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Long Division Of Polynomials - Some Help Needed Watch

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    Guys, I'm having a bit of a problem with a particular polynomial I'm trying to reduce, working in Z2.

    I'm trying to reduce the polynomial x9 - 1, and I know that it can be expressed as a product of three polynomials: f1f2f3.

    I know that f1 is definitely (1+x), which means that the other two polynomials must together make (1 + x + x2 + x3 + x4 + x5 + x6 + x7 + x 8).

    As a clue, I've been given that one of the other irreducible polynomials is of degree 2. Now I know that the only irreducible function of degree 2 is (x2 + x + 1).

    So in a nutshell, of the three polynomials I need, I know that f1 is (1+x), and I know that f2 is (x2 + x + 1).

    So, I need to divide (x2 + x + 1) into (1 + x + x2 + x3 + x4 + x5 + x6 + x7 + x 8) to get f3, and I'm going wrong somewhere.

    Rather than type it out here (it's a pain in the backside putting everything in tags, here's a JPG which shows what I've done so far.



    Now I have been given the answer, and I know that f3 should be (x6 + x3 + 1), but obviously this isn't what I'm getting.

    Can anyone be so kind as to point out to me where I'm going wrong?

    Thanks.
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    (Original post by Gazcobain)
    Guys, I'm having a bit of a problem with a particular polynomial I'm trying to reduce, working in Z2.

    I'm trying to reduce the polynomial x9 - 1, and I know that it can be expressed as a product of three polynomials: f1f2f3.

    I know that f1 is definitely (1+x), which means that the other two polynomials must together make (1 + x + x2 + x3 + x4 + x5 + x6 + x7 + x 8).

    As a clue, I've been given that one of the other irreducible polynomials is of degree 2. Now I know that the only irreducible function of degree 2 is (x2 + x + 1).

    So in a nutshell, of the three polynomials I need, I know that f1 is (1+x), and I know that f2 is (x2 + x + 1).

    So, I need to divide (x2 + x + 1) into (1 + x + x2 + x3 + x4 + x5 + x6 + x7 + x 8) to get f3, and I'm going wrong somewhere.

    Rather than type it out here (it's a pain in the backside putting everything in tags, here's a JPG which shows what I've done so far.



    Now I have been given the answer, and I know that f3 should be (x6 + x3 + 1), but obviously this isn't what I'm getting.

    Can anyone be so kind as to point out to me where I'm going wrong?

    Thanks.
    You're going wrong when you're subtracting  x^5 + x^4 + x^3 from the original equation. It should give you zero not  -x^4 - x^3
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    (Original post by Gazcobain)
    I know that f1 is definitely (1+x), which means that the other two polynomials must together make (1 + x + x2 + x3 + x4 + x5 + x6 + x7 + x 8).
    Firstly just a minor point to say f1=(x-1) not (x+1), but seeing as the rest of what you typed was based on (x-1), this is not the cause of your error.

    The mistake is in the second stage of your long division calculation. In the same way that you subtracted x 8 + x 7 + x 6 away from itself in the first stage to leave a highest term of x 5, in the second stage, when you multiply by x 3 to give x 5 + x 4 + x 3, you should be subtracting this from x 5 + x 4 + x 3 that is still left, not just the x 5.

    This will leave you now with highest term x 2 and then you need +1 to finish the division off as required.

    Apologies that I don't think this is overly clear, but I couldn't find a nice way to explain it Multiplying out the brackets will show this works though...
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    Cheers guys. Yeah I know it should strictly speaking be (x - 1), but as I'm working in Z2 then (x - 1) = (x + 1).

    I'll give that method a bash and see what I get. Thanks!
 
 
 
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