# Polynomial Functions: Properties and Factoring Video

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Polynomial Functions: Exponentials and Simplifying

### You're on a roll. Keep up the good work!

Replay
Your next lesson will play in 10 seconds
• 0:06 Polynomials
• 7:26 Lesson Summary
Save Save

Want to watch this again later?

Timeline
Autoplay
Autoplay
Speed Speed

#### Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Eric Garneau
Everything from projectile motion to trigonometric functions can be described by polynomials. Review factoring, polynomials and quadratic functions in this lesson.

## Polynomials

Have you ever noticed how boring a lot of functions are? For example, the function f(x)=1 â€¦ very boring. A straight line. We can probably make it a little more exciting by adding x to it, so our function is f(x)= 1 + x. That's still kind of boring; it's just a straight line, but at least it's got a little bit of variation to it. It doesn't just stay one - at some places it's two! We can make this function a little more interesting by subtracting x^2 from it. And we can make it even more interesting by adding to that (1/2)x^3. Now we're starting to get somewhere. Did you know that you can create a function, just by adding up terms like that, that looks a lot like sin(x)? You don't have to add too many terms to describe what happens to someone if they're shot out of a cannonball. You might not be able to describe what happens when they hit the ground, but you can describe how high they are and when just by having three terms.

We really like functions where we just start adding up terms. These functions are known as polynomials. In particular, a polynomial is a sum of powers multiplied by constants, so we write this as f(x) = (a sub n)x^n + (a sub n-1)x^(n-1) and so on and so forth. We keep decreasing n to get (a sub 2)x^2 + (a sub 1) x^1 + (a sub 0). Now all of these a sub ns - a sub 1, a sub 2, all the way on up to a sub n - I can write as a sub i, and these are all constants, like 2, or -3.14. Now the big trick here is that x is only to a power in each term, such as x^n or x, or it doesn't appear in a term, like this a sub 0 term. You will never see sin(x) or log(x) or anything that is not x to some power. The largest power in this equation, x^n, defines the order of this polynomial. In this case, the polynomial is an nth-order polynomial. If you have a polynomial like f(x) = x^34 + 2, you have a 34th-order polynomial.

So what do you do with polynomials? Well, most common polynomials that you'll see are quadratics. These can do things like describe what happens to the height of a person as he's shot out of a cannonball. We write these as f(x) = ax^2 + bx + c. Now, quadratics are really cool, not only because of what they describe, but also because we know how to solve them. For example, if you have 0 = ax^2 + bx + c, you can solve - given some constants a, b and c - for x. To do this, you use the quadratic formula.

To unlock this lesson you must be a Study.com Member.

### Register to view this lesson

Are you a student or a teacher?

#### See for yourself why 30 million people use Study.com

##### Become a Study.com member and start learning now.
Back
What teachers are saying about Study.com

### Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.