College Algebra | Section 1 | Introduction to Polynomials | Victor Rodriguez | Skillshare

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College Algebra | Section 1 | Introduction to Polynomials

teacher avatar Victor Rodriguez, Teaching Math and Science

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Lessons in This Class

10 Lessons (1h 8m)
    • 1. 1.1 Introduction and Definitions

      5:49
    • 2. 1.2 Like Terms and Term Degrees

      7:04
    • 3. 1.3 Adding and Subtracting Polynomials

      8:45
    • 4. 1.4 Multiplying Polynomials (Pt 1) Monomial X Monomial

      5:36
    • 5. 1.4 Multiplying Polynomials (Pt 2) Monomial X Polynomial

      2:50
    • 6. 1.4 Multiplying Polynomials (Pt 3) Polynomial X Polynomial

      4:54
    • 7. 1.5 Dividing Polynomials (Pt 1) Monomial Divided by a Monomial

      5:11
    • 8. 1.5 Dividing Polynomials (Pt 2) Polynomial Divided by a Monomial

      5:37
    • 9. 1.5 Dividing Polynomials (Pt 3) Polynomial Long Division

      11:15
    • 10. Class Worksheet Review

      10:29
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About This Class

In this class I will introduce you to the concepts and definitions surrounding polynomials. You will learn how to add, subtract, multiply and divide using various combinations of polynomials and monomials. The last video in the series is a review of some of the course materials. 

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Victor Rodriguez

Teaching Math and Science

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Hello, I'm Victor. I am an astrobiology researcher and a math and science enthusiast. I love sharing my knowledge with the world. 

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Transcripts

1. 1.1 Introduction and Definitions: Hello, everyone. Welcome to the course. Introduction to polynomial is this is a sub course of a larger topic college algebra and to kind of tell what's going on with the course numbers. The course numbers are in sync. Sequential order. This is introduction of polynomial is which is one and of course, the each video goes 1112 and so on. Eso If you're getting into this video, this is the first video in the series. Andi This way you can kind of start here and then you can get get into more advance college algebra topics. But this is it, uh, welcome and let's get started says introduction of polynomial. Now, what is a polynomial? Polynomial has one or more terms. Now a term is a, um no term is a number of variables enough numbers and variables without any addition or subtraction within them. So so five x times three is a term. However, five x plus three is not a term. So now let's go into some definitions. Now, these are the parts of a polynomial. A variable represents is represented by a letter or symbol, so you could have like X or s er, ze or it could be Ah, Greek letter. Um, pretty much any symbol can be Ah, variable. Then what I'll do is I'll write that on there so I can kind of show you kind of an example of in this in here of each each, um, definition. So in this case of variable would be variable would be the exits. In this case, no label those V now, the next definition is exponents. No, of course. Um, the next opponent is the so, um value. Multiply by itself, X number of times. Now, this X number of times could be, um, any amount now, of course. This right here is the exponents. That's the exponents right there. No, In this case, if you don't know how exponents were, this is nine x times nine X. Now, that was a three. That it would be nine. Next times nine. Next times idexx. Just a little, uh, definition here. Now, one thing I won't talk about with this course is there's a link to a worksheet with all of this stuff. After at the end of the course, you can either download it and work work as you go through it, or you can work on it later on. And, um, you can actually email it to me and I will actually correct and give you any advice or any issues that might find and kind of help you along with this course. Now the third definition is coefficients. Now the now the coefficient is the this number right here. Label that see it's a number in front of the variable. It will multiply the variable, So if you have three X, you meet you means you have three exes. It's three times X, a view of three apples. It's three apples and finally you have Constance. Now the constants are unchanged values that is, in this case, three in five numbers that don't change. And that's a, uh, and that's a pretty basic introduction into the parts of polynomial. 2. 1.2 Like Terms and Term Degrees: Now I'm gonna talk about, like, terms, actually, 1st 1st what I'm gonna do is I'm gonna define life terms like terms are the same variables and the same exponents. Now, I'm gonna give you an example of this. You have seven X squared. Why? Which is seven times X squared times? Why, of course, it's one term because there's no pluses and minuses. And then you have three x squared plus two two y. Now the constants don't matter. But what we do have is the same exponents right here in the same terms. So we have X squared X squared, and we have white. And why? And then now the reason for like terms, as you'll learn later is so that you can add them together. You cannot add polynomial, or you cannot add terms that are not like these. Uh, these can be added together. However, if this was seven x squared, why in three x two. Why those could not be those could not be are added together because they're not like terms . Now we're gonna talk about degrees of terms. The degree of the term is a Some of the exponents within the term of all the variables a degree of term. The degree of a term is the sum of the exponents of the variables within a term. Now here's an example. Okay, let's say you have eight x to third minus seven x to the eight plus five x Why minus three Now First, let's identify how many terms are in this polynomial. And of course, the best way to figure out the terms is by looking at the plus and minus signs. Here is one term and in a second term, then 1/3 term and in the fourth term. So this is a four term polynomial. Now we can then look at the actual degree of each polynomial. Now you're adding up all the variables that the exponents in the variables. So this has one variable the X with with 2/3 so that this is 1/3 degree polynomial. This one here has 12 or one variable to the eighth. So this is an eighth degree polynomial. Now, here you have to polynomial. However, you don't see any exponents. The thing is, when you see no exponents, you always you always assume a one because this is really exit a one and why to the one So he add those together. Now, of course, you don't add this isn't a variable, so you ignore that. But what you do is one plus one. You see, it's too. So this is a two degree pound on you. Now, here's a little tricky thing. You just have a constant in this term. Now the rule is it's a some of the exponents of the variables. Now there's no variable. So there's no exponents ad, so that zero So this is a zero degree polynomial now, the reason you need toe, uh, understand this is because of the rule of descending order, and we're already down, and I'll explain it a little right terms in descending order according to degree, You always right, um, polynomial in descending order in the now to take this example right here, I'm gonna, um, rewrite and we are going to rewrite. It is this is the eighth degree polynomial. So that one goes first. You take the highest to the lowest descending order, so it would be negative. Now you always after You don't want to lose this because this isn't, you know, eight extra third minus seven X to the eighth. What this really is This is negative. What you're doing is you're combining these terms, so you're combining eight X to the third with negative seven x d eight. So you you don't want to lose the negative. So you have a seven x to the eight and then we can check that one. No, and then the next one is the eight to third. So plus a yes, do third. So you can check that one off, and then we have the second degree, which is a plus five x. Why? And then finally, the zero negative three. And this just makes things a lot easier to deal with when you're you'll see it later on, when we start dealing with adding and subtracting polynomial. 3. 1.3 Adding and Subtracting Polynomials: So now that now that you have an introduction into polynomial is we're going to go into different operations now we're now in this video. I'm going to start talking about adding and subtracting polynomial now with addition and subtraction. The reason we're doing it together is because addition and subtraction are the same thing. If you're subtracting something, you really just adding a negative number or a negative value of some kind. So really, it's all addition. It's just whether you're you're adding a positive or negative number. That's the best way to really think about it now. First I want to show show some steps on how you would add, add and subtract polynomial. First it would remove for a disease. Then you combine like terms. Now, as you learn before, like terms have the same variables and exponents within two term, and you can go back and look at that. If you, uh, I miss that, an answer written in descending order as we did. As you saw in the introduction to polynomial policy, you take the degree of the polynomial of each term of the polynomial, and then you put him in order in descending order based on degree. This is how you add and subtract Palin animals. Now, I'm gonna take those steps, and I'm gonna put him into a an equation that we can solve. Now you have to. Polynomial is here. You have, um, negative X to this square plus five X minus five. And then you're adding it to get adding it to on negative three X minus four X squared minus phi ar minus X minus five. Now, in this situation, the first thing we do is we remove the parentheses. In this case, we can do this by simply doing very right. And what we will do is little right now, because now we will, Right? Negative three X squared plus five X minus five. And then now, this is a tricky part, because this was a This is a plus. But if this was a minus, this would flip all these terms. All these are signs. So this was a minus. These would become plus plus plus plus, um, because ah, minus a negative is is the is the equivalent of Adam. But that's not the situation here. So we can We can simply bring the signs down. Negative X to third minus four X squared minus X minus five. Now that the parentheses air gone now it's a matter of combining like terms. So now we need to find the like terms. And, of course, like terms are the same variables in the same exponents. Now, each sign, plus or minus sign divides the each term. So what we're looking for are like terms. First, the best way is to start. 01 thing you could do start from left to right and find a coefficient coefficient with X squared. So here's an X squared right here in an X squared right here. What? This? What this means is that you're actually taking negative three exes squared minus four X squared, which negative three minus four is negative. Seven. So takes care of those. Then we go to the five X. So to find the like terms, we find the other ex, uh, the other term that has the same variables right in the same exponents, which, if there's no exponents there, you assume one. So this is extra one in Exeter, one saying variables so we can combine those together. Now, in this case, we have plus five x minus X, which, if you see an X with no coefficient or no you know coefficient, you assume one. So it's five x minus one x, which is for X. You know that takes care of that. Now we go to the negative five. Now there's no variable, however, there is a constant, and you can come in. Those constants can then be combined. You can combine constant with constant. They are like terms because they have the same variables and the same exponents. The variable is that there is no variable and the exponents is one. So you combine these together negative five in negative five negative five minus five is negative. 10. Now it's very methodical the way I'm doing it, but this will allow you to understand exactly what's going on. So now that takes care of these. Now. The final one is Justice X to third, which that because there's no combining left, there's no like terms. You could move that one down negative three negative X to the third. Now we're not. We're not done yet because the final step is to rewrite it in descending order. So to do this, let's be a little methodical with this. This term is a two term or two degree phone or two degree term. This one has one degree because this is there's a exito one there. This because there's no variable is zero, and this would be three 33. So now, now that we figured out the degrees we can then bring this and rearrange it so we would start with a three. So be negative X to third. Then we do the Tu minus seven X squared and then the one which is plus four X And then we do the negative and this would be the answer, and this would be the case for addition or subtraction. Doesn't really matter If this was a subtract a negative sign there. This was a subtraction. You would do something you would distribute. So that's it for adding polynomial is the next video will be on multiplying polynomial 4. 1.4 Multiplying Polynomials (Pt 1) Monomial X Monomial : So in our last video, we did adding and subtracting pile. No meals this time will be doing multiplying polynomial. Now the reason we're only doing multiplication and not multiplication and division is because it's not like adding and subtracting. It's an actual reverse function, so division ist so what? What we'll do is multiplying. Right now. There's three types of multiplication. You have a mono meal times a my no, my no meal, and that's a a polynomial with only one term mono mono poly as many model was one. So my don't be old times my Romeo, that's number one. Number two. You can have a mono meal, sometimes a polynomial, and in three you have a polynomial times a polynomial, and each operation will give you different results. Now, my only old times, my no meals pretty straight forward now thing to remember with a mine on the old time pop times a polynomial and is to distribute a little cramped there but distribute And then, for a polynomial times a polynomial. Remember, foil. Now first we're gonna do a multiplication of mono meal times Amano meal. Now we have negative three X square. Why one term my no meal times negative two x to the fifth. Why Z now? Now the thing to remember is that, you know, right up here or down here, I'll start right here, multiply coefficients. And when variables are the same and exponents now what this means is well, what we'll do is we'll start with three the three x square in the fight in the two X If it not only canoes, you multiply these no negus multiply coefficients. Now we can only multiply that the like the like, um, variables. So what we have is three x negative three X squared times negative two x to the fifth. Now a negative times negative is a positive. So let's start with the coefficients that turned negative three times negative two is six. And then we have the exes, which, if you when you multiply exponents, what you do is you add, you add the actual numbers. So it's 5 to 7 and then you go on to the why, which is why times why which is now. If you remember variables that have no exponents, you can think of an exponents of one, so we'll just put one right there. too. So if you take white times why you add the exponents 1 to 1 and one is two. So what you end up with is why square? And then finally, there's no Z over here. So you can bring that one right now. And then This right here is the answer six x 27 Why times y squared time. See? And that's how you now this is a my Romeo times. Um I know meal. 5. 1.4 Multiplying Polynomials (Pt 2) Monomial X Polynomial: Now it gets a little, uh, more A little more conflict more complex when you when you talk in terms of, uh, my no meal times a polynomial. So this time we're doing mono meal times a polynomial three x, and this is amount of mono meal one ter times X to third minus. All right, X plus two. Now this is where this distribution comes in. The idea is we are going to distribute the three X throughout the entire polynomial and you go the way you do it is, you go one by one. Multiplying amano meal by a polynomial is the equivalent of doing a mono meal. Times of my normal maino mealtimes. Marano meal, mono mealtimes Not mine. No meal three times and you're distributing it throughout the system. The the equation. So let's do that. 1st 3 x times three to third you multiply the coefficients, which this there's a There's a one there, so three times one is three in an X square x Times X it 1/3. If you remember, there's one right there, so it's exited fourth because you add the exponents when you multiply them. So that's Exeter for and then Now we're done with that one, and then we can move on to this one. So then we multiply again. We multiply in this case negative five times three. That's negative. 15 and an ex. There's one right there, so x times X is X. You add the 21 in the one is too, so X squared. And now we're done with that one than 3rd 1 The third term in the polynomial will take three X Times two, which you multiply the coefficients so you get six and then the X. Because there's no X here. There's nothing to multiply with, so you can just bring it right over like that. And in this case, that's the answer. You distribute it throughout the system throughout the equation. 6. 1.4 Multiplying Polynomials (Pt 3) Polynomial X Polynomial: now, of course. Um, in every in every situation you want toe, rewrite them in descending order. Once you're done with this, if the multiplication causes a non descending order, um, degrees, you rearrange them and rewrite them polynomial times a polynomial. Now, for this one, we have to do first, outer inner last. Now what this means is, in essence, I'm gonna write a little No, over here. Let's call this A, B, C and D right, a a plus b times c plus d. What you're really doing is first the hours. So what you do is you first take a and you go eight times, see, and then eight times D and then you take B times C and then be times D. What we'll do is we'll start out with three x times two x, which three times to it's six in an x Times X is X squared. So then now we've done that one. This checks off this and then now we do three x times Negative seven which three times negative. Seven. This negative 21. They have 21 then the X. There's no X over here so that its X and then so that checks off that one. Now we're going to be to this section now it's negative. Five. It's B times C. This is be this sissy negative. Five times two x, which negative five times two is negative. 10. And there's no X over here, so you can just bring this X and then it's negative. Five times negative seven, Which is positive. 35 Now the final step in here A lot of time, especially when you get, um, polynomial that have many terms in them, like six terms in eight terms and what have you or of your combat? If you're doing multiple multiple occasions, you'll find out that you will have a pretty messy situation. And so the last step in this case will be to combine the light turns. Now you have the X Square, the X squared term right here, So six x, where there's nothingto, combine it with, you know, essentially what you what? You've come up. You've gone from multiplication to If you go back to my last videos now you're going to adding Palin or adding terms here. So it's six x squared in there. Uh, takes care of that. And then you have these, like turns, which negative 21 plus negative or plus names of 10 is negative 31. And that takes care of those to you. And then finally you have plus 35. And that's the answer for that. So those, of course, this these operations can get pretty complex these air simple versions of this. But this is the introduction to multiplication of polynomial. And then in the next video, I'll be doing dividing, uh, polynomial. 7. 1.5 Dividing Polynomials (Pt 1) Monomial Divided by a Monomial: So now in this video, in the last video we did, multiplication now will be doing dividing. So there are dividing polynomial. There's three types of polynomial division. You can have my no meal divided by my no meal. You can have a polynomial divided by a mono meal, and then you can have a polynomial. Bye bye polynomial. Now the one thing with this one, what you can remember is Long Division. Then, when dividing when dividing polynomial, Where you want to do is produce coefficients. Keep the same variables. It's attract exponents in an order larger to smaller or, in other words, descending order. Now, for a first, we're going to start with a mono meal divided by a mono meal. So for dividing a polynomial, we'll do. Right now. We'll do a mono. Mealtimes are divided by a mono, so what we'll do is six x to the fire to the fifth. Why Z in the six x 10 to the fifth times Why Times e divided by eight X to the third. Why, I swear. First I wanna look at the X is the X sit next six exit 1/5 and the eight x to third. Now, where you wanna think about here is we're gonna We're gonna reduce this. And one thing you want toe take note of is you have negative six and, uh, over eight. So you can think of a of native 62 the eighth, which we can reduce that. So what we'll do is we're gonna start reconstructing over here. If you reduce this. The common denominator is too. So it's actually becomes negative. 3/4. So you take negative free and then four and then the exes, we subtract exponents. No. Five minus three is to So on the top, we do X to the third. We're taking five minus three for the exponents. That means this goes away. Were left we're left with is up with extra third in that and then just the coefficient right there. So there we go. To the why which in this case, what we'll do is we'll do a you know, as you remember, This is a one. So one minus two is negative one. So you could think off. If you think of on the top here, you could put negative one. Why now? Here's a thing. And then Of course, this one cancels out, but here's the thing. A negative. If you have a negative why or negative exponents in the, uh, numerator you can actually take it, make it positive and put it in the denominator, Which, of course, that one that I put on there you can kind of erase because it's obvious. And now you have for why. And finally you take disease, which there's There's no Z to cancel out here, so you can actually move this right over here, so that equals that. 8. 1.5 Dividing Polynomials (Pt 2) Polynomial Divided by a Monomial: Now we're gonna do a polynomial divided by Amano meal. You want toe separate terms. That might not make any sense right now, but I'll show you exactly what I mean. Okay, let's say we have negative eight x to third minus seven X square. Why to the fifth plus two x Why divided by two x two ex wife my polynomial times or divided by am I know meal in this case, what we'll do now What I mean by separate the terms and I'll do a little example over here with the A, B, C and D. Let's say you have a term plus beater right here. Over. See? Well, this can be rewritten as a oversea plus be oversee. That is, that is actually the same thing. So what we're gonna do is we're gonna rewrite this in the way we're gonna rewrite it. ISS negative A X. It's 1/3 over two X. Why? Minus seven X square. Why? To 1/5 over two X. Why plus to X. Why over to X? Why now? Now, In this case, what we're actually doing is we're simplifying what we need to do, because now it's a mono meal divided by binomial My no meal two divided by my normal in another maino meal divided by mono meal So now we can do go back and do the original model Well divided by a mono meal steps which in this case, here we have, um we're gonna reduce the coefficients. So eight divided by to is four. So we'll set up this right here. So are negative. Eight divided by two is negative. Four next over here, we can put this right here. Plus, this just makes a little a little more are a little easier. No, uh, lose track of things in ways. So eight negative eight divided by two is negative for And then, of course, it to goes away because it to would be one. And then you go x 2/3 divided by X is X square and that it's goes away And then you have the There's no why up here. So this why just days over here and then you plus negative seven divided by two. Now, in this case, there was nothing in common to give you a whole number coefficients. That will actually be that the way it is. Negative seven over to, but we have X squared, divided by X, which means which is s because you're subtracting the exponents and this one goes away. And then it's why to the fifth divided by why which is why to before So you can get so you have that native seven x y to the fourth, divided by two. Then finally, you have this one which you don't even have to do anything to get to know. The answer is because anything over itself is one. So two x Why divided by two X Y is one. So we can just write one right here. And that's the way you would do. Um, amano me or a polynomial divided by a minor wheel. 9. 1.5 Dividing Polynomials (Pt 3) Polynomial Long Division : finally in our division section is the, um and this is this will be the last thing will be doing for our introduction of polynomial is polynomial divided by a polynomial which results in long division polynomial provided by parliament. Now what? What I'll do here is I put polynomial one and polynomial to right here just for a reference polynomial one would be the numerator, and the denominator would be put polynomial to now I'm gonna write polynomial s p p one and p two polynomial own division p two becomes divisor and p one becomes the dividend powers are written in descending order with a zero X powers to be replaced to replace missing powers. Now, that's ah might be a little confusing, but it makes more sense when you actually see it in practice. Now, for this example, I'm gonna dio x to the six by this X to the third minus x divided by X square minus X. Now, of course, this is P 12 So this becomes the, um He, too becomes the divisor and p one becomes the dividend. No, this is where you'll understand that long definition that I put out there no, for contrast. I'm gonna put the division bar or the division wrap it like that and what we'll do is writing in descending order. Now first, what we'll do is we'll take this has if you look at the exponents, it's two and one. So this one we can write X Square minus Yes. Now this is where that zero X thing comes in. We start with the six X or or the X to the six. And then now we need there. There's our exit third, but what we need to write in here is three x to the fifth exit of fourth because they were the missing ones. So this is where the zero X comes in. So what we do is you put plus zero X to the fifth plus zero x to the fourth. And then we can put next to the third minus X to the third. And then we have another gap. Because we have three, there's no too. And then there's one. So plus zero x square and then minus x so isn't too confusing. Oh, separate that So it doesn't look like it's all melted together. Now it's up. It's a lot like, um, division, like regular long division. What we're doing is we're fine. We're finding a term. Now there's two. There's two pilot or two terms here. So we're gonna start with two terms here. We're gonna find the term that will get this here to six. Okay. And, you know, it may not make a little sense, but no extra for okay. And now this is where the multiplying polynomial comes comes in. Exeter fourth comes next to the to is Exodus six. Exit for exit 1/4 times Native X is negative X to the fit now X to the six minus X to the six. And what you do here is you actually, um, you can put parentheses here and in minus this will distribute everything so minus x 26 If you take this, that zero and then if you take this is a negative and negative. So this makes it a positive. So zero mine or plus or zero X to the fifth plus x to the fifth is one is one exit. If it so it's exit. If it no, you can actually take this and take that zero right off. And that's your answer, and you can kind of see the intuition over here where it's it's kind of. It's very similar to long division long division that you do with regular numbers. So now we want to see what what will get us to extra what? Well, first, what we do is we bring this down. So is plus zero x to the fourth. And now we're going to see what what will get us to exit 1/5 which would in this case would be extra third because exit 1/3 time's exodus. Second is exit if it so again we do exit. There are Exit fifth. So at extra third time's Exodus Second exit, 1/5 and an extra third times negative X to the one or vaccinated of X is negative X to the fourth. So now again, remember the parentheses in the negative because we're actually subtracting it and you have yourself. It'll set up exit fifth minus exit. Affect is zero if we can just not right there and then x zero exit 1/4 minus or plus extra fourth is because negative negative makes it plus is negative for and this is something you really gotta remembers is that distributing the negative signs throughout the pound, the term or the polynomial otherwise you will mess up the whole thing. So the answer now is excited for then Now we bring this one down. It's negative X to the third and then we do it again. This time, um, you know what will get what will get extensive second t exit 1/4 which is Exodus Second X square times X square is X before No. So now what you do is you. You do that. You do the math again. X squared times X square is exit 1/4 X squared times negative X is negative X to third. Now, this case is pretty obvious. You can actually, um, figure out that this would be zero, but let's do it anyways. Next exit, exit 1/4 minus exit 1/4 0 negative exit. 1/3 plus next to the third is zero. So this could be zero. And then you could bring the stone plus zero. Yes, I swear. Now what you do is now what I do in this case, Nothing. Zero. If you put zero here, it will get you to this number now, you don't need to put it in here, but for, um, for Methodist Schism to be methodical about it, we'll put the zero right here. No, finally we bring down. So then 00 x swear or X, I swear you get yourself zero, and then you bring down the s. Now, in this case, the fight, because you can't go any further. The X right there is a remain remainder. So what I'm gonna do is I'm gonna take this and I'm gonna rewrite it over here. Then we can drop the zero. What you do is you take the remainder, which in this case is X over X square minus X. And this right here is your answer. Exit 1/4 plus extra third plus Xs X squared plus X over X squared minus X. And that's how you do Long division. Um, long division polynomial. So that's the end to the introduction to polynomial. Make sure to download the worksheet, working out, email it to me, and then I can give you some pointers on things that you may have missed. And also make sure you re watched the videos. If you have, um, any confusion or anything as well. Thank you. 10. Class Worksheet Review: So this is a review portion of the course and I will be taking three questions. Three problems, one from each worksheet. Now, the first worksheet I'm gonna work working on is the adding and subtracting polynomial, This is number 10 on there. So I'm gonna write that on the board and insulted and go through the steps. The first goal here is to remove the parentheses. So we will re rewrite this and we can do is we can do three minus six and to the fifth minus eight and to the fourth those. And then this is the tricky part is the, uh when you have a negative for when you're subtracting, you have to distribute the negative side throughout, which flips all the signs. So if you have a positive ago, it goes and negative is if it's negative, you get a positive or plus the minus. So what we do is we distributed all through and this this becomes plus six and to the fourth plus three and my r plus A and to the faith. So now we can solve this at we'll start with Thea. Um, Constant. There's only one constant, so we can 23 and then six sent to the fifth. There is another fifth these can combine. So minus a plus or minus six plus eight. That's eight minus six as to so you end up with plus to and 2 50 So these two or white though then a into the fourth six into the fourth can be combined, which gives you negative. Eight plus six is negative. Two so minus to end to the fourth. That way Tsos out. Then finally you have this one all by itself, plus three en. And then, of course, the last step is to rewrite it in ascending order. So we have, um this is 1/5 degree. This is 1/4 degree. This is Ah, first degree in. This is none so that we can go from fifth that's to and to the fifth and then minus two into the fourth minus or plus three. And and in plus three. And that's the answer to number 10 on the adding and subtracting polynomial now for the multiplying worksheet. I took a question number 20 and I didn't write that now on the board. Okay, so we will know combined, these will multiply teas. So you have. You start with seven. Okay, times seven k times k squared. That's seven, que 2 30 and then you have seven k times. Two k negative. Two K. That's 14. That's negative. 14 que square. And then you have seven k times. Seven. That's 49. Okay, so that now we're working on this one here. So negative. Three times k square K squared is negative. Three. Okay, Squared negative. Three times two k is positive. Six. Okay. And then finally negative. Three times seven is 21 native. 21. Now we're We can combine this together so we start with the highest degree, which is Thea seven k to the third, which that eliminates that one. And there's no others. And then we combined the squares, which is 14 k native, 14 k plus minus, um, three. So it's on a negative 17. Okay, square. And that eliminates that one. And then we have 49 6 which is plus 55. Okay, Better eliminates these in a negative 21 eliminates that, and there's the answer for that. Finally, we get into the long division and the reason I put ah whole page just on long division is because a lot of people have trouble with this part of polynomial. So I decided to make whole worksheet on it, rewritten with and plus to here. And then we're gonna go to the and to the third plus seven into the second. Now, this one is actually kind of easy because you don't have to put those 30 X is zero X is in here. There's no zero needed, plus 14 and plus three. And now we can solve this. Now what? What? What would we need to get to end three into? The third is we would have to square multiply this by at and square, so we would start right here. You don't start here. You start here and two and square. So in square, times two is two and square and square times and is into third. Now you parentheses. All this and what you're doing is you're subtracting so into third months into the 3rd 0 and, um seven and squared minus because this makes us minus minus two and square is five and squared. And then now the second. Now, now we move on. We bring this down plus 14. And so now we look for what? Now? What do you have to do to get to end Square is in this case, we have to get to five n Square, which means we would have to multiply this by five n five anti or five end times two is 10 and five n times n is five and square and in parentheses. This minus. Put that there. And now we can solve this. This is zero, and this is four n because we're subtracting here, bring down the three plus three. Now we can do it again. How do we get four end? We would have to multiply by 44 times to is okay and then four times en is for again we parentheses. Subtract. There's five negative five and this is zero. And then, of course, what you do is you take the negative five over the, um, Dubai, Uh, the end plus two. And there's your answer and squared. Plus five plus four. Remember that. Plus, here or um, plus native five and plus two. Or instead of putting negative plus here in negative five, you can also take this negative five and just put it the negative sign right there in the B , minus five and five over and plus two. So if you have any questions, make sure you send me an email or a note or anything. And hopefully this course was, uh, helpful for you. Thank you.