All 8 Uses

adieu
in
Ivanhoe
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Adieu, my dear friend; I have said enough to explain, if not to vindicate, the attempt which I have made, and which, in spite of your doubts, and my own incapacity, I am still willing to believe has not been altogether made in vain.†
Chpt Ded.

adieu = goodbye
Once more adieu; "vale tandem, non immemor mei".†
Chpt Ded. *
To buy his favour I extend this friendship:
If he will take it, so; if not, adieu;
And, for my love, I pray you wrong me not.†
Chpt 6
Adieu—and having taken the medicine which I shall send thee by the hand of Reuben, compose thyself again to rest, that thou mayest be the more able to endure the journey on the succeeding day.†
Chpt 28
Meanwhile, adieu.†
Chpt 32
Rowena waved a graceful adieu to him of the Fetterlock—the Saxon bade God speed him, and on they moved through a wide glade of the forest.†
Chpt 32
—Adieu, my Prince, till better times.†
Chpt 34
She hastened to bid Rowena adieu.†
Chpt 44

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)