# Thread: division of polynomials in rings

1. ## division of polynomials in rings

Hey,
I don't know how to do the following question:

For which integers $\displaystyle n$ does $\displaystyle x^2+x+1$ divide $\displaystyle x^4+3x^3+x^2+6x+10$ in $\displaystyle \mathbb{Z}/n\mathbb{Z}[x]$?

I'm afraid I'm not entirely sure what to do for this question. I think my confusion for this question begins with the definition of $\displaystyle \mathbb{Z}[x]$. Is $\displaystyle \mathbb{Z}[x]$ the ring of algebraic integers? Should I be looking an integer $\displaystyle x$ that makes the 2 above polynomials equal to 0, then find out what integer multiples make the 2 polynomials equal to 0 as well?

Thanks for your time.

2. Originally Posted by Silverflow
Hey,
I don't know how to do the following question:

For which integers $\displaystyle n$ does $\displaystyle x^2+x+1$ divide $\displaystyle x^4+3x^3+x^2+6x+10$ in $\displaystyle \mathbb{Z}/n\mathbb{Z}[x]$?

I'm afraid I'm not entirely sure what to do for this question. I think my confusion for this question begins with the definition of $\displaystyle \mathbb{Z}[x]$. Is $\displaystyle \mathbb{Z}[x]$ the ring of algebraic integers? Should I be looking an integer $\displaystyle x$ that makes the 2 above polynomials equal to 0, then find out what integer multiples make the 2 polynomials equal to 0 as well?

Thanks for your time.

I think they meant $\displaystyle \left(\mathbb{Z}/n\mathbb{Z}\right)[x]=$ the ring of polynomials over the ring of integers modulo n.

Divide the right polymomial by the left one with remainder (long or synthetic division) and find out what conditions on n

would make the remainder vanish.

Tonio

3. Thanks, mate. Makes sense now.