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Current time:0:00Total duration:15:59

Video transcript

in this video I want to introduce you to the idea of a polynomial 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 exponent 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 essentially saying you have many terms poly bean tends to mean many this is a binomial if I were to say 4x to the 3rd minus 2x squared plus 7 this is a trinomial I have three terms here and let me give you just to get 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 one this is not a polynomial 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 doesn't it has a it has a negative and a fractional exponent in it or if I were to give you the expression 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 one-half power so all of the exponents on our variables are going to have to be non-negative so once again this is 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 the degree of the polynomial and that's essentially the highest exponent or the that's the 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 polynomial I'm going 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 second degree 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 vs. 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 I guess last part to dissect the polynomial properly is to understand the coefficients of a polynomial so let me write a I don't 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 7 X 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 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 this really 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 negative in front of it it's going to dominate everything else so it really gives you a sense for how quickly or or how fast the whole expression would grow or decrease in the case if it's 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 the coefficient on this term right here is negative five you have to remember you have a minus five so we consider negative five to be the whole coefficient the coefficient on this term is a seven there is no coefficient here it's just a constant term of negative five and then the coefficient on the x squared term is one the coefficient is one it's implicit you're assuming it's one times x squared now the last thing I want to introduce you to is just the idea of the standard form of a polynomial standard standard formula 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 lists 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 7 X to the fifth then what's the next smallest degree well then I have this x squared term I don't have an X to the fourth or next to the third here so then it'll be plus one I don't have to write one plus x squared and then I have this term minus 5x and then I have this last term right here minus five 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 nearly your your mathematical career so let's just simplify a bunch of polynomials and we've kind of touched on that 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 had 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 I could just it's not like 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 where I say like degree has to also have the same variable but we also in this example we only have the variable X so let's add let's say I have this x squared term and I have 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 for the X terms so 4x + 7 X 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 1 x squared I could just write negative x squared but I want to show you that I'm just adding negative 2 to one there then I have plus 4x plus 7x is 11 X and then I finally have my constant term minus 12 and I end up with a 3 term second-degree polynomial the leading coefficient here the coefficient on the 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 it more let me do a complicated example so let's say I have 2a squared B minus 3a B 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 3a B 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 a minus out here so minus 2a squared B squared minus 4a squared B and the 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 an a squared B have to be very careful here well no that's a B squared now a squared B squared well here I have an a here I have a a squared B a squared B 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 a B squared term now do I have any other a B squared terms here any other a b squared no other a b squared so I'll just write it minus 3a B squared and then let's see I have an a squared B squared term here do I have any other ones well you're sure the next term is that's an a squared squared term so let me just write that plus 5a squared B squared minus 2a squared B squared all 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 to of something minus four of something is going to be negative two 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 3a B squared and then we can add these two terms if I five of something minus two of something I'm going to have three of that something plus 3a squared B squared and then finally I have that last term plus 5b squared now here and 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 as 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 a B squared terms first 2a squared B and then you have the minus 3a v squared and then we have just the V 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 part this blue part right here the area there is x times y x times y and then what's the area here it's going to be x times Z so plus x times Z 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 I could just write say plus two times x times Z and here we have a polynomial that represents the the area of this figure right there now let's do this next one what's the area here well I have an A time's a be a B this looks like an A time's the be again plus a B that looks like an a B again plus plus a B plus a B and this is I think they've drawn it actually a little bit a little bit strange well I'm going to ignore this C right there maybe they're telling us maybe they're telling us that this that this right here is C because that's what 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 a B here which I'll assume for this purpose of this video and then we have that last a B and then we have this one a times C one a times C this is the area of this figure and obviously we can add these four terms this is for a B and then we have plus a C and I made the assumption that this was a bit of a typo that that's see what they're 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 of the whole rectangle which would be 2 X Y and then we could subtract out the area of these squares so it's each square has an area of x times X or x squared and we have two of these squares so it's minus two x 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 a B and then the area over here looks like it will also be a B so plus a B and the area over here is also a be also a B so the area here is three a B well anyway hopefully that gets us pretty warmed up with polynomials