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# Long Division Of Polynomials - Some Help Needed Watch

1. 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.
2. (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 from the original equation. It should give you zero not
3. (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...
4. 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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Updated: December 11, 2010
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