Polynomials Intro - Video Tutorials & Practice Problems

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Introduction to Polynomials

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Everyone. So in this video, we're going to start talking about polynomials. We're gonna be working with polynomials a lot later on. In the course, we'll be doing things like adding and subtracting and multiplying them. So I wanted to give you a brief introduction as to what they are in this video. And it turns out that when we looked at algebraic expressions, we've already seen the book four. So we're just gonna go ahead and give them a name. So let's get started. So a polynomial, the definition is that it is an algebraic expression. It's a type of algebraic expression where variables have only whole number exponents. And I actually want to say that it's a positive whole number exponents. So in other words, when we saw algebraic expressions, we saw something like this where we have six X cubed and three X squared and five X. All of these things here. If you look at the exponents, they all have positive whole numbers, we have 32 and five when it doesn't have an exponent, it just means sort of like uh an invisible one that's here. All of these things have positive whole number exponents So this is the definition of a polynomial. So we've seen these types of expressions before now, we can just call them polynomials. All right. So uh one of the things I want to mention here is that your exponents can't have negatives or fractions in them. So for example, and when we talked about exponents, we saw some expressions that kind of looked like this like two X to the negative three power, this is not a polynomial because it has a negative exponents. So that's really the whole thing always just look at the exponent, make sure it's a positive whole number. All right. So this expression is an example of a polynomial. But you may see your books refer to these things as monos trinomial and binomials. And really the whole thing comes down to the prefix. Um So the word, the prefix monno mono means one. So in other words, this is just an ex a polynomial with one term, the prefix by in binomial think bicycle means two and the prefix try like think tricycle uh means three terms, right? So the whole idea is that the a polynomial is kind of like an umbrella term, polynomial is like all encompassing. But if it has one term, it's a monomer. So for example, if I only just had the six X cubes, that is a monomer. But if I had the six X cubes and the three X squared, and if I had that expression that's a binomial. But if I had the whole entire thing over here, that's a trinomial and all of these things are polynomials. All right. So that's kind of like the, uh, umbrella term. All right. So, the very first types of problems that you might see is just actually figuring out if an expression is a polynomial. Because if, so we're gonna have to do things like add or subtract or multiply them or something like that. So that's what we're gonna do in this example problem. And if so we're gonna identify what type it is. If it's a binomial or a monomer or trinomial or whatever, let's go, we can get started with the first problem here. We have 3/4 X plus X cubed. Now remember the definition of a polynomial is that we look at the exponents and it has to be a positive whole number. So let's take a look at the different parts of this expression. We have 3/4 X. So this 3/4, does that break the rule? Well, remember the rule only applies to the exponent. It mentions nothing about the terms that are attached to the variables. So in other words, we can have negatives and fractions as numbers, but we just can't have them as exponents. So 3/4 X means 3/4 X, the first power. And that's a positive whole number exponent and X cubes that's also a positive whole number exponents. In other words, this thing has whole number exponents and it definitely is a polynomial. So how many terms does it have we remember from algebraic expressions? Terms are just separated by the pluses and minus signs in your algebraic expressions. So there's two terms here. So there's the 34 X and the X cubed. So if it's a two term polynomial, what type is it? Is it a mono by or trinomial or a bi means two? So this is a binomial. All right, pretty straightforward. Let's take a look at the second one here. The second one is five divided by Y. So what happens is this a polynomial? Well, you might think, well, this is just Y to the first power. But remember this is a fraction and remember from rules of exponents, one way we could rewrite this expression is we could say that this is five to the negative one power. So because this has a negative exponents, this is not a positive whole number. This is not a polynomial. So it turns out this expression is not a polynomial. Uh Even though it only has one term, it actually doesn't fit the definition of a polynomial. So it's none. All right. So these types of a expressions um are not polynomials. All right. So now let's move to the last one here. We have two X cubed Y squared. So in this situation here, we actually have an expression with two variables. We've got an X and we've got a Y, is this a polynomial? Well, remember the rule says nothing about having only one type of variable. It only just says that the exponents have to be positive hole numbers. In this case, we have X to the third power. That's a positive hole number and Y to the second, those are both positive hole numbers. This is a polynomial. So how many terms does it have? Well, in this case, I don't have any pluses or minus signs. Everything's all just sort of multiplied together. So this is actually all, just one term. And that just means that this is a monomer. All right. So even these types of expressions are still polynomials. All right. So that's it for this one. Let me know if you have any questions. Let's move on.

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Standard Form of Polynomials

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Now you've seen the basics of polynomials. A lot of problems will ask you to simplify polynomials and write it in a specific form called standard form. That's what I'm gonna show you how to do in this video. I'll show you how to write polynomials in standard form. And really, it just comes all down to the order of the terms in the polynomial. Let's go ahead and take a look here. So what standard form actually means is that all the terms of your polynomials should be written in descending or decreasing order of exponents. So what does that mean if you look at this expression or this polynomial? Three X squared plus five X plus four? Notice how the exponents actually keep decreasing. First, we have an X squared, then we have a five X to the first power, right? There's an invisible one there and then you have plus four and one way you can kind of think about four, remember is that you have X to the zero power. Um And X to the zero power is just one. So it's kind of weird. But um basically you can see here that the order of the exponents keeps on decreasing 2 to 1 to then zero. And the other thing is that all like terms have to be combined. In other words, you can't simplify the expression any further. So if you look at this expression, I can't combine anything with the three X squared or the five X or the four. So it turns out that this expression is already in standard form, all right, couple of other things that you should know um one is called the degree of the expression. And that's basically just the highest exponents of the variable that you see in the polynomial. So in other words, the highest exponent that we see that's attached to a variable is the X squared. It's the two. So that means that this is called a second order or a second degree polynomial. And last, but not least there's just a new vocab word. Uh We know that the numbers not attached to variables out by themselves are called constants. We know numbers in front of variables multiplying them are called coefficients. And there's a special name for the, the number that goes way out in the front that gets attached to the, the variable of the highest exponents. And that's called the leading coefficient. It's like the number that leads the entire expression. All right, that's the leading coefficient. That's basically it. So lots of problems are gonna ask you to now write in expressions in standard form and that's what we're gonna do in this problem, we're gonna identify the degree and the leading coefficients. Let's get started here with this first expression. So in other words, we have one half X and remember that's one half X to the first power over here plus X to the third power. So we have to write this in standing uh and in standard form. And that means that we have to write it in decreasing order of exponents or descending order. So we have an extra to the third power and an X to the one power that's backwards. So I just have to flip the two expressions and when you move expressions rubs be very careful what happens to the sign. In other words, I'm gonna rewrite this as X cubed plus one half X, all right, because it's basically just the X to the first power. So does this have decreasing order of exponents? Yes, it does. And are all the like terms combined? I can't combine anything with the X cubes and the one X. So this definitely is a simplified expression. So it's in standard form. So what's the degree? The degree is really just the highest exponents that gets attached to a variable and that one is three. So in other words, the degree over here is three. And what about the leading coefficients? Well, the leading coefficient is the number that gets attached to that variable with the highest exponents. And over here what you'll see is that basically there wasn't a variable. But remember the, it's always kind of like an invisible one there if you don't see a number. So the leading coefficient in this case is actually just one. OK. So let's take a look at this expression was a little bit more complicated. There's more terms I noticed that there's some terms with X, there's some with X squared. Uh and there's actually just some constants in here. So I have to write them in descending order, but be very careful when you do this because you basically have to keep track of the signs. In other words, I have a negative three X squared and then X squared. So I'm gonna move that. I have an F five X and A two X. I'm gonna move that and I have a negative seven. So I wanna make sure that all the X squared go first. So this is gonna be negative three X squared um plus X squared and I'll have the plus five X plus two X and then I have the minus seven. So when you pick these numbers up around and move them, remember you're always doing this with the sign that goes in front of them. All right. So now what happens is we have descending order of exponents, I have uh exponents two and then one and then I have zero over here. So this definitely has descending order, but it's not as simplified as it could be because I could still combine all the like terms. So that's what I have to do in the second step. So if I combine this expression over here, negative three X and one X, it's kind of like negative three apples and one apple, this actually just becomes negative two apples. Uh when happens to the five and the two that becomes the seven X and then the negative seven just becomes negative seven. So now all the like terms have been combined and this definitely now is in standard form. So what's the degree of this polynomial? What's the highest variable, highest exponent of a variable that we see? It's just the two over here. All right. So in other words, we have two and then what's the leading coefficient? What's the number that goes in front of that term? Uh It's actually just this negative two over here. So that's the negative two. All right. So really, that's all, that's it for this one. Folks. Let me know if you have any questions and we'll see you in the next video.

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Adding and Subtracting Polynomials

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Welcome back everyone. So now that we've seen the basics of polynomials, lots of problems will now ask you to start manipulating them or doing something with them like operations. So in this video, I'm gonna show you how to do adding and subtracting of polynomials. And I'm gonna show you that we've already basically done this when we looked at algebraic expressions. So there's really not a whole lot that's new here. Let's just go ahead and jump right into it. So when we had algebraic expressions and we wanted to simplify them, the easiest way to do that was to combine the like terms. For example, if I had two X plus three and four X plus eight, I group the Xs with the X's and the constants with the constants, and I basically end up with something like this. And the idea here is that these things are actually just polynomials if you just look at them. So we've basically already added polynomials before. So the whole thing is just like algebraic expressions. The way we add and subtract polynomials is where, where you just perform the operations. And we just combine all the like terms all right. So let's just jump right into the first example over here, we have five X squared plus two, X plus three and X squared plus seven X plus eight. The idea here is that this thing just looks exactly like this. It's just longer and more complicated. But the idea is that we're just gonna take all the X squared terms, add them and combine them. We'll take the X terms and add them and combine them. And then the, the constants if we have any and add them and combine them. So let's just do it. So if I had five X squared plus X squared in this expression, then I combine the like terms and this becomes six X square, I have two X and seven X, this combines to nine X and if I have the three and the eight, this combines down to 11, all right. So the whole idea is that you're just gonna match up these things and then just go ahead and add them. And usually what you're gonna see is that your answers are gonna be written in standard form. So in other words, we're gonna have decreasing order of exponents and all the like terms combined. But this is how you add and add a polynomial and the subtracting a polynomial is very, very similar. So for example, if I have three X squared plus two X plus four minus five X plus 10 minus X squared, then I basically just have to do the operation and combine all the like terms. The one thing that is different though is it, you have to distribute negative signs into parentheses if you have any, it's really important. A lot of students will mess this up. So just be very careful when you do this, you don't get the wrong answer. Um Basically, you're just gonna distribute this negative into everything inside of this parentheses. So let's just do that. So the three X squared, so the three X squared plus two X plus four doesn't change. And then we distribute the negative into the five X and we get negative five X, distribute the negative into the 10 and they get negative 10 and then negative into negative X squared. That's the negatives will cancel. So it's really important there. That's why this step is super important. And now we basically can just drop the parentheses of everything because we're just adding a bunch of terms together. But the idea is the same, we're just gonna take the X squared and sort of combine them together. So X squares with X squared, uh XS with XS and the constants with the consonants. So if you do that, what you end up with getting is a three X squared and then I have plus X squared over here. So that means plus X squared becomes four X squared. Uh There's two X and then I have a negative five X so that will actually be two minus five, which is negative three X and then I finally have plus four with negative 10. So this will actually just become negative six. All right. And so this expression is also in standard form with all the like terms combined. All right. So that's how to add and subtract polynomials. Just be really careful when you do uh subtraction because it might get a little tricky. Um But otherwise, that's it for this one. Thanks for watching.