"I need a book."
If you think of the translation of potrzebuje as 'having need of', maybe the use of the genitive case makes more sense
This comment helped a lot!!! "I am in need of a book" would help me, tremendously, to remember that this verb "potrzebowac" would be genitive. Even though the sentence does not read "potrzebuje z ksiazki", it helps my mind wrap around this case. Thanks for the comment.
While your version, being in biernik (accusative) is quite widely used (1880 results in google for the exact phrase, vs 4080 for "potrzebuję książki"), the linguists would say that the only case which can be used with "potrzebować" is dopełniacz (genitive), thus making "Potrzebuję książki" the only really correct option.
I am still baffled that anyone would say this with the Accusative. Sounds clearly wrong to me.
Why is it książki and not książky? I'm using the grammar book "Polish Verbs & Essentials of Grammar" (see pg 18) and the declension for the genitive case of książka should be książky because książka is a hard stem feminine noun. Is this noun an exception to the rule? Why/Why not?
the translation should be 'Potrzebuję książkę' not 'książki' because książki is the plural form. Książka means book and książki means books
To put it short: no.
Potrzebować comes with genitive (dopełniacz). Singular genitive of książka is „książki”. And yes, it is the same as nominative plural (genitive plural is: „książek”).
Your surname suggest that you might be of Polish nationality and that you know the language, if that's the case consider why we say: „potrzebować czasu” and why not „potrzebować czas”.
We call genitive „dopełniacz” (to complement, to make a whole) in Polish because it indicate that something is lacking. When we are in need of something we lack something.