Polynomial and rational functions

Factoring quadratics

Just saying the word "quadratic" will make you feel smart and powerful. Try it. Imagine how smart and powerful you would actually be if you know what a quadratic is. Even better, imagine being able to completely dominate these "quadratics" with new found powers of factorization. Well, dream no longer. This tutorial will be super fun. Just bring to it your equation solving skills, your ability to multiply binomials and a non-linear way of thinking!

Completing the square and the quadratic formula

You're already familiar with factoring quadratics, but have begun to realize that it only is useful in certain cases. Well, this tutorial will introduce you to something far more powerful and general. Even better, it is the bridge to understanding and proving the famous quadratic formula. Welcome to the world of completing the square!

Quadratic inequalities

You are familiar with factoring quadratic expressions and solving quadratic equations. Well, as you might guess, not everything in life has to be equal. In this short tutorial we will look at quadratic inequalities.

Adding and multiplying polynomials

"Polynomials" sound like a fancy word, but you just have to break down the root words. "Poly" means "many". So we're just talking about "many nomials" and everyone knows what a "nomial" is. Okay, most of us don't. Well, a polynomials has "many" terms. From understanding what a "term" is to basic simplification, addition and subtraction of polynomials, this tutorial will get you very familiar with the world of many "nomials." :)

Dividing polynomials

You know what polynomials are. You know how to add, subtract, and multiply them. Unless you are completely incurious, you must be wondering how to divide them! In this tutorial we'll explore how we divide polynomials--both through algebraic long division and synthetic division. (We like classic algebraic long division more since you can actually understand what you're doing.)

Synthetic division

In this tutorial, we'll learn a technique for dividing one polynomial by another--synthetic division. As always, we'll also explore why it works!

Factoring and roots of higher degree polynomials

Factoring quadratics are now second nature to you. Even when traditional factoring is difficult, you know about completing the square and the quadratic formula. Now you're ready for something more interesting. Well, as you'll see in this tutorial, factoring higher degree polynomials is definitely the challenge you're looking for!

Polynomial graphs and end behavior

In this tutorial, we will study the behavior of polynomials and their graphs. In particular, we'll look at which forms of a polynomial are best for determining various aspects of its graph.

Binomial theorem

You can keep taking the powers of a binomial by hand, but, as we'll see in this tutorial, there is a much more elegant way to do it using the binomial theorem and/or Pascal's Triangle.

Simplifying rational expressions

You get a rational expression when you divide one polynomial by another. If you have a good understanding of factoring quadratics, you'll be able to apply this skill here to help realize where a rational expression may not be defined and how we can go about simplifying it.

Rational functions

Have you ever wondered what would happen if you divide one polynomial by another? What if you set that equal to something else? Would it be as unbelievably epic as you suspect it would be?

Asymptotes and graphing rational functions

Partial fraction expansion

If you add several rational expressions with lower degree denominator, you are likely to get a sum with a higher degree denominator (which is the least-common multiple of the lower-degree ones). This tutorial lets us think about going the other way--start with a rational expression with a higher degree denominator and break it up as the sum of simpler rational expressions. This has many uses throughout mathematics. In particular, it is key when taking inverse Laplace transforms in differential equations (which you'll take, and rock, after calculus).

Advanced structure in expressions

This tutorial is all about *really* being able to interpret and see meaning in algebraic expressions--including those that involve rational expressions, exponentials, and polynomials. If you enjoy these ideas and problems, then you're really begun to develop your mathematical maturity.

Fundamental Theorem of Algebra

This tutorial will better connect the world of complex numbers to roots of polynomials. It will show us that when we couldn't find roots, we just weren't looking hard enough. In particular, the Fundamental Theorem of Algebra tells us that every non-zero polynomial in one-variable of degree n has exactly n-roots (although they might not all be real!)