I understand that a CRC verifies data integrity by producing a checksum, which is the result of polynomial long division. I've heard hash values referred to as hash checksums, so my question is whether hash functions use some sort of polynomial division as well? I know they break the data up into block ciphers, so my guess would be that the hash functions create some relationship between the polynomial check value and how it's divided into the different blocks. Can someone let me know if I'm way off base here? I'm a crypto newbie so apologies if I'm way off.
I've heard hash values referred to as hash checksums, so my question is whether hash functions use some sort of polynomial division as well?
No, they do not.
Actually, the original meaning of "checksum" has nothing to do with polynomial division at all; instead, it was just a simple sum used to detect errors. The usage has been generalized to mean anything added to the data to detect random errors; a CRC does that, and a hash of the data could be used as well.
These both work well about detecting random errors. However, since we're in crypto stack exchange, we also get to worry about deliberate changes made by an intelligent attacker; in that case, neither CRCs nor hashes help (as the attacker can easily compute the CRC/hash of the modified data). If this is the problem space you're in, you need to use either a Message Authentication Code or a Signature; they both depend on secret information that the attacker does not know.