Le livre rendu à la bibliothèque est déjà sur l’étagère.

Rendu is a past participle of rendre, and here it works a lot like an adjective modifying livre.

So le livre rendu à la bibliothèque is a shortened way of saying something like:

• le livre qui a été rendu à la bibliothèque

In English, we also do this sometimes:

• the returned book
• the book returned to the library

So in this sentence, rendu à la bibliothèque describes which book we are talking about.

In French, a past participle used this way usually comes after the noun it describes.

So:

• le livre rendu à la bibliothèque = the book returned to the library

This is very normal in French. English sometimes puts this kind of description after the noun too:

• the book returned to the library

If you put rendu before livre, it would sound wrong here.

This is a very common question.

French usually uses rendre for returning something to someone / giving something back.

So:

• rendre un livre à la bibliothèque = to return a book to the library

By contrast, retourner often means:

• to go back
• to turn over
• sometimes to return in other contexts, but not usually for giving back a borrowed item

So for a library book, rendre is the natural verb.

The preposition à here marks the place or institution something is returned to.

• rendre un livre à la bibliothèque
• literally: return a book to the library

French uses à in many cases where English uses to.

Also:

• bibliothèque is feminine singular
• so à + la bibliothèque becomes à la bibliothèque

Here, rendu agrees with livre, which is:

• masculine
• singular

So the correct form is simply rendu.

If the noun changed, the participle would change too:

• le livre rendu → masculine singular
• la revue rendue → feminine singular
• les livres rendus → masculine plural
• les revues rendues → feminine plural

So there is agreement here; it just happens that the masculine singular form has no extra ending.

Est is the present tense of être (is).

The full main clause is:

• Le livre ... est déjà sur l’étagère.

That means the main statement is about where the book is now.

So the sentence structure is:

• Le livre rendu à la bibliothèque = the subject
• est = the verb
• déjà sur l’étagère = the rest of the predicate

In other words:

• The book returned to the library → subject
• is already on the shelf → what is being said about it

French adverbs like déjà often come after the verb être:

• est déjà

So:

• est déjà sur l’étagère

This is the natural French word order.

English also does something similar:

• is already on the shelf

So this part matches English fairly closely.

Because sur means on, and a book placed on a shelf is thought of as being on it.

• sur l’étagère = on the shelf

French usually uses sur with étagère in this context.

Using dans would mean in the shelf, which would suggest being inside the furniture rather than resting on a shelf surface.

This is because of elision.

When la comes before a word beginning with a vowel sound, French usually drops the vowel and uses an apostrophe:

• la étagère → l’étagère

The same thing happens with:

• le ami → l’ami
• je aime → j’aime

So l’étagère is just the normal shortened form before a vowel.

Yes. A more explicit version would be:

• Le livre qui a été rendu à la bibliothèque est déjà sur l’étagère.

This means the same thing, but it is longer and more explicit.

French often prefers the shorter participle structure when it is clear:

• le livre rendu à la bibliothèque

That is similar to English preferring:

• the book returned to the library

instead of:

• the book that was returned to the library

French uses articles more regularly than English.

In this sentence:

• Le livre = the book
• à la bibliothèque = to the library
• sur l’étagère = on the shelf

A native English speaker may wonder whether French could drop them, but in normal French, these articles are expected.

Also notice:

• le = masculine singular
• la = feminine singular
• l’ = le or la before a vowel sound

So the article changes depending on gender and pronunciation.