Uruk-hai is _a_ plural form. Uruks is an a anglicization. 'S' is the English plural marker. 
Uruk is a Black Speech word. 
Uruks is not.
  
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		| Unfinshed Tales, Index: Uruks Anglicized form of Uruk-hai of the Black Speech; a race of Orcs of great size and strength.
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		| Letter 78 Urukhai is a figure of speech. There are no genuine Uruks, that is folk made bad by the intention of their maker...
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 Note that Uruks (the anglicized plural) is used in place of Uruk-hai.
We have three uses of 
-hai, Uruk-hai, Olog-hai, Oghor-hai it is never used as a singular. 
-hai would seem to mean something like 
folk, race.
  
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		| 'Who's blame's that?' said the soldier. 'Not mine. That comes from Highter Up. First they say it's a great Elf in bright armor, then it's a sort of small dwarf-man, then it must be a pack of rebel Uruk-hai; or maybe it's all the lot together.' | 
	
 It is quite clear that this is referring to events at Cirith Ungol. Frodo is the 'small dwarf-man'. Sam was mistakenly thought to be an Elf-warrior. The 'rebel Uruk-hai' would be the Orcs who might have fled from Cirith Ungol after the fight over Frodo and his mail. If you have any evidence that Saruman's Uruk-hai were involved I would be interested in reading it.
  
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		| 'We are the fighting Uruk-hai! We slew the great warrior. We took the prisoners. We are the servants of Saruman the Wise, the White Hand: the hand that gives us man's-flesh to eat. We came out of Isengard, and led you here, and we shall lead you back by the way we choose. | 
	
 This does not show that the Uruk-hai originally came out of Isengard. What it does show is that _these_ Uruk-hai came out of Isengard. These Uruk-hai were in the service of Saruman. Sauron had his own Uruk-hai in his service and had for a long time:
  
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		| In the last year of Denethor I the race of uruks, black orcs of great strength, first appeared out of Mordor, and in 2475 they swept across Ithilien and took Osgiliath. | 
	
 [ December 15, 2002: Message edited by: Tar Elenion ]