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Polynomial Functions: Properties and Factoring

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  • 0:06 Polynomials
  • 3:00 Quadratic Polynomials
  • 3:40 Factoring Quadratics
  • 7:26 Lesson Summary
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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

Graph of a polynomial function
Function Example Graph

Have you ever noticed how boring a lot of functions are? For example, the function f(x)=1 … 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.

Quadratic Polynomials

A polynomial is a sum of powers multiplied by constants
Polynomial Sum of Powers

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.

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