Polynomials review

In this video I want to introduce you to the idea of a polynomial. It might sound like a really fancy word, but really all it is is an expression that has a bunch of variable or constant terms in them that are raised to non-zero exponents. So that also probably sounds complicated. So let me show you an example. If I were to give you x squared plus 1, this is a polynomial. This is, in fact, a binomial because it has two terms. The term polynomial is more general. It's essentially saying you have many terms. Poly tends to mean many. This is a binomial. If I were to say 4x to the third minus 2 squared plus 7. This is a trinomial. I have three terms here. Let me give you just a more concrete sense of what is and is not a polynomial. For example, if I were to have x to the negative 1/2 plus 1, this is not a polynomial. That doesn't mean that you won't ever see it while you're doing algebra or mathematics. But we just wouldn't call this a polynomial because it has a negative and a fractional exponent in it. Or if I were to give you the expression y times the square root of y minus y squared. Once again, this is not a polynomial, because it has a square root in it, which is essentially raising something to the 1/2 power. So all of the exponents on our variables are going to have to be non-negatives. Once again, neither of these are polynomials. Now, when we're dealing with polynomials, we're going to have some terminology. And you may or may not already be familiar with it, so I'll expose it to you right now. The first terminology is the degree of the polynomial. And essentially, that's the highest exponent that we have in the polynomial. So for example, that polynomial right there is a third degree polynomial. Now why is that? No need to keep writing it. Why is that a third degree polynomial? Because the highest exponent that we have in there is the x to the third term. So that's where we get it's a third degree polynomial. This right here is a second degree polynomial. And this is the second degree term. Now a couple of other terminologies, or words, that we need to know regarding polynomials, are the constant versus the variable terms. And I think you already know, these are variable terms right here. This is a constant term. That right there is a constant term. And then one last part to dissect the polynomial properly is to understand the coefficients of a polynomial. So let me write a fifth degree polynomial here. And I'm going to write it in maybe a non-conventional form right here. I'm going to not do it in order. So let's just say it's x squared minus 5x plus 7x to the fifth minus 5. So, once again, this is a fifth degree polynomial. Why is that? Because the highest exponent on a variable here is the 5 right here. So this tells us this is a fifth degree polynomial. And you might say, well why do we even care about that? And at least, in my mind, the reason why I care about the degree of a polynomial is because when the numbers get large, the highest degree term is what really dominates all of the other terms. It will grow the fastest, or go negative the fastest, depending on whether there's a positive or a negative in front of it. But it's going to dominate everything else. It really gives you a sense for how quickly, or how fast the whole expression would grow or decrease in the case if it has a negative coefficient. Now I just used the word coefficient. What does that mean? Coefficient. And I've used it before, when we were just doing linear equations. And coefficients are just the constant terms that are multiplying the variable terms. So for example, the coefficient on this term right here is negative 5. You have to remember we have a minus 5, so we consider negative 5 to be the whole coefficient. The coefficient on this term is a 7. There's no coefficient here; it's just a constant term of negative 5. And then the coefficient on the x squared term is 1. The coefficient is 1. It's implicit. You're assuming it's 1 times x squared. Now the last thing I want to introduce you to is just the idea of the standard form of a polynomial. Now none of this is going to help you solve a polynomial just yet, but when we talk about solving polynomials, I might use some of this terminology, or your teacher might use some of this terminology. So it's good to know what we're talking about. The standard form of a polynomial, essentially just list the terms in order of degree. So this is in a non-standard form. If I were to list this polynomial in standard form, I would put this term first. So I would write 7x to the fifth, then what's the next smallest degree? Well, they have this x squared term. I don't have an x to the fourth or an x to the third here. So that'll be plus 1-- well I don't have to write 1-- plus x squared. And then I have this term, minus 5x. And then I have this last term right here, minus 5. This is the standard form of the polynomial where you have it in descending order of degree. Now let's do a couple of operations with polynomials. And this is going to be a super useful toolkit later on in your algebraic, or really in your mathematical careers. So let's just simplify a bunch of polynomials. And we've kind of touched on this in previous videos. But I think this will give you a better sense, especially when we have these higher degree terms over here. So let's say I wanted to add negative 2x squared plus 4x minus 12. And I'm going to add that to 7x plus x squared. Now the important thing to remember when you simplify these polynomials is that you're going to add the terms of the same variable of like degree. I'll do another example in a second where I have multiple variables getting involved in the situation. But anyway, I have these parentheses here, but they really aren't doing anything. If I had a subtraction sign here, I would have to distribute the subtraction, but I don't. So I really could just write this as minus 2x squared, plus 4x, minus 12, plus 7x, plus x squared. And now let's simplify it. So let's add the terms of like degree. And when I say like degree, it has to also have the same variable. But in this example, we only have the variable x. So let's add. Let's see, I have this x squared term, and I've that x squared term, so I can add them together. So I have minus 2x squared-- let me just write them together first --minus 2x squared plus x squared. And then let me get the x terms, so 4x and 7x. So this is plus 4x plus 7x. And then finally, I just have this constant term right here, minus 12. And if I have negative 2 of something, and I add 1 of something to that, what do I have? Negative 2 plus 1 is negative 1x squared. I could just write negative x squared. But I just want to show you that I'm just adding negative 2 to 1 there. Then I have 4x plus 7x is 11x. And then I finally have my constant term, minus 12. And I end up with a three term, second degree polynomial. The leading coefficient here, the coefficient on the highest degree term in standard form-- it's already in standard form --is negative 1. The coefficient here is 11. The constant term is negative 12. Let's do another one of these examples. I think you're getting the general idea. Now let me do a complicated example. So let's say I have 2a squared b, minus 3ab squared, plus 5a squared b squared, minus 2a squared b squared, plus 4a squared b, minus 5b squared. So here, I have a minus sign, I have multiple variables. But let's just go through this step by step. So the first thing you want to do is distribute this minus sign. So this first part we can just write as 2a squared b, minus 3ab squared, plus 5a squared b squared. And then we want to distribute this minus sign, or multiply all of these terms by negative 1 because we have the minus out here. So minus 2a squared b squared minus 4a squared b. And a negative times a negative is plus 5b squared. And now we want to essentially add like terms. So I have this 2a squared b squared term. So do I have any other terms that have an a squared b squared in them? Or sorry, a squared b. I have to be very careful here. Well, that's ab squared, no. a squared b squared. Oh! Here I have an a squared b. So let me write those two down. So I have 2a squared b minus 4a squared b. That's those two terms right there. Let me go to orange. So here I have an ab squared term. Now do I have any other ab squared terms here? No, no other ab squared, so I'll just write it: minus 3 ab squared. And then let's see, I have an a squared b squared term here. Do I have any other ones? Well, yeah sure, the next term is. That's an a squared b squared term, so let me just write that. Plus 5a squared b squared minus 2a squared b squared, right? I just wrote those two. And then finally I have that last b squared term there, plus 5b squared. Now I can add them. So this first group right here in this purplish color, 2 of something minus 4 of something is going to be negative 2 of that something. So it's going to be negative 2a squared b. And then this term right here, it's not going to add to anything, 3ab squared. And then we can add these two terms. If I have 5 of something minus 2 of something, I'm going to have 3 of that something. Plus 3a squared b squared. And then finally I have that last term, plus 5b squared. We're done. We've simplified this polynomial. Here, putting it in standard form, you can think of it in different ways. The way I'd like to think of it is maybe the combined degree of the term. Maybe we could put this one first, but this is really according to your taste. So this is 3a squared b squared. And then you could pick whether you want to put the a squared b or the ab squared terms first. 2a squared b. And then you have the minus 3ab squared. And then we have just the b squared term there. Plus 5b squared. And we're done. We've simplified this polynomial. Now what I want to do next is do a couple of examples of constructing a polynomial. And really, the idea is to give you an appreciation for why polynomials are useful, abstract representations. We're going to be using it all the time, not only in algebra, but later in calculus, and pretty much in everything. So they're really good things to get familiar with. But what I want to do in these four examples is represent the area of each of these figures with a polynomial. And I'll try to match the colors as closely as I can. So over here, what's the area? Well, this blue part right here, the area there is x times y. And then what's the area here? It's going to be x times z. So plus x times z. But we have two of them! We have one x times z, and then we have another x times z. So I could just add an x times z here. Or I could just write, say, plus 2 times x times z. And here we have a polynomial that represents the area of this figure right there. Now let's do this next one. What's the area here? Well I have an a times a b. ab. This looks like an a times a b again, plus ab. That looks like an ab again, plus ab. I think they've drawn it actually, a little bit strange. Well, I'm going to ignore this c right there. Maybe they're telling us that this right here is c. Because that's the information we would need. Maybe they're telling us that this base right there, that this right here, is c. Because that would help us. But if we assume that this is another ab here, which I'll assume for this purpose of this video. And then we have that last ab. And then we have this one a times c. This is the area of this figure. And obviously we can add these four terms. This is 4ab and then we have plus ac. And I made the assumption that this was a bit of a typo, that that c where they were actually telling us the width of this little square over here. We don't know if it's a square, that's only if a and c are the same. Now let's do this one. So how do we figure out the area of the pink area? Well we could take the area the whole rectangle, which would be 2xy, and then we could subtract out the area of these squares. So each square has an area of x times x, or x squared. And we have two of these squares, so it's minus 2x squared. And then finally let's do this one over here. So that looks like a dividing line right there. So the area of this point, of this area right there, is a times b, so it's ab. And then the area over here looks like it will also be ab. So plus ab. And the area over here is also ab. So the area here is 3ab. Anyway, hopefully that gets us pretty warmed up with polynomials.