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Mathematics III—Semester A

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

  • Credit Recovery Enabled
  • Course Length: 18 weeks
  • Course Type: Basic
  • Category:
    • Math
    • High School

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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.

Technology Requirements

You're reading this on our website, so you already meet the first requirement for this course: some kind of computer with some kind of internet access. A scientific or graphing calculator would be helpful, but isn't 100% required.

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.

Recommended prerequisites:

  • Mathematics I—Semester A
  • Mathematics I—Semester B
  • Mathematics II—Semester A
  • Mathematics II—Semester B

    • Sample Lesson - Introduction

    Lesson 2.07: Deriving the Quadratic Formula

    Here's lookin' at you, hamburger.

    (Source)

    Classics are classics for a reason. Whether it's Casablanca, Michelangelo's David, or hamburger, fries, and a milkshake, these things warm our hearts and upset our stomachs like nothing else. (We were actually taking about Casablanca on that last one. "Here's lookin' at you, kid"? Seriously? Barf.) 

    In the math world, one of these classics is none other than the quadratic formula. It sits in the Algebra Hall of Fame, right next to the Pythagorean Theorem and y = mx + b, draped in ribbons and adorned with jewels. 

    But what makes it such a classic? With Michelangelo's David, it's the detail and precision of the artwork; with Casablanca, it's the superb acting and the heartwarming romance; with a hamburger, it's the fatty, meaty, delicious flavor. 

    And with the quadratic formula? Well, that's what we'll find out in this lesson—right after we scarf down another burger.

    Sample Lesson - Reading

    Reading 2.2.07: Let There Be the Quadratic Formula

    Ah, the quadratic formula: the longest collection of letters, symbols, and numbers you've memorized to date. Unless we're counting your dad's Amazon password, that is. ("No, Dad! You ordered me that Xbox, remember?") 

    We already remember how to do things like complete the square, and we're really hoping that you at least remember the quadratic formula. Negative b plus or minus the square root of b squared minus—yeah, that thing. 

    The thing is, this formula didn't just fall from the sky and land squarely in our laps, error-free and ready for us to use. It's actually derived from the process of completing the square. Want more deets on deriving the quadratic formula? Take a look at this.

    Recap

    The quadratic formula can be proved by applying the process of completing the square to a general quadratic of the form ax2+ bx + c = 0. Assuming we complete the square correctly and properly, we'll end up with this beaut: 

      

    Sample Lesson - Activity

    1. True or false: The quadratic formula is derived by completing the square.

    2. The first step in one derivation of the quadratic formula has been done. In order to complete the square so we can eventually isolate x, we must get a coefficient of one on that x2 term. What must we do to get the x2 to have a coefficient of one?

    3. We've decided factoring out an a from the two terms on the left is the way to go. What would that look like if we started with ax2 + bx = -c?

    4. The first two steps of one derivation of the quadratic formula have been done. The next step is completing the square. What value should go inside the parentheses after the term?

    5. Continuing from the previous problem, there's one more thing we need to do to our equation. Because we added something to one side of the equation, we must add the same thing to the other side. What must be added to the -c on the right side?

    6. We're almost ready to solve for x, but we need to complete the square by turning our perfect square trinomial into the square of a binomial. Which expression below is the correct next step?

    7. Suppose we had reached the following step while trying to derive the quadratic formula. To solve for x, we'll take a square root, but first, we need to isolate the squared term. What would our equation look like after doing so?

    8. Continuing from the last problem, which expression below has correctly taken the square root of both sides?

    9. If we get x all by itself, we'll have a Frankenquadratic formula. It is a version of the formula, just not the one we're used to. Which expression would we get if we isolated x from the previous problem?

    10. The previous problem's equation is a real mess. We need to clean it up by getting some common denominators on the right side. What would it look like after doing so?

    11. From the result of the previous problem, we can rearrange the fractions inside the radical in a more familiar pattern. Which expression shows the correctly simplified denominator?

    12. Last step! We just need to add our two fractions together, and we've got ourselves the world famous quadratic formula. Which expression is the correct quadratic formula?