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Does the remainder theorem only work for polynomial equations being divided by a binomial of the form $\ x-a\ $?

Are there any limitations on the remainder theorem?

I realize in polynomial division, it can be the case that the remainder includes x terms. That being the case, I assume that the remainder theorem can only work for a binomial divisor; I just would like to make sure there is not something I am over looking.

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    $\begingroup$ It doesn't work all that nicely for polynomials with repeated roots, e.g. $(x-a)^2$. However, it does work perfectly for polynomials with distinct factors. $\endgroup$
    – Calvin Lin
    Jun 11, 2013 at 2:08

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Over any (coefficient) ring, one can divide with remainder by any monic polynomial (or any polynomial with unit leading coefficient). Hence, from this viewpoint, there is no limitation at all.

If you wish to go further, one can split into coprime (comaximal) factors, then use CRT (Chinese Remainder) to (Lagrange) interpolate values. To handle nonlinear powers one can use values of derivatives also, similar to Taylor series (e.g. search on "jet spaces").

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    $\begingroup$ Can you explain how to calculate the remainder when $x^k - 1$ is divided by $(x-1)^n$? $\endgroup$
    – Calvin Lin
    Jun 11, 2013 at 3:28
  • $\begingroup$ @Calvin Use the division algorithm, or compute a taylor expansion at $\,x = 1.$ $\endgroup$
    – Key Ideas
    Jun 11, 2013 at 3:36
  • $\begingroup$ The question being, how do you use the remainder-factor theorem, which is what OP is referring to. I know the remainders exist, and we can calculate them in a variety of ways. The remainder factor theorem gives us an easy way to calculate it for linear factors. $\endgroup$
    – Calvin Lin
    Jun 11, 2013 at 3:37
  • $\begingroup$ @Calvin Well of course one has to generalize it a bit to handle more general cases. Whether or not one calls the generalization by the same name is a matter of terminology. $\endgroup$
    – Key Ideas
    Jun 11, 2013 at 3:42
  • $\begingroup$ When you say generalize it a bit. What do you mean? I did not comment when I first read your post because I'm unfamiliar with a lot of what you said (coefficient rings and comaximal factors), so I was doing some research. I think the comment Calvin Lin first made here are my sentiments as well. My question is referring to the limitation on using the remainder theorem when the divisor is a polynomial greater than the first degree. There are cases when the remainder from poly division contains an x term and it seems the remainder theorem doesn't account for this; is this a limitation? $\endgroup$
    – Klik
    Jun 11, 2013 at 14:49

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