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A bit of background to my question: last year I signed up for a refresher course in Maths at a university, because I seem to have forgotten most of what I learned in school about Maths but feel interest and need to get better at Maths though.

I take a course grouped into various segments. At the end of that (probably) multi-year journey stands the option to continue the course and study Maths at university level (to go for an associates degree or bachelor.)

In higher courses we will learn Calculus (which I never learned in school) and of course other advanced subjects. I read feedback from other students that being "algebraic competent" is very important to grok the more advanced topics, I also read on this website for example that "algebraic competency" is very important for Calculus and the lack thereof is often cause for struggling later on.

So, what would that "algebraic competency" be? Being able to manipulate even complex equations with ease? Or … something else? And how can I learn it?

Since this is my second try at the subject of Maths, and I also have more resources available to me than when I was young, I now have the option to already learn more about subjects that I can see will give me trouble later on.

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    $\begingroup$ The best people to ask are the math faculty at the university you are dealing with. $\endgroup$ – Gerry Myerson Jan 8 '18 at 2:47
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From my experience, Calculus teachers will expect you to know how to do some "basic algebra". They might give you a quick refresher at the start of the first semester but if at that point things are still fuzzy, you will feel that the course is heavy and hard to follow. So for example, being able to:

  • Isolate a variable ($2x+1=3x\Rightarrow x=1$)
  • Know your exponent and log laws
  • Know basic math symbols and notation such as $\mathbb{R}$, $\Rightarrow$, $\infty$, etc.
  • Knowing what a function is and their properties
  • Know your basic functions, their domains, ranges and graphs (not quite algebra but still useful)
  • Knowing vocabulary such as "constant", "variable", "factor", "polynomial", "degree", "term", etc.
  • Being able to use factorisation (of polynomials)

will all come in handy. You don't have to know everything from the start, but the more proficient you will be with this, the more smoothly Calculus will go.

Let me give you an example. In the first Calculus course, you will have to calculate limits like one: \begin{equation} \lim_{x\rightarrow\infty}\dfrac{x^2-x}{x-1} \end{equation} In order to do this, you have to see that $x^2-x=x(x-1)$ which will allow you to cancel out the $x-1$ on the numerator and denominator and get to the final answer (which is $\infty$). But if you are not used to do factorization, your eye might not see it and you might get stuck on the problem longer than normal (as in someone who is "algebraically competent").

The list I gave you already covers a lot but probably isn't complete. It's a good place to start.

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