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Long Division and Remainder Theorem

  1. Sep 14, 2016 #1
    NO TEMPLATE BECAUSE MOVED FROM ANOTHER FORUM

    Hello,

    I've been trying to figure out how it works for complicated problems, I know how to use long division, but I'm not understanding how this process is done for a problem like I have.

    Instructions: Write the function in the form ƒ(x) = (x - k)q(x) + r for the given value of k, and demonstrate that ƒ(k) = r

    Problem: ƒ(x) = -4x³ + 6x² + 4,
    k = 1 - √3 <--- when plugging in q in divisor, signs change as you can see in my picture.​

    My Work:
    So as you can see from my first attached file, I knew that if I wanted to cancel out 4x³ in the dividend, I would multiply 4x³ on top (quotient) with -1 (divisor). But I also now have to multiply 4x³ with √3. I don't know how to answer it? That's where my question mark is.

    In my second attachment, you can see I put what I think is right. Then I drop down the 6x² because that's what you do in long division. So the 4x³ cancels out, but then I have two different degrees in the next part. I don't think I can do anything with that.

    Another thing is that you've probably read the "Remainder Theorem in the question, I've already done that here so that's fine. I get a remainder of 0. This is the remainder I should get when I long divide.

    Remainder Theorem
    ƒ(1 - √3) = -4(1 - √3)³ + 6(1 - √3)² + 12(1 - √3) + 4
    Solve: equals 0, so remainder is 0.

    Then, for the answer, it's asking to put into form of, ƒ(x) = (x - k)q(x) + r. I already have (x - k) which is the (-1 + √3), I would multiply that with q(x), which is what I'm stuck on for getting the quotient in long division. Lastly, I would add the remainder (r) which there is none.
     

    Attached Files:

    Last edited by a moderator: Sep 14, 2016
  2. jcsd
  3. Sep 14, 2016 #2

    SammyS

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    A screen shot of your 2nd division attempt:
    upload_2016-9-14_21-30-23.png

    You need to divide by x - k, which in this case is ##\ x-1+\sqrt{3\,}\ .\ ## You left out the ##\ x\ .##
     
  4. Sep 15, 2016 #3
    I think I know what you're saying but here is what I have so far.
     

    Attached Files:

  5. Sep 15, 2016 #4

    SammyS

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    upload_2016-9-15_11-29-3.png
    You have unbalanced parentheses and an x2 in the wrong place, or an extra x2.
    upload_2016-9-15_11-29-53.png
    I suggest you leave the coefficient of x2 intact, that is to say, write the above line as:

    ##\ -(-4x^3+(4-4\sqrt{3\,})x^2)\ ##

    With long division, keeping track of all those signs gets to be a pain. Do you know synthetic division ?
     
    Last edited: Sep 15, 2016
  6. Sep 15, 2016 #5
    Haha, it's funny how you mention synthetic division just now. I asked my professor today where I left off with you and he said to use synthetic division. Hey, thank you very much for your help, I do appreciate it, it helped me understand more.
     

    Attached Files:

  7. Sep 15, 2016 #6

    SammyS

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    You are welcome, and thanks for posting your final result. I'll display the final image below:
    img_0045-jpg.106002.jpg
     
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