This is the 5126th most frequent Polish word.
przekonanych
"Przekonanych" means "convinced" (plural, genitive case) in English.
The word 'przekonanych' here is used in the genitive plural form of 'przekonany', agreeing with the subject 'uczniowie' to denote their state of conviction.
Uczniowie byli przekonanych, że zdadzą egzamin.
The students were convinced that they would pass the exam.
In this sentence, 'przekonanych' functions as a descriptive participle attribute referring to 'those convinced', specifying results.
Sprawdzanie wyników przekonanych było konieczne.
Verifying the results of those convinced was necessary.
Here, 'przekonanych' is used in the genitive case to describe the people who had doubts about the decision's correctness.