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Current time:0:00Total duration:7:21

Video transcript

what I want to do in this video is think about how expressions are formed and then words we use to describe the different parts of an expression and the reason why this is useful is when you hear other people refer to some expression and say oh I don't agree with the second term or the third term has four factors or why is the coefficient on that term six you'll know what they're talking about and you can communicate in the same way so let's think about what those words actually mean so we have an expression here the first thing I want to think about are the terms the terms of an expression or what a term is and one way to think about it is a term a term of an expression are the things or the terms are the things that are getting added and subtracted so for example in this expression right over here you have three things that are getting added and subtracted the first thing you're taking 2 times 3 you're adding that to 4 and then from that you're subtracting 7y so in this example you have 3 you have 3 terms the first term is 2 times 3 the second term is just the number 4 and the third term is the number or as is e is 7 times y now let's think about the term factor and when people are talking about a factor especially in terms of an expression they're talking about the things that are getting multiplied in each term so for example if you said what are the factors of the first term the first term refers to this one right over here 2 times 3 and there's two factors there's a 2 and a 3 and they are being multiplied by each other so here you have two factors in the first term what about the second term the second term right over here the second this was the first term the second term here has only one factor just the four it's not being multiplied by anything and the third term here once again has two factors it's the product of 7 times y so we have two factors here we have a 7 and a y and this constant factor here this number 7 that is multiplying the variable also has a special name it is called the coefficient of this term coefficient coefficient coefficient and the coefficient is the non variable that multiplies the rest of the term that's one way of thinking about it so here's 7y even if with seven XY or seven and XYZ or seven X Y Z squared that non variable that's multiplying everything else we would consider to be the coefficient now let's do a few more examples and in each of these I encourage you actually right now to pause the video think about what the terms are how many terms are there in the in each expression how many factors in each term and what are the coefficients so let's look at this first one it's clear that we have three things being added together this is the first term this is the second term and this is the third term so this is the first term the second term this is the third term and they each have two factors this first one has the factors 3 and X the second one here has the factors x and y and this third one is has the factors Y & Z now what are the coefficients here will remember coefficients is a non variable multiplying a bunch of other variables and so here the coefficient in this first term right over here is a 3 now you might be saying well what about the coefficients on these terms right over here depending on how you think about it one way to say is well X Y is the same thing as 1 times X Y so some people would say that hey you have a coefficient of 1 here on the XY or it's implicitly there that it wasn't written but you're multiplying everything by 1 and that might be subject to a little bit of interpretation one way or another now this one is really interesting because if we look at the bigger expression if we look at the whole thing if we look at the whole thing it's clearly made up of three terms the first term is X Y Z the second term is X +1 that thing whole thing times y and then the third term is 4x and if you look at that level if you look at the first term and you say well how many factors does that have well you would say that it has three factors x y&z how many factors does the second term have well you could say well it has two factors one factor is X plus y and then the other factor is why it's multiplying x + or sorry the first factor is x + 1 and the second one is why it's multiplying this expression this expression itself the smaller expression itself is one of the factors and the other one is why and then this third one also has two factors a 4 and an X and if someone said here what's the coefficient on this term you say hey look the coefficient the coefficient is the 4 now let's look at this one over here actually before I look at that one what was interesting about this is that here you had a little smaller expression itself acting as one of the factors so then you can go and then zoom in on this expression right over here and you can ask the same question on this smaller expression how many terms does it have well it has two terms an X and a 1 those are the two things being added or subtracted and each of them have exactly one factor so when we're giving these there's you can keep nesting these expressions to think about when you talk about terms or factors or factors of terms you have to really specify what what part of the nesting you're thinking about if you're talking about the terms of this whole expression there's 1 2 3 but then if you go you could look at this sub expression which itself is a factor of a term and say oh well there's only two terms in this one now let's look at this one how many how many terms well once again there's clearly three actually let me add one more cuz I'm tired of expressions with three terms so I'm just going to add a one year so now we clearly have four terms this is the first term second term third term fourth term and how many factors are in each of them well this is interesting you might say well a lot of the factors are the things that are being multiplied but here I'm dividing by Y remember dividing by Y is the same thing as multiplying by its reciprocal so it would usually be considered to have three factors here where the factors are 3x + 1 / y if you multiply 3 times x times 1 over Y you're going to get exactly what you have right over here so you would say this has three factors if someone asks what's the coefficient here well you'd say well that 3 is the coefficient here how many factors do you have and this is a little bit tricky because you might say well isn't 5x squared times y isn't that equal to 5 times x times X times y and you'd be right and you'd be this would be very tempting to say that you have four factors but the convention the what the tradition that most people use is that they consider the exponent of with X as a base as just one factor as this is just one factor so traditionally people will say this has three factors it has a 5x squared and a Y x squared is just considered a factor and once again what's the coefficient it's the five so with that in mind how many factors here well you have three factors here you have an X you have a y squared and you have a Z to the fifth and then finally this last term it's a constant term how many factors does it have was just one it's just got a one sitting there it's not being multiplied by anything