In studying for the Algebra Qualifying Exam, these are some exercises you should really really know. Most are either common questions on past qualifying exams or popular homework problems chosen from Hungerford’s Algebra.
Since these exercises are so fundamental, solutions to many of them can be found either in John Dusel’s notes, or in Kayla Murray’s notes, or somewhere online like Math Stack Exchange (MathSE). If you find a solution online, you should send me a link so I can post it here. Otherwise, if you think it’ll help you study, you can type up a solution and send me a PDF to post here. Or better, you can type up your solution on MathSE so that other algebra students can easily find it, add to it, comment on it, etc. The MathSE community is going through a bit of a phase right now, though, so it would be a good idea to read over this brief guide to posting on MathSE before writing up your solution there.