All 7 Uses

adieu
in
Pride and Prejudice
(Edited)

...in his manner of bidding her adieu...there was a solicitude, an interest which she felt must ever attach her to him with a most sincere regard.
p. 149.9

adieu = goodbye
Adieu to disappointment and spleen.
p. 152.5 *
...and in the clamorous happiness of Lydia herself in bidding farewell, the more gentle adieus of her sisters were uttered without being heard.
p. 227.9

adieus = goodbyes
They entered the woods, and bidding adieu to the river for a while, ascended some of the higher grounds;
p. 242.8

adieu = goodbye
Adieu! (ending a letter)
p. 262.7
Mr. Wickham's adieus were much more affectionate than his wife's.
p. 312.6

adieus = goodbyes
The letter, perhaps, began in bitterness, but it did not end so. The adieu is charity itself.
p. 348.7

adieu = closing words

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)