Study Guide

Polynomials

Polynomials Introduction


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's 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'll 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'll 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's easy to slip up, especially if you're 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.

Polynomials Resources

Websites

Purplemath: Negative Exponents
Still not entirely sure about the connection between negative exponents and fractions? For more help with negative exponents, check out this page at PurpleMath. You'll have only a fraction of the fun on another website (although exponentially more fun at Shmoop).

S.O.S. Math: Quadratic Polynomials – Factoring by Guessing
You won't be asked very often by your algebra teacher to guess. However, trial and error is a perfectly acceptable method for factoring quadratic polynomials. Click "Next" at the bottom of the page to see how to factor by completing the square or by using the quadratic formula. Quadratic formula, by the way, is what you should feed babies if you want them to grow up to be math wizzes.

Video

Math Rap – Scientific Notation
Exactly what it sounds like. If you're looking to get jiggy with the algebra, here's your chance. This unique video will help you firm up your understanding of scientific notation while providing a beat you can groove to at the same time. This is totally going to be your jam.

Math Rap – Factoring
If you enjoyed the above video, you…uh, may or may not enjoy this one. The title is deceiving. This is about the furthest thing from rap we can possibly imagine. However, if you're interested in some lighter fare, this little ditty might do the trick. Or it might make you throw up. You have been warned.

Khan Academy: Multiplying Monomials by Polynomials
Let this video help you make sense of how to smush together those monomials and polynomials without getting tragically lost. Does multiplying monomials by polynomials give us monopolynomials? That's our favorite game.

Degree of a Polynomial with Multiple Variables
For a refresher on finding the degree of a polynomial with multiple variables, you can't go wrong with this vid. Just don't let the speaker's calm, soothing voice put you to sleep. You have homework to do tonight.

Games and Tools

SliderMath.com: Simplifying Polynomials
See if you can simplify these polynomials and figure out which multiple choice answer is correct before the dolphin does. Yes, there's a dolphin, and he's apparently good at this sort of thing. Who would have guessed?

Quia: Polynomial Jeopardy
Play by yourself or challenge a friend to a game of Polynomial Jeopardy. Practice adding, subtracting, and multiplying polynomials, and see how many points you can rack up. When you're done, Alex Trebek will come by and shake your hand. That's awesome. We love that guy.

Quia: Evaluate the Polynomial
Fill in the given values for x or y and evaluate each of the polynomials. If you're successful, you'll make the brain wearing the shoes very happy. Collect coins and fill up your piggy bank! Yeah, there's a lot going on.

WebMath: Multiply Two Polynomials
You can plug in any two polynomials and this page will give you the answer. It doesn't show you the work it took to get there, however, so please only use it for checking your answers. This is a handy tool, but you don't want to be one...if you know what we mean.