You have:
    4x^4 - 3x^2 + x + 2 = (2x + 3)(ax^3 + bx^2 + {\color{red}cx}+d) + R

    Open the brackets:

    \begin{align*} 4x^4 - 3x^2 + x + 2 &=(2a)x^4+(2b)x^3+(2c)x^2+(2d)x+(3a)x^3+(3b)x^2+(3  c)x+(3d)+R \end{align*}

    \begin{align*} \implies 4x^4 - 3x^2 + x + 2 &= (2a)x^4+(2b+3a)x^3+(2c+3b)x^2+(3c+2d)x+(3d+R) \end{align*}

    Equating the coefficients:

    2a=4

    2b+3a=-3

    2c+3b=0

    3c+2d=1

    3d+R=2

    The correct solutions to these equations will give you the value of a,b,c,d \ \textrm{and} \ R.
    Last edited by sbhatnagar; December 6th 2011 at 05:06 AM.
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  4. #4
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    Re: Dividing polynomials

    Hi guys, thank you very much for your answers (and sorry for my very late reply).

    I've now learned how to do synthetic division, and in the process of doing so also learned about the general form, which ended up solving my problem with this! I was going wrong because I wasn't accounting for the "missing power" in the original equation. It jumps from x^4 to x^2. I've solved the problem by adding 0x^3 in between them -- does this sound right? It's produced the right answer, at any rate.

    Thanks again.
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  5. #5
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    Re: Dividing polynomials

    Another way to divide 4x^4 - 3x^2 + x + 2 by 2x + 3, more like "standard" numerical long division:

    2x divides into 4x^4 2x^3 times. Multiplying that by 2x+ 3 gives [tex](2x+3)(2x^3)= 4x^4+ 6x^3. Subtract [tex](4x^4+ 0x^3)- (4x^4+ 6x^3)= -6x^3. That leaves us -6x^3- 3x^2+ x+ 2. Now, 2x divides into -6x^3 -3x^2 times. Multiplying that by 2x+ 3 gives -6x^3- 9x^2. Subtract (-6x^3- 3x^2)- (-6x^3- 9x^2)= 6x^2. That leaves 6x^2+ x+ 2. 2x divides into 6x^2 3x times. Multiplying that by 2x+ 3 gives 6x^2+ 9x. Subtracting (6x^2+ x)-(6x^2+ 9x)= -8x. That leaves us -8x+ 2. 2x divides into -8x -4 times. Multiplying that by 2x+ 3 gives -8x- 12. Subtracting (-8x+ 2)- (-8x- 12)= 14.

    That is, the quotient is 2x^3- 3x^2+ 3x- 4 with remainder 14.

    (Captain Black, I was under the impression that "synthetic division" could only be used to divide by something of the form x+ a.)
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