Your spacecraft has just touched down in the middle of a bizarre planet overrun by a strange species called "polynomials." They look a little funny, have unusual rules and customs, and don't smell the greatest. But hey, you're a mathronaut, and it is your job to befriend them and communicate that we mean them no harm. Hopefully their intentions are similarly peaceful.
A polynomial is an expression consisting of numbers and letters or, as we like to call them in the algebra galaxy, constants and variables.
In this unit, we'll probe deep into what makes a polynomial tick. According to numerous first-hand reports, they've spent decades abducting and probing us, so why shouldn't we probe them back? We'll evaluate them, add them together, pull them apart, subject them to a highly experimental process called "factoring," and then solve them. It's icky, and we may need to wade through a ton of alien guts, but we'll learn a lot in the process.
Fine, maybe we do mean them a little harm.
In order to perform these experiments on polynomials in as gentle a manner as possible, we'll need to remind ourselves of a few basic rules about exponents. Once the process is done, we will close up shop with a few words on scientific notation. See how we hang that tantalizing carrot out there to ensure you stick with us until the end?
Snap on a pair of rubber gloves and fasten your safety goggles, because the secrets of polynomials are about to be revealed.
Now let's talk a bit about polynomials minus the alien analogy.
Polynomials are a particular kind of expression, and we can do with them the usual things we can do with expressions: we can evaluate, add, subtract, multiply, and divide them. We can also factor them, which is like factoring integers, but different. We shall tell you how so.
Before talking about polynomials, however, we'll discuss exponents and expand our previous definition of whole number exponents so we can use any real number as an exponent. The definition of polynomials only uses whole number exponents, but it's good to take a look at other exponents so we can see what polynomials are and what they aren't. Sorry if this is getting too existential for you.
Be careful: When working with polynomials, the best thing you can do is work slowly and carefully. It is easy to slip up, especially if you are doing these problems while standing on a waxy surface.
Problems involving polynomials are usually more tedious than difficult, and it's not hard to make a careless mistake and get a math test back with points off. Then you look at what you did wrong, slap your forehead, and go, "Argh! Stupid, stupid, stupid...I know how to do that!" This can be the Silly Mistake section, but if you're slow and careful, it can totally be the Sweet, I Know How To Do This section. Now stop hitting yourself.