All 8 Uses

adieu
in
A Gentleman in Moscow
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But experience is less likely to teach us how to bid our dearest possessions adieu.†
p. 14.6

adieu = goodbye
1926 — Adieu†
p. 146.1
It was with this troupe that he had come to Moscow in 1913 where, having fallen in love with a bookseller in the Arbat, he had bid the circus adieu.†
p. 222.9 *
Adieu.†
p. 302.5
As a result, I am afraid that I must now bid you adieu.†
p. 404.9
So many years before, he had bid adieu to a whole household.†
p. 438.8
Then a few years later he had bid adieu to a suite.†
p. 438.8
Now, he was to bid adieu to a room that was one hundred feet square.†
p. 438.9

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)