okay you guys welcome to math 102 our first lesson in math 100 is section 5-1 add and subtract polynomials so first we need to decide what exactly is a polynomial that is a single term or the sum of two or more terms containing variables with whole number exponents and always written in descending order if we have a polynomial so an example of a polynomial written in descending order would be 7x cubed minus 9x squared plus 13x minus 6 so this would be an example of a polynomial in descending order so the thing that makes it descending are these exponents right here so notice we have an X cubed and then we have an x squared the exponent here is a 1 when you don't see the exponent and then there is no variable in the last term that would be considered our constant term so this is what descending order looks like now if I want to actually do what that says and simplify that polynomial that's where we're headed before we get to simplifying what we want to do is I want to think of this as 7x cubed I'm going to change this minus to a plus just for a moment I know that some students prefer to write theirs all as addition I particularly do not usually do this because I don't like to see two signs directly beside each other like these two I don't like to see a minus and a plus a minus and a plus I always turn those into you what the original equation says but for the sake of where we're headed I just wanted to point this out when you think of this set term here it is a negative 9x squared all right so there's a couple of other words that we need to look at before we move on all right one of those words is the degree right each term has a degree and so a term looks like this and it's a degree is just that in all right so this ax n that would be considered a term and its degree is in if we're talking about the degree of the constant term okay now remember I said that that negative 6 up there was our constant because it does not have a variable a constant term always has a degree of 0 so if we look at our our polynomial that we have sitting up here I'm going to separate it into each term the coefficient for each term and the degree for each term just so that we can see what all of those things mean the coefficient said that word so I feel like I need to make sure you understand what a coefficient is a coefficient is the number part of the term so in this case the number part of our term if I had a next to the end there's our term okay there's our term so the number part would be the a would be that coefficient just abbreviate there alright so the a is the coefficient there all right so let's take our little polynomial that I have here and we're going to make a little table and in our first column we're going to identify each term of the polynomial so our first term is 7x squared our second term is negative 9x squared making sure to take the sign with it our third one is 13 X and then our last one is negative 6 so there's each of the terms of our polynomial okay the next thing that we'll look at is the coefficient for each term now remember I said the coefficient is the number part of the term so the coefficient here would be 7 negative 9 13 and even though negative 6 is a constant that is still the number part so that negative 6 is our coefficient and then we're going to look at the degree of each term all right so of this first term the degree remember I said the degree was the exponent the end value so in this case our degree is 3 here it is to remember when there is not an exponent on the variable we understand that it is 1 and then our constant doesn't have an exponent and so the statement I wrote above the degree of the constant term is always 0 so this table is something that you will see in MyMathLab in one of your assignments so you can revisit that if you need to alright our next little idea is there are three types of polynomials okay polynomial is like a family for instance my last name is Wirtz and to all of my children have the name works so they're all in the works family and so polynomial is like a family and then we get specific if we talk about in my children's case their first name would make them specific so one of these would be considered a monomial and then we have a binomial and then we have a trinomial now I have three children and so this would be like having my three children a lead a Brielle and Michael so monomial binomial and trinomial are three different kinds of polynomials what makes them different is a monomial is exactly one term like two X or 5x squared right just one term a binomial has two terms a trinomial has three terms now our polynomial at the beginning of this page that we've been going over in our notes is none of those because our polynomial that we've been working with so far has four terms so when it has four terms it no longer gets a particular name you just call it a polynomial if it has five terms you call it a polynomial so it's kind of like if I had one more kid I wouldn't give them a name they would just be words which is silly but that's kind of how we do it in MAPP all right all right and so we talked about the degree of a term now let's talk about the degree of the entire polynomial not just the degree of each term the degree of the polynomial and so that would be the greatest degree of all the terms so if we look back up at our table that we completed just a moment ago so I'm going to scroll back so we can see that table right so if we look at there's a degrees that we have listed here obviously 3 is our highest degree that's our greatest degree so if I wanted to know the degree of the polynomial it would be the highest if the ones that I have listed here so in this case it will be three the degree of this particular polynomial is three the highest of the degrees of each term all right so we've kind of laid some groundwork with some vocabulary and so now let's actually get to adding and subtracting all right so when we add our polynomials or when we subtract them we need to make sure that we have like terms all right so we can only add and subtract like terms and so in order for things to be considered like terms they have to have exactly the same variables all right the other thing that they have to have is that those variables have to be raised to the same powers all right so let's look at a couple of these to see what like terms would look like alright so I'm just going to make a little list here things that might be like terms would be 3x squared and negative 2x squared those would be considered like terms that exactly the same variable raised to the same power if I had something like 12 X Y squared and I had three XY squared those would be considered like terms we the same variables raised to the exactly the same powers all right so things that are not like terms just to give you an example of that all right if I have 3x squared and 3x those are not like terms the 3 has nothing to do with them being like terms has everything to do with the variable and the exponent on it they both do have an X but the first one is x squared and the next one just has an X to the first power so these are not like terms so let's actually take a problem where we have some like terms we're going to have to identify and then we will add and subtract if we need to all right so let's start with negative 9x cubed plus 7x squared minus 5x plus 3 we're going to take that polynomial and we're gonna add this second polynomial to it so 13 x squared plus 2x squared minus 8x minus 6 I'm sorry 13 X cubed all right so we are going to add these polynomials now the first thing I'm going to do is identify what my like terms are now I'm just going to use a different color to underline so we can decide what are the like terms alright so here we have an X cubed and an X cubed so those would be considered like terms we have here an x squared and an x squared so those would be considered like terms this negative 5x and this negative 8x would be considered like terms and then our last one this 3 and this negative 6 neither one of those has a variable at all so they would be considered like terms so when I get ready to do my adding of terms we could only add the like terms all right so I'm going to show you two different ways MyMathLab is the first way that I'll show you because MyMathLab what it does is it's going to take this first polynomial and it's gonna put the other polynomial up under it making sure to line up our like terms notice that my like terms are lined up on top of each other and then I'll come over here and put a plus all right so now I'm actually going to do it so add the coefficients the numbers in front so negative nine plus thirteen we give me 4x cubed positive 7x squared plus 2x squared give me plus 9x squared negative 5x plus negative 8x when the signs are the same we add and keep their sign so that's something that hopefully if you took math 98 you're clear on some of you may need a little help with your integers let me know if you need some extra help there and then our last two would be 3 plus negative 6 my sons are not the same this time so I subtract and take the sign of the larger one so this would be my solution right here there are no like terms in my answer so we leave the answers just like that now MyMathLab calls this vertically it just says to work the problem vertically that is your option if you want to stack it on top of each other like that some people don't like to stack it vertically and that's okay if you don't want to write it vertically what I do want is to see your work okay so the other way to do that all right this way is stacking it vertically all right if I choose not to write it vertically then I'm going to put like terms beside each other so let me slide this up just a bit all right so if I'm not doing it vertically then I would write my like terms beside each other from the original problem so those two I underlined in green they are like terms then I have plus 7x squared plus 2x squared those are like terms minus 5x plus now notice here I have a plus negative 8x I have to care be careful with my sons and then my last two terms are a plus three plus negative 6 all right so now I'm ready to combine my like terms these first two are like terms so I get 4x cubed do you add their coefficients sounds are not the same so we subtract take this on of the larger one okay now I would add these two like terms I get 9x okay these two like terms my sons are the same so you get negative 13 X and then over here my last two the sons are not the same so I subtract and keep the sign of the larger one so notice our answer here is exactly the same as we got when we wrote it vertically except I have a small error on my board here we go that should have had a 9x squared the variable does not change nor does it exponent when you are adding and subtracting so notice that the two things that are circled in red are exactly the same thing no matter which way you write it vertically or if you write it beside each other doesn't matter to me all right so one last little idea that I want to show you is if we are subtracting sometimes that's where it gets a little crazy and so in MyMathLab they're going to give you some problems that looks thing like this they're gonna say subtract and then that will list for you a polynomial and they will say subtract it from and then they'll list another polynomial alright so there's our two polynomials alright so if I used to write it vertically like MyMathLab likes you to do then here's what I have to decide my two polynomials here they are there's my first polynomial it does include that six right there sorry my five kind of went away all right and then here is my second polynomial all right now synthesis subtract this first polynomial from the other one then I have to start with the ten x cubed polynomial okay that one must go on top okay because of this word from then it has to go on top because of the word from okay and so that I'm gonna is does say subtract and when I put this other one down here I'm going to put a big parenthesis in front because I am subtracting which is gonna matter a little bit alright now for most of us actually for all of us because I do want us all to show our work that is something that I'm going to require excuse me all right so our next step is this little negative that's sitting out here we need to do the distributive property which means we distribute the negative through the parentheses so over here to the right I'm going to rewrite the problem okay our top polynomial is not going to change its going to stay exactly the same our bottom polynomial now it changes to negative 3x cubed minus 8x squared times that negative out front makes it a positive okay and then we had this minus 5x but times the negative out front makes that a positive and then our 6 is a positive but when I multiply it by the negative out front it becomes a negative now I'm ready to do what it says vertically so I have 10 X cubed minus 7x cubed I'm sorry minus 3x cubed gives me 7 X cubed negative 5 x squared plus 8x squared gives me positive 3x squared 7 X plus 5x will give me plus 12x and then negative 2 minus 6 will give me negative 8 so my solution here is 7 X cubed plus 3x squared plus 12x minus 8 you will see problems like this in MyMathLab so please be very careful and make sure that you write them correctly the second polynomial does need to go on top you're welcome to watch other videos there are some more videos in your my math lab that are provided for you but I would like for you to also watch all of the videos that I provide on our YouTube channel so let me know if you have any questions I look forward to seeing you in our team's meeting