polish vs brush up

polish is a noun but brush up is not a noun.

Word	Noun	Adjective	Verb	Adverb
polish	Yes	No	Yes	No
brush up	No	No	Yes	No

As verbs, polish and brush up are synonyms defined as:

polish and brush up: bring to a highly developed, finished, or refined state

Other synonyms of polish include polish up, round, round off.

polish (noun)	brush up (noun)
the property of being smooth and shiny
a highly developed state of perfection; having a flawless or impeccable quality
a preparation used in polishing

polish (verb)	brush up (verb)
bring to a highly developed, finished, or refined state	bring to a highly developed, finished, or refined state
improve or perfect by pruning or polishing	refresh one's memory
make (a surface) shine