Mathematics III—Semester A

It's like Math II, only completely different-er.

Shmoop's Mathematics III course has been granted a-g certification, which means it has met the rigorous iNACOL Standards for Quality Online Courses and will now be honored as part of the requirements for admission into the University of California system.


We understand—you're hankering for some good math. You went through Mathematics I, but you just weren't satisfied. Two semesters of Mathematics II, and you're still hungry for more. Well, we've got just what you need. That's right, this is Mathematics IV: A New Hope.

No, wait, sorry. We're on Mathematics III: Revenge of the Polynomials. We still need to work on our arithmetic.

We've got a platter of problem sets, a fully catered set of click-throughs, a smorgasbord of solvable examples, an almost alarming number of activities, and more—all aligned to the Common Core. Here's what you'll be doing with all of them in Semester A:

  • We'll kick things off by kicking back with our pals, the polynomials. We'll review the fundamentals—adding, subtracting, and factoring them every which way—then dive into the Fundamental Theorem of Algebra. (Whoof, sounds important, doesn't it?) Plus, we'll hang out with Pascal and his buddy, the Binomial Theorem.
  • The polynomial fun doesn't stop there. Have you noticed that we've been tap-dancing around the idea of polynomial division? Probably not—our dancing is just that good. Well, now's the time to start dividing polynomials together, creating rational expressions in the process.
  • Once you've gotten a taste of rational expressions, the obvious next step is to slap an equal sign down and see what kinds of equations you can make. Surprising no one, they're called rational equations. We'll learn how to deal with them, as well as radical and absolute value equations.
  • A picture's worth a thousand words, so next we'll graph all the equations we've been working with.
  • At this point we'll switch gears and mess around with sequences and series. They keep going and going and...
  • We'll wrap up the semester by introducing two new types of functions: exponential and logarithmic functions. Whether you've got values that grow super quick or molasses slow, this tag-team duo has you covered.

Just so you know, Mathematics III is a two semester course, and you're scoping out Semester A. If you want a taste of what's to come, click here.


Looking to veg out for a bit? Turn off the boob tube (we love you, TV, really) and turn on the Shmoop tube instead. Here's a sample of the kind of video content we've packed into this course, alongside our witty written lessons.


Unit Breakdown


  1. Mathematics III—Semester A - Introduction to Polynomials

    You may have dealt with polynomials a bit in Mathematics II, so get ready for the reboot with a few bonus features. Not only will we gain some new tips and tricks to help us deal with these expressions, we'll learn and—yes—prove a few theorems along the way. Yeah, we're ready to get down and dirty.

  2. Mathematics III—Semester A - Polynomial Division and Rational Expressions

    Do you have fond memories of long division? We're sure you do. Now you'll be doing it with polynomials, which we'll then use to build rational expressions. There'll be some heavy duty factoring here, so make sure you've got those skills up to snuff.

  3. Mathematics III—Semester A - Polynomial, Radical, and Rational Equations

    This is where we'll make our acquaintance with polynomial, radical, and rational equations. All the skills needed to solve these equations will be covered with a fine-toothed comb. We'll even take a moment for a detour into the real world to do some modeling. What would a unit be without some savory apps?

  4. Mathematics III—Semester A - Polynomial, Rational, and Root Functions

    The title of this unit may look virtually identical to the last one, but the material couldn't be more different. Now we're going to treat polynomial, rational, and root equations as functions involving x and y. That means we're good to go with graphing them. Once we have a graph in hand, nothing can stop us.

  5. Mathematics III—Semester A - Sequences and Series

    We'll enter new territory here with sequences and series. We'll provide you with some neat formulas to work with them, and introduce you to a mystical new way to prove theorems: proof by induction. Get hyped.

  6. Mathematics III—Semester A - Exponential and Logarithmic Functions

    This is a pretty hefty unit. Logarithms and exponents rule the day here, so we've got to play by their rules. (E&L take these things very seriously, and don't like it when we break them.) Then we'll throw this duo into equations and solve them in every which way we could possibly imagine: algebraically, graphically, and with technology.

  7. Mathematics III—Semester A - Geometry, Trigonometry, and Modeling

    This is where we'll study the strange interplay between geometry and trigonometry. It turns out we can use trig to find the area of a triangle, and the laws of cosines and sines are just other tools in our belt for finding missing sides and angles in triangles. Then we'll enter the third dimension and do a little modelling with 3-D solids. Get ready to strike a pose. No duck face allowed.

  8. Mathematics III—Semester A - Trigonometric Functions

    Get ready for a face-to-face meeting with trigonometric royalty. After introducing the unit circle and a brief review of the basic trig ratios, we'll be ready to up the ante by graphing trig functions, and shifting and flipping them all over the coordinate plane. We'll finish things off with some applications of what you can actually do with your newfound trig connections.

  9. Mathematics III—Semester A - Building Functions

    This unit is for the pragmatist in all of us. We're going to do our best Bob the Builder impression and build some functions. Using all the functions we've learned about so far, we'll toss them in a pile and mix-and-match them to our heart's content. It may sound like a frightening, genetic engineering project gone wrong, but we promise we know what we're doing.

  10. Mathematics III—Semester A - Statistics

    We're doing a 180 here by moving into statistics. We're talking everything from summary stats to the normal distribution and everything in between. By the end of it all you'll never look at data (maybe even the world) the same way again. We're 100% confident.

  11. Mathematics III—Semester A - Probability

    After a unit on statistics, it's only natural that we move into probability. These two topics are a lot more similar than you might think. Probability is all about taking what we know about a situation and making our best guess as to what'll happen next. Sometimes we'll be wrong, but that's just how this game works. Given that we live in a world so random that a butterfly in Kazakhstan might cause a war in Timbuktu, having the math to predict events is pretty cool.

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