Okay, welcome. And in this section here we're going to talk about polynomials. Okay, and we're going to define what that is right here in this bullet. a polynomial is a algebraic expression, using literal numbers with more than one term. Alright, let's just go through this one term is used. It is a monomial, a monomial, meaning one.

Mono means one. So I have one term. An example of that would be a two a three X, five y squared or four A B. All right. I have one term and if you remember from previous discussions, we may have a term but in this example, for a B, for a be a factors within that term. Alright.

Let's go on to the next bullet right here. a binomial has two terms. Okay? example would be a plus b, or three x plus five y. Here's my terms right there in there. So in this expression, A plus B, A is one term, and B is another.

And this expression over here, three x is a term, and five y squared is a term. So we have two terms and that expression It's called a binomial by means to. All right, we have special binomials with the same terms, but opposite signs are called conjugates. And what do we mean by that? Well, A plus B and A minus B, notice that the signs are different. We have a plus and a minus.

So when we have that, they're called conjugates. Right there. And the last one we see here is a trinomial. Try means three. Okay, an example would be a squared to a b, plus b squared. I have three terms and again Each term contains could contain factors.

Alright. So with that said, I'd like you to take a couple of minutes, stop the presentation and do the exercise that I give you on the bottom here. It what I'm asking you to do is identify each of the following as monomials binomials or trinomials. All right, and I will do one for you one or two. Okay, I'm going to pick this one here. And that one is what that is a binomial, because I have two terms.

Right over here is a trinomial. Why? Because I have three terms. Okay. So what I'd like you to do is stop the presentation here. Take a few minute, do the exercise.

And as always, the answers are on the next slide. Okay, here we are with the answers. I did this one for you. It's a monomial, binomial binomial. monomial, one term in there, but two factors, binomial I term term trinomial term term term, trinomial turn term term. Another trinomial term term term, but we know that in this example here, we have factors in each one of those terms.

And right here we have a monomial. And we have how many things factors in that term 1234. All right, we're done here. We're going to go on to the next topic. All right here we're going to talk about a little bit more about polynomials. And just before we start, it's some general practice to write terms in descending order.

So for instance, what we mean by that is, if you look at the first one, first one, look at the powers of x, how they descend, x to the third power x squared x, this one over here to a cubed, three A squared, and then six and the same thing on the last one here, x cubed, y, x squared, and y squared, x and y but what we've done is or what they do is they Pick the highest power of the exponent, and they descend in order with that. So even though I've, I've got y here, which is just y to the One Power, they've selected the X in their in descending order with the x, x cubed, x squared and then x. And then they just fill in the other literal number to make sure that the expression is correct. Okay. The next bullet here is when I multiply a polynomial by a binomial, I multiply each term in the polynomial by the monomial.

So for instance, I want to multiply x plus y by two y. Well, this is what I Want to do right there? All right. So what do I do? I multiply this by that, and this by that. So I get to x y times x plus two x y times y.

When I combine the terms, I get two x squared, because x times x is x squared plus two x y, because when I multiply the y's, I get an x y. So I can say, therefore to x y times x plus y equals to x squared y plus two x y squared. I'm going to I'm going to clear the slide so you guys can see that and here's my answer right here. All right. Okay. So take a look at it.

And there's I'm sure there's going to be some exercises. In a few slides down, we've got a couple of more to talk about. And again, we're here to help you. Alright, right. In the previous slide, we talked about multiplying a polynomial by a monomial. Now we're going to divide a polynomial by a monomial.

Okay, so what we do is we divide each term in the polynomial by the monomial. So my example here is I want to divide s six x squared plus 12 x by three x right here. So basically, I put this in the numerator, this in the denominator, okay. Then I can break them up. And if you remember from previous discussions, we can break them up in the sense where I can say six x squared over three x plus 12 x over three x right here. And now what I can do is I can cancel out the terms, but the factors in each one of these terms, for instance, here, and here.

Alright, so let me clear the slide. And three here, goes into six, how many times to all right, if I divide, let me get that to there. If I divide x into x squared, I'm left with x. All right. 12 divided by three is four. And my x cancel out.

So what is my answer to x plus four? All right. So the reason I brought this up here into two distinct terms here, and here is so I can take my factors and cancel them out or use the division in the fact doors. All right. So let's clear the slide here. All right, let's look at this bottom part here the slide where I'm going to multiply two polynomials.

Meals together. All right. And what we want to do here is we want to multiply each term in one polynomial by each term in the other. All right, I'm going to do an example here for you. So the example that I give you is I want to multiply x squared plus x x plus three by x minus one. So the way we set this up here is what I show you.

We take the larger one, meaning the one with the with, with the more terms and we placed that on top right there. And then we take the other polynomial with with the less number of turns, and we put that to the left, like I'm showing you here. All right. Then we want to multiply each term and we start With the left hand turn, okay. All right. So x times x squared is x cubed.

X times a plus x is a plus x squared. x times plus three is three x. All right, let me clear the slide. All right, now we're gonna go to the term over here, minus one times x squared is a minus x squared minus one times a plus x is a minus x minus one, times a plus three is a minus three. Let me clear the slide again. Now we're going to combine or add the terms.

So I bring down my x cubed, because that's there's nothing there. It's assumed to be zero plus x squared minus x squared, that is a zero plus three x minus x plus two x. There's assumed to be a zero there. So we've just have a minus three. So when I simplify the terms, I get this because I really don't need the zero, do I? Alright, so I have x cube plus two x minus two Sorry.

Now it may look complicated, but if you do a couple, it should kind of kind of click in and it shouldn't be that tough. Okay, you need to try a couple of these. And I'm sure there's going to be Sigrid examples, down, down on the slides here. All right, here's some examples I really would like you to do. And I really think you should take the time and do them. Okay.

That's the only way you're going to learn once you do a couple of these, they don't look as bad as they seem. Just I'll start you off on a couple of them here. Ah, a glitch. Just look at this one here, a squared plus two A B plus four. So all I need to do is multiply that term here and that term there. All right, how about this one over here?

I got to multiply this term This term, this term and this term and add them, get the terms and then add them. Or if you remember in the other example, what I can do here is I can do x plus y times x minus y, and start from the left and go from here to here. If for instance, x squared, then right there, I get a plus x, y. All right, and then I go six y, and then what I'll do is right here, I, I'll start over here, and I go minus x minus y times minus x. So I get what? A minus x y And then I get a plus y squared.

And I kind of did it for you can finish the rest. All right. So again, take your time, do the problems. And as always, the answers are on the next slide. I'll do a couple more for you over there. I used to have a student said, Well, I need to strain the brain.

So you need to strain the brain. And I'll help you out in the next slide. See over there. Okay, here are the answers. And I think I'm going to do a couple here for you while you can see the answers, but I'm going to do a couple hour I think we'll do this one here. All right.

So what I'm going to do is I'll do like one or two of them. We'll start with that one here. So let me clear the slide. And I'm going to whiteout, the slide here and we're going to do that This one here. All right. Now the way I get it before, the way I showed you before is we can do it this way.

I can say x plus y over x plus y. All right? So we start from the left x plus x. I'm sorry, x times x is x squared. x times plus y is a plus x, y. All right, start over here. And we, we move over to the left, so we still have a plus xy.

Y times y is y squared. Now I add or combine my terms. That's a special Were right there. Remember that's assumed to be zero. So we have x squared. So x y plus x, y is two x y.

And that's assumed to be zero there. So we have y squared. And there is my answer x plus two x y plus y squared. Okay, just want to mention there was an arrow right there, that should have been a plus not a minus, so I fixed it. So we're good. Um, let's see which one do we want to do now?

All right, let's, let's do this one right here. Okay, x plus y plus x times x minus y. So let's go to the screen. Let's wipe it out. All right, so I set it up here. So again, x times x x squared x times plus y is a plus x y.

Okay, now going in the bottom minus y times x is a minus x y. All right? A minus y times a plus y is a minus y squared. Okay assumed to be a zero, they're assumed to be zero here, x squared plus x y plus a minus x y they zero minus y squared. Okay, so now I simplify it. We've got x squared minus y squared and there's my answer.

Okay, this will end this lecture on polynomials. We've got a couple of more sections to go here, but we're going to end it now it went a little longer than what I had hoped but it's okay. Not a problem. And with that, we'll see you on the next section.