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.
Best Answer
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").