I am a freshman college student taking a required algebra course. This content should be very easy, but I have struggled with the more complicated applications of algebra since I was young. So I have decided to familiarize myself with the foundations of algebra to give myself a solid foundation before reattempting learning the algebra, more out of interest than necessity - I know I could learn the steps by rote. (I'm more of a big-picture thinker. The issue comes down to getting myself together to solve the actual practical questions during the exams).
The content includes factoring, rational equations, and solving polynomials, the only subjects I'm really concerned with. What types of logic do these algebraic operations and problems stem from and use? How are they/were they proven? I have experience with basic first order, Boolean, predicate, propositional, and modal logic. Where should I begin? General mathematical logic? Boolean algebra? I'm overwhelmed. Any book recommendations, links, even just a general overview of what subjects I should go for would help.
Thank you in advance!
The steps of algebra are justified by the fact that $\mathbb R$ is a field. The properties of fields are what we assume (as axioms), and all steps of algebra are justified by them. When I teach algebra to my A-level students, I first introduce them to this idea briefly, but then they get used to autopilot.
Have a look at section 1.2 of these notes, and see how a lot of things you do automatically are proved.
Think about some of the exercises in exercise 1.5: why is $x+x = 2x$? How can you solve $2x+1=3$ justifying every step?
Also take a look at how to solve a quadratic in this mind set by looking at examples 3.9(iii).