Like humans, zoo animals must have a dentist to __ their teeth

The correct answer is option number 1st.

To +V1 is called as the infinitive form if the verb.

After `to' we use first form if the verb.

Infinitive form if the verb is used when we want to express the purpose,to answer why.

Ex-He studied hard to make his parents proud.

Why he do study -to make his parents proud.

Infinitive form is used as a subject of verb,as a object of verb,as the complement of the verb,as a object of preposition,to qualify noun,to qualify a sentence.

Here it is used as a `Adverb'.It qualifies verb `Studied'.

In the given example, options -`to be filled,filling,filled'are wrong.Because after `to' we can't use 3rd form of the verb,rest options are grammatically incorrect.