I have just now, purely by accident, stumbled upon another clue; I add it to this discussion here for the sake of completeness.
Synthetic division can be used to divide any polynomial by any other polynomial. If, however, the divisor is linear, synthetic division reduces to what is called "division by Ruffini's rule." The "Horner scheme" is a general algorithm for performing calculations with polynomials. The Horner scheme can be used to divide a general polynomial by a linear polynomial; when used this way, it is an implementation of division by Ruffini's rule and apparently identical to synthetic division. The Horner scheme seems to be a known term in German. In fact, the German wikipedia article on the Horner scheme is longer than the English article. Here are the two links.http://en.wikipedia.org/wiki/Horner_scheme
The German wikipedia article includes an example of polynomial division with a divisor of order 2. This leads me to suspect that what is known in English as synthetic division might be known in German as "das Horner-Schema".
* Ich entschuldige mich, denn ich auf Englisch schrieb. Leider kann ich mathematische Fachsprache nicht.