Yes, the great Mrs. Mazz said that algebra 1 is a quite rigorous course it requires a great deal of thought and perseverence.
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No, algebra 1 is defiantly not the same as math course 3.
Linear algebra concerns vector spaces whether finite- or infinite-dimensional. Abstract algebra, or modern algebra, includes linear algebra, along with many other kinds of objects, such as groups, rings, fields, lattices, and so on. In part, it was an attempt to put mathematics on a more rigorous footing. Please see the links.
Probably not much; of course you'll learn new things (just as you did in algebra), but they will be introduced gradually (just as they were in algebra).
yes you would have to because that would be the full course of algebra just at a slower and easier pace
Depends. Most of us would not know what "Algebra Foundations 2" or "Algebra 1B" are, since the name of such a course is probably determined by the school or local district. Compare the curricula of both courses to see if they are the same.