Learn Polish: Musimy rozdzielić te dwie grupy. - We have to separate these two groups.

The base verb dzielić means "to divide" or "to share". The prefix roz- adds the idea of separation, dispersal, or moving things apart in different directions. Together, rozdzielić means "to separate" or "to split apart" physically, which perfectly fits the context of pulling two groups away from each other.

What case is te dwie grupy, and why?

It is in the accusative case because it is the direct object receiving the action of rozdzielić. Since grupa is a feminine noun, its plural falls into the "not masculine-personal" (non-virile) category. Therefore, it uses the matching non-masculine forms: the pronoun te (these) and the numeral dwie (two).

Why is the perfective verb rozdzielić used here instead of its imperfective form?

The focus here is on successfully completing the action of separating the groups this one time. If you wanted to say "We have to separate these groups every week" (an ongoing or repeated action), you would use the imperfective equivalent, rozdzielać.

Could I use rozdzielić się to say the groups separated on their own?

Yes! Adding the reflexive pronoun się changes the meaning from separating something else to separating yourselves. If the groups split up on their own, you could say Te dwie grupy rozdzieliły się. In our sentence, there is no się because we are the ones doing the separating to them.