I suspect that everyone does polynomial division the same way, the way we learned in high school. However, this can lead to large exponentially increasing errors. Problems will occur if the denominator has a large magnitude root and the quotient is long. For example, if 1000 is a root of a long polynomial and it is deflated, expect large errors. If the polynomial has high enough degree, expect infinity at some point in the deflated polynomial.
For an explanation of why this occurs and what you can do about it see:
cnx.org/content/m43398/latest/, "Exponential errors in polynomial evaluation, deflation, & division"
P.S. Understanding this article is helped immensely if you know the program Matlab.
While you are there, you might check out my other article:
cnx.org/content/m15595/latest/, "Information in the Spectrum of the Polynomial Coefficients"
Both articles are posted on Connexions which is a project of Rice University. Connexions allows anyone to publish, on the web, anything about any subject. There are no referees to shoot down your paper. Your paper does not need to be new information. If you feel that you are knowledgeable about some well researched subject, then write a summary paper. Connections will publish it for you on the web. Connexions makes it easy for anyone to become a published author. Do you have a paper in you?