Learn Portuguese (Portugal): É possível que eles tenham depositado o cheque. - It is possible that they have deposited the cheque.

The expression É possível que (It is possible that) introduces uncertainty, which requires the subjunctive mood. Because the action of depositing the cheque already happened, we use the Perfect Subjunctive (tenham depositado) to express a past action wrapped in present doubt.

How exactly is tenham depositado built in this sentence?

It uses the Present Subjunctive of the helper verb ter — which is tenham to match eles — followed by the past participle of the main verb depositar, which is depositado.

Does the word depositado need to be plural (depositados) because the subject is eles?

No. In compound tenses formed with the auxiliary verb ter, the past participle never changes to match the subject. It always stays in the default masculine singular form, which is depositado.

If the sentence were about a certainty, like "I know they deposited the cheque", how would the verb change?

If there is no doubt, you drop the subjunctive. The phrase "I know that" is Eu sei que, which triggers the normal indicative past tense. The sentence would become: Eu sei que eles depositaram o cheque.