All 9 Uses

adieu
in
Madame Bovary
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Adieu!†
Chpt 2.13

adieu = goodbye
Adieu!†
Chpt 2.13
And there was a last "adieu" divided into two words!†
Chpt 2.13 *
The theatre rang with cheers; they recommenced the entire movement; the lovers spoke of the flowers on their tomb, of vows, exile, fate, hopes; and when they uttered the final adieu, Emma gave a sharp cry that mingled with the vibrations of the last chords.†
Chpt 2.15
The adieux were sad.†
Chpt 3.3
Adieu!†
Chpt 3.4

adieu = goodbye
adieu!†
Chpt 3.4
Suddenly she seized his head between her hands, kissed him hurriedly on the forehead, crying, "Adieu!" and rushed down the stairs.†
Chpt 3.5
He swung it gravely, then handed it to Charles, who sank to his knees in the earth and threw in handfuls of it, crying, "Adieu!"†
Chpt 3.10

Definitions:

(1) (adieu)
a farewell remark (an alternative to goodbye)

Adieu is typically more formal than bye. It comes from the French, à Dieu which literally means "to God" — as in I entrust you to God's care.
(2) (meaning too rare to warrant focus)