"They work on an assignment from chapter three in this chemistry textbook."
Translation:Mereka mengerjakan tugas dari bab tiga di buku kimia ini.
In the strictly formal style ('baku') you're not allowed to drop the 'me-'.
In everyday colloquial speech, it's perfectly fine to drop it.
"Mereka mengerjakan tugas ..." (formal)
"Mereka kerjakan tugas ..." (colloquial, informal)
"Mereka kerjain tugas ..." (colloquial , using an unofficial suffix)
The 'me-' is dropped in imperative sentences, like this:
"Kerjakan tugas ini !!!"
It's wrong because /bekerja/ is specifically the intransitive form of the verb 'work' and because intransitive verbs can't take an object in Indonesian. They have to have a preposition following the verb if there's going to be an object after the verb. (What makes this more confusing to me is that /belajar/ can be either transitive or intransitive. Not everything that is grammatically intransitive with a /be-/ or /ber-/ prefix on it is semantically intransitive, but most verbs with /ber-/ on the front end will be intransitive verbs. Just know that there are exceptions to this usual pattern.
/kerjakan/ or /mengerjakan/ is Causitive (or Benefactive, but not in this instance). The following is an awkward way of stating it, but if it helps you, think of /mengerjakan/ as 'to cause to be worked'. Yeah, that sounds weird, but in English we still have a similar example of this for the verb (or root) 'work' which now shows up only in a few narrow contexts.
Have you ever heard of "wrought iron" (a.k.a. "cold iron")? It's iron that has been "wrought" or 'worked' on a forge and/or with a hammer and anvil... but not heated up so much nor treated in such a way that the iron has formed one of the crystal lattice structures needed to qualify it as having become a form of steel. Essentially, it's 'worked iron' or 'iron which has been worked on (or worked over).'
This sounds archaic to modern English-speakers, but you could ask the question "What have you wrought?" which literally means 'What have you worked.TRANSITIVE.PAST?' But nowadays, you'd more likely hear "What have you done?", "What did you do?", "What did you cause?", or "What did you make?" instead of "What have you wrought?"
In some regions of the English-speaking world, some folks say "Work this math problem," and that's a transitive use of 'work'. Other folks add a preposition after it, so that they say it as "Work on this math problem," which makes the statement look intransitive in form, but both statements mean the same thing.
Have you ever had a physical trainer or a coach who promises to "work you hard"? That'd be another transitive usage of "work" in English. Colloquial, perhaps, but still transitive.