Polynomials Functions- Properties and Factoring
Everything from projectile motion to trigonometric functions can be described by polynomials. Review factoring, polynomials and quadratic functions in this lesson.
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Graph of a polynomial function
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 n s - 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 n th-order polynomial. If you have a polynomial like f(x) = x ^34 + 2, you have a 34th-order polynomial.
A polynomial is a sum of powers multiplied by constants
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.
The other really cool thing about quadratic polynomials is that we can factor them, usually pretty easily. When we say we’re going to factor them, we’re going to take the whole equation, f(x) = ax ^2 + bx + c , and we’re going to find two things that multiply together to make that whole. So, for example, 2 x ^2 + 7 x + 3 can be written as (2 x + 1)( x + 3). We have factored 2 x ^2 + 7 x + 3 into two things that can be multiplied together. When we factor a polynomial, specifically a quadratic, we’re going to look for things that we can multiply together to make that quadratic. Specifically, we’re going to look for things that have the form some number times x plus some other number. We’re going to see how many of these we have to multiply together to make our polynomial.
Let’s take a look at an example. Let’s factor x ^2 + 4 x - 12. The first part that I’m going to factor is this number that goes in front of the x . To find the number that goes in front of the x , I’m going to look at the x ^2 term.
The x ^2 term is just 1 x ^2, so 1 can be factored as 1 1. That’s our only option. So each one of these 1s is going to go in front of the x in each of our two terms. To get the second number in our terms, we’re going to look at the constant in our quadratic.
In this factoring example, the terms must have a product of -12 and sum of 4
This -12 can be factored as -12 1, -1 12, -2 6, -6 2, -3 4, and -4 3. So any one of these combinations could go in for the second number in each of these terms. So which one is it? To determine which one it is, we look at the middle term in our quadratic, this 4 x . Now 4 x has to be the sum of our two numbers. If we add these two together, 1 and -12, we get -11. 12 - 1 is 11, and so on and so forth. Now sure enough, one of these numbers is what we need, which is 4. We know that the two constants that are in our two terms are -2 and 6. So I can rewrite this as ( x - 2)( x + 6). Now let’s go ahead and check: ( x - 2)( x + 6) is x ^2 + 6 x - 2 x - 12, which is x ^2 + 4 x - 12.
So the tricks to factoring quadratics: Use the a in front of your x ^2 to determine the coefficients of your factored terms. You’re going to use c to determine the options for your constants, and you’re going to use b to determine which of those options to use. Now obviously, it can get very complicated very quickly, so you should remember that not everything can be factored, and factoring takes a lot of practice.
To review, polynomials can be used to describe almost anything. The quadratics are our favorite types of functions because we can factor them, and we can solve them using the quadratic equation.
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