Are there any limitations to the remainder theorem?

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.

• 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. Jun 11, 2013 at 2:08

• Can you explain how to calculate the remainder when $x^k - 1$ is divided by $(x-1)^n$? Jun 11, 2013 at 3:28
• @Calvin Use the division algorithm, or compute a taylor expansion at $\,x = 1.$ Jun 11, 2013 at 3:36
• 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?