How is synthetic division called in Germany?
I think it would be referred to either generally as Polynomdivision / Partialdivision, which is the general term for polynomial division, or alternatively as das Horner-Schema. We have that name in English too; Horner's Scheme/Rule/Method, and in both languages as far as I understand it's not exactly the same as saying "synthetic division" in English, but it's the underlying principle behind synthetic division. To my knowledge (and after doing some research), there isn't a more specific term for synthetic division in German.
From the English Wikipedia page on Horner's Method:
"...synthetic division is based on Horner's method."
Looks indeed to be easy: Synthetische Division