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Thread: Polynomial Remainder Theorem

  1. #1
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    Question Polynomial Remainder Theorem

    Hi,

    I hope someone can provide me with some clarification.

    I'm trying to find the remainder for a dividend of x^2+9x-7. I already have information regarding the quotient and divisor: (x + 3) and (x + 5). It's unknown which one is the quotient or divisor, but I don't think that's relevant information anyways. One method of solving the remainder would be to isolate the remainder ("r") in the given equation: (x + 3)(x + 5) + r = x^2 + 9x - 7. This works for me, and I get the correct answer of r = x - 22.

    However, I don't understand why I ALSO can't use the remainder theorem to solve this. While I understand that a solution with a variable is impossible to solve with the remainder theorem, how would I know this prior to using the theorem? I thought that when factors have a degree of 1, that you could find the remainder by simply using the remainder theorem - clearly this case is different. Please help!!!!

    Sincerely,
    Olivia
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  2. #2
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    Re: Polynomial Remainder Theorem

    performing synthetic division with x = -3 ...

    Code:
    -3].......1........9........-7
    ..................-3.......-18
    -------------------------------
    ...........1.......6.......-25
    note the remainder, -25 = x - 22


    performing synthetic division with x = -5 ...

    Code:
    -5].......1........9........-7
    ..................-5.......-20
    -------------------------------
    ...........1.......4.......-27
    note the remainder, -27 = x - 22
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  3. #3
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    Re: Polynomial Remainder Theorem

    It seems like an odd question because the remainder should be of a lower order than the divisor.

    If (x+3) is the divisor, then your remainder x-22 = (x+3) - 25. So the quotient ought to be (x+6) rather than (x+5) giving a remainder of -25.

    Similarly, if the divisor is (x+5), then the remainder x-22 = (x+5) - 27. So the quotient ought to be (x+4) rather than (x+3) giving a remainder of -27.
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  4. #4
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    Re: Polynomial Remainder Theorem

    Yes, the fact that the remainder isn't lower than divisor is what makes it confusing. So are you suggesting that the question is not explained correctly? (i.e. I need to change quotient in order for me to produce x - 22 ?)

    Also, how would I know in advance that I can't use the remainder theorem for this problem?
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