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Polynomials - Overview

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.


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Now that we've reviewed some of the basic concepts of functions in general, we're gonna walk through and talk about some special types of functions that we're gonna encounter again and again. And now the first type we're going to talk about is polynomial functions in rational functions. But let's start with the simplest possible function, and that's a constant function. So a constant function outputs exactly one real number for every input. So our function ffx will just be some numbers. See where C is a real number. And even though these air the simplest types of functions, they can be really confusing. Because, let's say I ask, what is f of to? Well, you say Okay, I'm supposed to everywhere I see an ex plug in to, but they're no exes. I only have that the function is equal to the constant. But that's because for every input in this case to output the same number C. If I wanted f of zero, I would get see whatever that real number is. So note that the domain of a constant function well, I can plug in any real number and get the same constant output, So the domain is all real numbers and now the range of the constant function, the set of outputs where there's Onley, one output. So the range of a constant function is just the one number C, and we can write that Justus the set containing the number seat. There's Onley, one output for any input, which the input could be any real number. We have exactly one output see, no matter what I poke in. So if I plug in negative one, I get see if I plug in 100 I'm going to get C. So it's very, very simple. But it can be a little tricky, because again, there are no exes toe plug numbers into. But that's because every output will always just be c mhm. So stepping up just a little bit is another type of function that I'm sure you've seen over and over again. And that's a linear function. So a linear function, as the name suggests, is just a line. The graph of a linear function is online, but a linear function has the following form. F of X is equal to mm x plus be we're here in is a real number called the slope and then be is also just a fixed real number called the Y intercept. And I'm assuming this is review For a lot of you. The slope is just kind of the rise over the run because the graph of this function, if we look at the graph so it's gonna look something like this. So the slope is just how much I rise when I go over one unit like that, and then the Y intercept is where I cross the y axis. So that's where these names were coming from. And like I said, the graph of the function of a linear function is just a line and now as long as and is not zero. So the slope is not zero. The function f is 1 to 1, and the inverse well, we could just solve. We do our same tricks we saw for Why are we switch X and Y and sulfur y in terms of X? And the inverse ends up just being X minus B over him. And here we see definitely am needs to be not zero because we were dividing by zero here. So that is as long as in is not equal to zero. The function are linear. Function will be one toe one. So there will be a unique output for every input. And then the inverse of a linear function is again a linear function on it looks like this. Okay, So moving on from linear functions, we'll jump up and talk about power functions. So power function Well, it's called a power function because it really is just a power of X. So it looks something like this. Ffx is equal to see X to the A and here, see and A are both really numbers. And we're going to require that c is not equal to zero. Because if C is equal to zero, this is actually just a constant function. It's just the constant function equal to zero. So a special case of a power function occurs when this A is actually a non negative integer. And this is the case. We're gonna look at a little bit more so special cases when a is a non negative. So I'm including zero non negative integer. Okay, So, like X squared is a power function or three. X cubed is another power function. And so in this special case. What we want to note is that F is 1 to 1. If in is odd, sorry, it is on now notice that makes sense, because I'm assuming that Okay, in general, a could be like a half or square root of two or whatever. But in a special case, when a is a non negative integer, if a is odd, an odd non negative integer so 1357 whatever at this 1 to 1 and then F is not one toe one if a is even. But in both cases, the domain meth is all real numbers. There's no problem plugging in any real number into a power function, and now the range is a little bit trickier. So the range will be all real numbers if is odd, and then if a is even, it's either going to be 02 infinity or negative infinity to zero, depending on whether C is plus or minus. But we'll explore that a little bit later when we look into actually the calculus of power functions, that's when that will come up again. But next we want to talk again. In the special case of power functions about what's called the end behavior of the power functions. Okay, so notice that I have this nice table here on I want to pick up just a couple of things that we're assuming. So we're assuming that we have a power function c x to the end. Now I'm using in here because I only want to consider the case where in is actually a natural number. So that would be a non negative integer, not including zero. So 12345 etcetera. And what we see is that we have four cases and the cases are based on whether or not in is an odd earn even number. And whether the number that we're multiplying, the power function by which we're assuming is not zero is either positive or negative. And we're looking at what's called the end behavior. And the easiest way to understand the in behavior is to actually look at a graph. So I want to just do a quick sketch of these four cases of power functions because this is going to be very important later when we look at what's called limits at infinity of different types of functions. So I'm just going to sketch a quick graph of each of the cases. I'm sure you've seen a lot of these before, but let's just sketch him again. So in the first case, this is an odd power function, meaning that the power is odd and C is positive. And now what this is saying is that the in behavior as X goes to positive infinity. So this way the function F. It's also going to posit infinity like this and then his ex goes to minus infinity. The function is actually dropping off to negative infinity like that. That's if she is positive. If she is negative, the graph is actually reflected over the X axis like this. So the in behavior is opposite Is X goes to minus infinity. F goes deposit infinity and his ex goes to positive Infinity, Afghanistan minus infinity. Now the cases for in being even, are a little bit different. So if she is positive, this is like our classic parabola. So it's a happy face because it's positive. The coefficients see is positive, So his ex goes toe plus infinity f goes to plus infinity in his ex goes to minus infinity f goes to minus infinity. And then if we take C to be negative, the graphic it's reflected over the X axis. So we have a sad face like this because our co efficiency is negative. Okay, and again, is X goes to minus infinity f is going to minus infinity and his ex goes to plus infinity f is also going to minus infinity. So this is the in behavior of thes power functions, and that's gonna play an important role later on in the course. So now that we have a handle on power functions, we can define polynomial and polynomial zehr kind of the bread and butter of what we're going to be doing now we're gonna consider more complicated functions as well. But a lot of the ideas will be best understood just by looking at polynomial. So let's go ahead and define what we mean by a polynomial so polynomial function in polynomial. Yeah, what this word actually means, Polly meaning many. No meal meaning terms. It's a mini term function, so polynomial function honest. It's a son of power functions. See x to the end. We're here. We're seeing that in is a non negative number. That's our non negative integer. So now I say non negative because I actually want to include zero because notice that if, uh in is equal to zero X to the zero will be one, and we'll just have cease or actually allowing constant terms. We're allowing X to the first terms. Ex the second terms X to the third terms, etcetera. So that's what a polynomial function is. And so a general example of the polynomial function is something like this. So it might look like two x cubed minus four x plus one. That's a example of a polynomial function. Now there's a couple important things that we need to identify about polynomial function. So the degree of a polynomial function is the largest power. Yeah, so in our example, let's identify the degree. So the degree is supposed to be the largest power of one of the terms. And notice that here that example is three. So this example is a degree three, and now the leading term is the actual power function with the largest end. Okay, so let's identify that in our example. So the leading term is going to be this entire power function to execute. That's the leading term. And then finally we have what's called the leading coefficient, and the leading coefficient is going to be the number that's multiplying the highest power of X in the leading term. So in this case, the leading coefficient is going to be to, and the leading coefficient is going to play a big role in the end behavior of the polynomial. But we're going to talk about that a little bit later. So just a couple of notes about polynomial Paula. No meals are not in general, 1 to 1. So most polynomial is don't have em verses. And then the next note is that Paul no meals have all real numbers as their domain. So there's no problem plugging any riel number into a polynomial and out putting a real number. So that is just a couple of key facts about Paul. No meals will see these time and time again. I will make a note that a linear function is an example of a polynomial function. Notice that a linear function is a degree, one polynomial. So let's talk about a few other features of polynomial is so we already mentioned that a linear function is a degree one polynomial, so a degree to polynomial also has a special name. It's called a quadratic function, and we can actually keep going if you're really up to date with your Latin. This is actually not too hard. So the degree three polynomial Well, quad. Okay, so then we have a cubic function, and then we'll just say etcetera. We have a quart IQ Quinn tick. So Degree four would be a core tech quintet will be a degree five. You could just keep going on and on and on. So those were just some special types of functions that you'll see thrown out. This is a quadratic function. This is a cubic function, but they're all these types of polynomial is They're all just sums of power functions where the powers are non negative integers. So this next point is actually something true about all functions in general. But it's discussed a lot with polynomial, so the zeros of a function are riel numbers in the domain of the function. So let's say our function is F, so the zeros of a function F are real numbers in the domain of Beth such that Well, the word zero kind of gives it away. Ffx is equal to zero. So we're looking for numbers X in the domain of the function for which F is equal to zero. And now these real numbers that we're just calling X. So the real numbers X in the domain of F such that f of X zero are actually what are called the zeros. So the numbers X that you plug into the function to get zero are called the zeros of the function. It's a little confusing, but you'll get the hang of it. Okay, so for a degree, one polynomial finding the zeros is really easy. Because remember that a degree one polynomial looks like this. Okay? And so again, we need to assume that the slope is not zero. So if the slope is not zero? Yeah, Then there is exactly 10 and it's given by Well, you just set this equal to zero and solve for X. Okay, so it's gonna be negative. Be over. Yeah, that's going to be the zero of the degree one polynomial the linear function. Okay, so you just plug in negative, be over em. you're going to get zero for the function, assuming that N is not equal to zero. So for a degree to polynomial, this is a really good review. So it's kind of fun to go up to people on the street and ask them to remember the quadratic formula. And a lot of people may or may not remember it, but it's even more fun to ask people. What is the quadratic formula used for? Well, the quadratic formula is used to find the zeros of a degree to polynomial. So remember, this is again called Hey Quadratic! And a quadratic has the following general form. Yeah, a X squared plus B x plus c. Okay, and again, we're going to assume that a is not zero otherwise, is actually not a quadratic if a zero then is really just a linear function. So if is not zero, the zeros are so it's negative. B. So it's negative. The coefficient in front of the X to the first term, plus or minus square it of B squared minus four times a. Sometimes the leading coefficient times seat or C is the constant er, all divided by two A. So you see. Of course, we need a to not be zero. Otherwise, this doesn't make sense. Just like here. We needed him to not be zero. And now you say Okay, well, this plus or minus just means that for one of the zeroes, there's there's actually possibly going to be two zeros, one of them. I'll take the positive square root here, and one of them I'll take the negative, but there may actually be no zeros, so notice that it could be possible for there to be a negative under this square root or this square root underneath the square root could be exactly zero. So there's actually three possible cases. So a quadratic could have zero, one or two zeros and what we see. How could we have zero zeros? Well, that's if this factor B squared minus four, I see is negative. We have exactly 10 If this factor B squared minus for a Z is actually equal to zero, and then we have two zeros. If this factor B squared minus four, A. C is greater than zero. And so because of the importance of this term underneath the square root, it actually gets a special name, so it's called the Discriminate. It It's the discriminate of the polynomial. Now there's also some special cases that can occur with quadratic functions. So quadratic functions, actually, in general polynomial functions can be factored, and this is often times a better way to find the zeros. And the quadratic formula is always going toe work. But in some cases we can factor the quadratic. So if the quadratic looks like this X squared plus bx plus c, what we're looking for is we're actually looking for numbers that I'll call Alfa Beta camera and Delta. So these air numbers that when I foil out so I multiply this term. By this term, I used to distribute property this term times this term, etcetera. So if I foil this out so first outside inside last, I should get exactly this. So if I'm lucky and I can actually guess what these numbers are now, that's not always easy. But what we're looking for, we're looking for Alfa and Gamma to multiply together to give us a We're looking for beta and Delta to multiply together to give us see, and we're looking for Alfa Times, Delta plus Beta times gamma thio equal be so If we can sort of guess from these conditions what these numbers are, then we can actually factor the quadratic. And so why does this help us give two zeros? Because if I set this equal to zero, there's this really important property of the real numbers that if I'm multiplying two things together in that product happens to be zero than one of these two factors has to be zero. So either Alfa X plus beta is zero, or there's that keyword Oregon Gamma X Plus Delta is Europe, but that means that either X is equal to negative. Beta over Alfa or X is equal to negative delta over gamma. And what you see is that these air actually exactly the zeros you will find in the quadratic formula. So if we can factor and I'll do an example in a second, then we can very quickly find the zeros of a quadratic and in general for higher degree polynomial. So let's do a quick example just to illustrate what I mean. In case you forgot, suppose that we had X squared minus five X plus six, okay, and I want this t equals zero. So here's my quadratic function. I'm setting equal to zero to find the zeros of the function. Now, notice that my A here is one. So that tells me that Alfa Times gamma has to be one. Okay, well, then I'll just let alfalfa in Yemen equal one. So this should be X plus or minus something, and then X plus your mind is something now see is equal to six. So that means beta times Delta. These two numbers that I'm adding should multiply together to give me six and then also will sense again. Alfa and Gamarra, one delta plus beta should be negative five. So I'm looking for two numbers that multiply together to give me six and add together to give me negative five. Well, that's negative to negative three. And now again by the zero product property either X minus two has to be zero or X minus three has to be zero. Okay, so then X equals two or X equals three. Those were the two zeros of this quadratic function, so I talked a lot about how to factor a quadratic function, her quadratic equation. But I mentioned that for higher degree polynomial. It's also very useful to be able to factor them. And so I included just a list of very common factoring techniques here that you may need to use sometime later in the course. And we definitely will use thes later in the course. So I would suggest maybe just keeping this is a reference. So when you see something like a difference of cubes X squared minus a squared, where is a real number? Know that it factors is X minus eight times, experts say. When you see X cubed minus a cube, know that it factors is X minus a times X squared plus X plus a squared. Now these other ones. This this next one is just kind of a general phenomenon of the difference of squares difference of cubes. But it's also useful to know how squaring X plus a expands and how explicit a cute expanse. So it's just something to keep in the back of your mind. So when you see something like X cubed minus eight, you should think right away. Oh, great, I can write. That is X minus two times X squared plus two x plus four. Because eight is too cute. So it's gonna be really nice if you could remember. Okay, Maybe you're taking a limit Derivative something. We'll see this later. You may see something like this. Well, being able to factor this is gonna allow you toe work through the problem a lot easier. So the last thing we're gonna talk about in this topic is what's called a rational function. Now, we're not going to say much about rational functions right now, because really, we're going to discuss them once we start talking about limits and in behavior of functions, actually in the in the course material for calculus. But let's just introduce what a rational function. It's so a rational function. And now it's sort of like a rational number. So remember, a rational number is a question of two integers. Irrational function is a question of to polynomial. Yes, let's say P and X and G of X. So our function looks something like this. P of X, divided by Q Pecs. So we have a polynomial and top with some degree. We have a polynomial bottom with some degree, and I'm sure you can remember from your pre calculus class that you learned a little bit about rational functions and really Cem useful facts regarding the degree of the numerator, the degree of the nominator. We're actually going to go in and prove some of those facts that you may remember from pre calculus. But for now, just know that irrational function is the question to Paula. No meals. And now the domain of F. Well, it's just gonna be the set of real numbers. Remember that the domain ever of a polynomial function is all real numbers, so there's no problem. Unless so, we need to think about the zeros of cute because a zero of Q is going to make the denominator zero, and we're going to be dividing by zero, which is a problem. So the domain of F is going to be all real numbers, except for those that make the denominator zero so X and are such that Cube X is not equal to zero, not to be the domain of a rational function. In general, rational functions will not be one toe one. Sometimes it will be, but in general not so in general. They will not have in verses. And like I said, we'll talk a little bit more about Thean behavior and finding the zeros, uh, you know, well, actually, spend some time thinking about how to graph rational functions, actually end the calculus course.