What material is covered in a college algebra course?

[MAronnax]

Ok, this is kind of a weird question, and probably requires a bit of explanation. I'm planning a to take a few calculus classes for the next fall semester, and I know that while college algebra isn't "technically" required on paper, it is if you want to do well in the class, understand concepts, etc. Now, I haven't taken it, and no colleges near where I live offer it during the summer. So, I've decided to do it on my own at the library, a few days each week.

Now, I've seen enough math textbooks to know that classes generally cover around 1/3 of the book, and the rest is passed up and left out, for whatever reason. I was wondering if there was anybody who had recently taken (or taught) an algebra class who could give me a list of concepts and ideas covered, so that I could maximize my time and efficiency, it would be greatly appreciated. I don't need EXACTLY what was done from day to day, just the overall stuff.

Anyhow, this is kind of a long shot, as I can't really think of where all else I would ask this sort of thing, so any feedback at all would be helpful.

[Sartorius]

My college algebra class was basically a review of Pre-AP Algebra II in high school. I'm not in a position to dig out my notes to see exactly what we covered, but I think if you have a decent grasp of mathematics, and since you're reviewing on your own, you should do fine.

[merc]

As I remember, my "precalc" course was just a lot of working with functions...

You probably need to know all of the common, "elementary" functions. Polynomials, rational functions, exponential and logarithmic functions, and trigonometric functions. Are you familiar with these?

Do you know their properties? E.g. log(a^b)=b*log(a), a^(bc)=a^(b) * a^(c), etc.

Could you long divide a rational function? How about find the zeros and asymptotes (vertical or otherwise) for a rational function? Are you familiar with "partial fraction decomposition"?

Do you know your trig identities? To be fair, you don't really need to know most of them except when you do that chapter or two, but there's some good ones to know and they'd most likely be covered, including their derivation.

This is all I can come up with off the top of my head, but if you have positive answers to all these, you're probably set for calculus.

[auteur52]

a^(b+c)=a^(b) * a^(c), etc.

Fixed.

[merc]

Whoops. I'm definitely not a math undergrad or anything...

[yeyui]

If you DO want to know the specifics, Go to mathclass.org, create an account, and add an MA 109 course from (university of Kentucky). This will give you access to the online homework currently used for our college algebra course. You can complete the assignments and find out instantly if you have the correct answers, and get the correct answers also. If you just want to check that you are up to date, so speak, complete the last assignment which is a review for final exam. Here is set of notes used a few years ago http://www.msc.uky.edu/ken/ma109/notes.htm if you need some information, and here a colleague is posting videos for the course:http://megan.arthur.googlepages.com/ma109 as she teaches it this summer using the current lecture "activities" (read: scripts) currently used. This second link is an ongoing work as the (8-week) course just started.

There is also a college algebra text by one of our professors here http://www.ms.uky.edu/~sohum/ma109_fa07/fa07_edition/ma109fa07.pdf. We stopped using this text as it is more, um... sophisticated than was felt necessary. Students found the course with this text very hard. Ironically, the grades dropped noticeably when we switched to the less sophisticated course. Anyway, if you want to get a really good [i]understanding[i] of algebra, instead of just strong skills, I think this text is worth the read. (I have taught from both this text and the new text.)

Now if you really just want a list of important topics, here goes:

Have a strong understanding of the concept of functions.
Solve linear and quadratic equations without blinking.
Solve systems of equations, especially systems of linear equations.
Know basically everything that there is to know about lines.
Be on a first name basis with parabolas.
Have strategies for solving higher order polynomial equations.
Understand the Fundamental Theorem of Algebra and the relationships between roots, zeros, factors, remainders, intercepts, etc.
Sketch polynomial, rational, exponential, logarithmic, and sinusoidal functions by hand.
At one point you will need to use polar coordinates.
Know and internalize the algebraic properties of exponentials and logarithms.
Be familiar with the (six common) trig functions and
Be comfortable with basic trigonometry.
Complete the square
Partial fractions


Polar coordinates are usually introduced as a new topic and is not covered in our algebra course. It is in our "pre-calculus" and trigonometry courses though. Likewise none the trigonometric topics are in our algebra course. Many places do not place much emphasis on hand graphing skills. My high school pre-calculus course did, however, and I feel that it is extremely beneficial to success in calculus. The last two are algebraic techniques that are useful in integral calculus which come up at the very end of a first semester calc course or near the beginning of the second. These are sometimes introduced as a new technique in this context, but often in college courses they will be assumed.

[notzeb]

A college algebra class should teach you group theory (simple groups, Sylow subgroups, free groups, group actions on sets), field theory (Galois theory, finite fields), and a little bit of ring theory (prime/maximal ideals, principal ideal domains).

But I don't see how any of that will help you with calculus, unless you're studying Lie Groups or something...

[jmorgan3]

A college algebra class should teach you group theory (simple groups, Sylow subgroups, free groups, group actions on sets), field theory (Galois theory, finite fields), and a little bit of ring theory (prime/maximal ideals, principal ideal domains).

But I don't see how any of that will help you with calculus, unless you're studying Lie Groups or something...

Ignore this guy. He's talking about advanced topics that only math majors take.

[notzeb]

Advanced topics that only math majors take would be something like category theory, profinite groups, or semi-simple Lie algebras.

I was talking about basic college algebra (i.e. Math 5 at Caltech).

I don't know what this "graphing"/"trig function"/"logarithm" nonsense you guys keep jabbering on about has to do with college algebra. It looks a lot more like highschool "algebra" than actual algebra to me.

[merc]

Many universities have a class called "College Algebra" which is exactly what everyone else is talking about. It's a low-level class for people who need to take a math class for a very non-math major, or need calculus but have had very little math in high school. Usually what you're referring to is called "Advanced Algebra" or just "Algebra."

The real problem here is that the word "algebra" has so many different (even if related) uses. How did we let that happen?

[notzeb]

Many universities have a class called "College Algebra" which is exactly what everyone else is talking about. It's a low-level class for people who need to take a math class for a very non-math major, or need calculus but have had very little math in high school.

Whaaaaaaaaaaaaaaaaaaaat

Why don't they call it "highschool algebra for college students"?? If I put the adjective college in front of the word algebra, then it should be understood as something different from highschool algebra? Right? Right???

[RghtHndSd]

A college algebra class should teach you group theory (simple groups, Sylow subgroups, free groups, group actions on sets), field theory (Galois theory, finite fields), and a little bit of ring theory (prime/maximal ideals, principal ideal domains).

Galois theory is not even included in an undergrad course.

Advanced topics that only math majors take would be something like category theory, profinite groups, or semi-simple Lie algebras.

And these are definitely topics for graduate courses, not undergrad.

[auteur52]

A college algebra class should teach you group theory (simple groups, Sylow subgroups, free groups, group actions on sets), field theory (Galois theory, finite fields), and a little bit of ring theory (prime/maximal ideals, principal ideal domains).

Galois theory is not even included in an undergrad course.

We're already off-topic so...

Galois theory is definitely covered in most undergraduate Algebra classes. At my college, the third quarter of Abstract Algebra is field theory/Galois theory.

[AgentD]

Galois theory is definitely covered in most undergraduate Algebra classes. At my college, the third quarter of Abstract Algebra is field theory/Galois theory.

I'm uncomfortable with the use of "most" in that sentence, unless you took a survey or something...

My experience did not include Galois theory, but it was offered to us as a "we have some free time at the end of the year, what shall we do" topic. We picked something else.

[Fafnir43]

Galois theory is definitely covered in most undergraduate Algebra classes. At my college, the third quarter of Abstract Algebra is field theory/Galois theory.

I'm uncomfortable with the use of "most" in that sentence, unless you took a survey or something...

My experience did not include Galois theory, but it was offered to us as a "we have some free time at the end of the year, what shall we do" topic. We picked something else.

Let's do a survey here! My university offers it in the third year.

[Kag]

Why don't they call it "highschool algebra for college students"?? If I put the adjective college in front of the word algebra, then it should be understood as something different from highschool algebra? Right? Right???

One of the algebra courses at my high school uses a textbook entitled College Algebra. In fact, a quick googling indicates that this seems to be the accepted meaning of the phrase.

[leigao84]

I for one absolutely thought the term college algebra meant abstract algebra. Most courses that are designed to catch students up to speed are probably called Pre-Calculus or Intro to Math for Dumb People / English Majors.

[majikthise]

Bah, you crazy USians and your crazy course names!
Us Brits are sensible, and do courses like Applied Partial Differential Equations 2 without having an APDE 1 (at least in Bristol).
We don't do Galois Theory until fourth year (MSc) either.

[Yakk]

A college algebra class should teach you group theory (simple groups, Sylow subgroups, free groups, group actions on sets), field theory (Galois theory, finite fields), and a little bit of ring theory (prime/maximal ideals, principal ideal domains).

Galois theory is not even included in an undergrad course.

What? That's 3rd year material, not even 4th. And rather fun.

Advanced topics that only math majors take would be something like category theory, profinite groups, or semi-simple Lie algebras.

And these are definitely topics for graduate courses, not undergrad.

Category theory should be at least touched upon. I ran into it when I took a course on algebraic topology (the group of loops of the donut! Or, why Tim Hortons is a very funny place.)

I didn't take that much upper-year algebra, so I can't tell you if profinite groups or semi-simple Lie algebras where covered.

[Bsidney]

The original post for this thread was not ambiguous: the question was what algebra would be needed for a (presumably first) course in calculus. Whether one thinks of this stage of learning mathematics as appropriate to children or post-docs isn't relevant, at least not to the poster of the question.

I've been teaching calculus to undergraduates in the U.S. for many years. It is quite doable for a committed student to "boot-strap" into it without the usual course preparation. Much will depend on the level of mathematical maturity expected by your instructor, but you will definitely want to be comfortable with polynomials of a single real variable, how to factor and graph them, and how to find their roots. Some familiarity with trigonometry is also expected, and with logarithmic and exponential functions. Be prepared to spend double-time on homework; half for the assignment, half for looking-up and digesting any precalculus skills or concepts you lack that are assumed for the assignment. Identifying another student in your class who is as hard-working as yourself, and who is open to studying together, can be a great help to both of you, even if one of you is a little more advanced in your learning.

A first course in calculus is all about functions. In a sense, calculus is the science of functions of a single real variable. Get to know functions, their types and behaviors, their shapes and eccentricities. Develop a knack for identifying even and odd functions (look it up!), when they are bounded, smooth, continuous. Get a graphing calculator and think of it as your telescope for "seeing" functions. Become a "function naturalist."

Calculus is arguably the most important intellectual achievement of our species. It doesn't come cheap, but it is accessible to everyone, and hugely rewarding.

Good luck.

[merc]

I think the original question has been answered pretty sufficiently, so I'm going to chime in and say that I did basic Galois Theory during second semester "Advanced Algebra," along with modules and a bit of linear algebra with canonical forms.

[Yakk]

Barring an instructor chiming in (hey look, there's one!), you might want to copy/paste the course description for the course you are taking, and the course you aren't taking.

[Sygnon]

Galois theory is absolutely standard material for undergraduate algebra sequences. I would have more trouble finding fellow students who did not have it during their undergrad to be certain.

[majikthise]

Maybe we'll touch upon it in next (3rd) years Algebra courses, but Bristols "Galois Theory" course is definitely in the MSc year. Now I feel left out.