Algebra II—Semester A
Nothing complex here...except complex numbers.
- Credit Recovery Enabled
- Course Length: 18 weeks
- Course Type: Basic
- Category:
- Math
- High School
Schools and Districts: We offer customized programs that won't break the bank. Get a quote.
Shmoop's Algebra II 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.
This course has also been certified by Quality Matters, a trusted quality assurance organization that provides course review services to certify the quality of online and blended courses.
You had so much fun in Algebra that you had to come back for more? Yeah, we don't blame you.
Algebra II has all the expressions and equations you've seen before…and then some. You're sure to see some old familiar friends along the way (we're looking at you, polynomials), along with a few unfamiliar faces (we're looking—or at least trying to look at you, imaginary numbers).
Semester A starts off with us venturing into the land of the imaginary. (Feel free to extend an invite to your imaginary childhood friend, Maurice.) After getting our bearings, we'll check out a familiar landmark, our old friends the polynomials. They've learned some tricks, though, including shoving themselves under radicals and stacking each other into rational expressions and equations. We'll finish up the semester by working with more sequences and series than you can shake a finite stick at.
With loads of readings, problem sets, and activities, we'll cover:
- working with imaginary and complex numbers;
- multiplying and dividing polynomials and rational expressions;
- creating linear, quadratic, exponential, radical, and absolute value equations;
- graphing and solving nonlinear equations and inequalities;
- using formulas for arithmetic and geometric sequences and series.
P.S. Algebra II is a two-semester course. You're looking at Semester A, but you can check out Semester B here.
Here's a sneak peek at a video from the course. BYOP (bring your own popcorn).
Unit Breakdown
1 Algebra II—Semester A - Number Systems
We'll start the course off by diving headfirst into some numbers. Sure, you've been dealing with numbers for your whole mathematical career, but here we'll delve into the structure of integers, rational numbers, and real numbers. A tour of the complex number plane may even be in order.
2 Algebra II—Semester A - Introduction to Polynomials
You may have dealt with polynomials in the past, but you ain't never seen 'em like this before. Not only will we gain some new tips and tricks to help us deal with these pesky expressions, we'll learn and even prove a few theorems along the way. Yeah, we mean business.
3 Algebra II—Semester A - Polynomial Division and Rational Expressions
Had enough of long division? We sure hope not. We'll show you how to do it with polynomials and then move on to doing arithmetic with rational expressions. There'll be some throwbacks to factoring throughout, so make sure those skills are still sharper than a fine cheddar.
4 Algebra II—Semester A - Radical and Rational Equations
This is where we'll get you acquainted with radical and rational equations. All the algebra you'll need to solve equations with these guys will be covered in gory detail. We'll even take a moment to use our newfound knowledge to model some situations. What would a unit be without some applications?
5 Algebra II—Semester A - Polynomial, Rational, and Root Functions
You'll find a smorgasbord of functions in this unit. We'll take everything we've learned about polynomial, rational, and root functions and use it to create some pretty accurate graphs of these guys. As always, we've saved plenty of room for some tasty real-world apps.
6 Algebra II—Semester A - Sequences and Series
We'll cap the semester off with a tour of sequences and series. We'll introduce you to some nifty formulas to work with them, and give you a brand-spanking new tool for your mathematical toolbox to help you prove statements—proof by induction. Try to contain your excitement.
Recommended prerequisites:
Sample Lesson - Introduction
Lesson 6.01: Review of Arithmetic and Geometric Sequences
Some of us are super organized. You know the type—they've got a thousand color-coded tabs in their binders, and a daily planner the size of War and Peace.
But for the rest of us, making some kind of list is essential. Ever try grocery shopping without a list? Talk about a nightmare. Were we supposed to get eggs or egg whites? Marbled rib-eye or ground chuck? Snickers or Kit-Kat?
There are different kinds of lists, though. Your grocery list for a typical Thursday is gonna be a lot different than your giant shopping list for the Superbowl party you're hosting, right? The pattern is different—you may only need one bag of chips for your Thursday night Grey's Anatomy marathon session, but you'll need to crank that up to like 50 bags of chips if the whole block is coming over to watch the game.
Don't worry, we have a point here. The math version of a list is called a sequence, and it works the same way—all your numbers get listed in a particular order. For the two types of sequences we're looking at in this lesson, they even follow a pattern that lets you predict what the next few numbers in the list are gonna be. They're like the ultimate grocery lists.
Sample Lesson - Reading
Reading 6.6.01: Math and The Chamber of Sequence
Refresher time. A sequence is just a list of numbers in a certain order. (Not to be confused with a series, which we'll look at in the next lesson.) Nothin' too fancy here: 1, 2, 3,… and 110, 213, 316,… are both sequences. But we're more interested in two specific types of sequence: arithmetic and geometric. They even sort of rhyme, for all you poetry lovers out there.
Don't Be Apathetic / When Your Sequence Is Arithmetic
When each term in a sequence adds the same constant value every time, we've got an arithmetic sequence on our hands. For example, the sequence 2, 7, 12, 17,… is arithmetic because we're just adding 5 to each term: 2 + 5 = 7, and 7 + 5 = 12, etc.
We call 5 the common difference, since that's the, uh, difference between each term. And the dots at the end mean we're gonna follow that same pattern forever. Click here for a quick reminder of how these puppies work. Also, FYI, the common difference can totally be negative. The sequence 86, 83, 80,… would have a common difference of -3, since we're adding -3 to each term.
But it ain't always practical to add the common difference a bunch of times. What if we only know the first 4 terms and we want to find the 100th? Nobody's got time for that nonsense.
Good news: there's a formula for that.
Here's the formula for the nth term of an arithmetic sequence, where d is the common difference between each term and a1 is just the first term in the sequence:
an = a1 + (n – 1)d
That's a lot of letters, but it's actually pretty simple. For example, if you're trying to find the 10th term in the sequence 1, 4, 7, 10…, you'd just plug in a1 = 1 (since that's the first term), n = 10 (since you want the 10th term), and d = 3 (since each term goes up by 3). Ain't no thang:
a10 = 1 + (10 – 1)(3) = 1 + (9)(3) = 28
In other words, if you start with 1 and add 3 every time, your 10th term is gonna be 28.
Let's Get Geometric
But arithmetic sequences are only one side of the coin. They're pretty chill and straightforward, but there's also a more dramatic version: in a geometric sequence, each new term is the previous term multiplied by the same constant every time. We call this guy the common ratio instead of the common difference, since we're multiplyin' instead of addin'. Take a peek at this sequence.
3, 6, 12, 24,…
See the pattern? The common ratio is 2, since we're multiplying each term by 2 to get to the next one (3 × 2 = 6, 6 × 2 = 12, etc.). Here's a review of how to identify these dudes in the wild.
What's that? You want a formula for this version too? We've got you. To find the nth term in a geometric sequence where r is the common ratio between each term and a1 is the very first term (again), use this bad boy:
an¬ = a1rn – 1
So if you wanna find the 8th term of that sequence from earlier (3, 6, 12, 24,…), just plug in a1 = 3, n = 8, and r = 2. Voila:
a8 = 3 × 28 – 1 = 3 × 27 = 3 × 128 = 384
These guys tend to grow or shrink a lot quicker than their arithmetic cousins.
Recap
The two basic types of sequence are arithmetic and geometric. An arithmetic sequence adds a constant called the common difference to each term, which gives us the next term (and it can totally be positive or negative, btw). The formula for the nth term in an arithmetic sequence is an = a1 + (n – 1)d, where a1 is the first term and d is the common difference.
A geometric sequence, on the other hand, multiplies each term by a constant called the common ratio. And yep, it's got a sweet formula too: the nth term in a geometric sequence is an = a1rn – 1, where a1 is the first term and r is the common ratio.
There's even a trick for figuring out if a sequence is arithmetic or geometric. Take a bunch of pairs of consecutive numbers. If you subtract 'em and get the same difference for each pair of terms, your sequence is arithmetic. If that doesn't work, try dividing—if the ratio between each pair of terms is the same, then it's geometric for sure.
Sample Lesson - Activity
- Credit Recovery Enabled
- Course Length: 18 weeks
- Course Type: Basic
- Category:
- Math
- High School
Schools and Districts: We offer customized programs that won't break the bank. Get a quote.