Problem : Why are there three types of RNA polymerases in eukaryotes and not in prokaryotes?

Eukaryotic RNA polymerase comes in three forms (I, II, and III). Each of these forms catalyzes the transcription of a different class of mRNA. Prokaryotic RNA polymerase comes in one form that catalyzes the transcription of all forms of mRNA.

Problem : How do eukaryotic polymerases recognize promoter regions?

Eukaryotic promoters are incapable of recognizing promoter regions on their own the way prokaryotic RNA polymerase can. Instead, eukaryotic prolymerases rely on additional transcription factors that bind to the promoter sequence and recruit RNA polymerase.

Problem : What is the name of the RNA polymerase II promoter element that is similar to the -10 region of prokaryotic polymerase promoters?

The TATA box.

Problem : What is the name of the most important transcription factor for RNA polymerase II? What is its function?

TATA-binding protein binds to the TATA promoter region and helps recruit RNA polymerase II.

Problem : What is the additional structural unit makes eukaryotic DNA less accessible to transcriptional machinery?

Eukaryotic DNA is found wrapped around nucleosomes.