Terms, factors, & coefficients

What I want to do in this video is think about how expressions are formed and the 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 6, 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. And the first thing I want to think about are the terms of an expression or what a term is. And one way to think about it is 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 three terms. The first term is 2 times 3. The second term is just the number 4. And the third term 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? This was the first term. The second term here has only one factor, just the 4. 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. And the coefficient is the nonvariable that multiplies the rest of the term. That's one way of thinking about it. So here's 7y. Even if it was 7xy or 7xyz or 7xyz squared, that nonvariable 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 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. This is 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 has the factors y and z. Now, what are the coefficients here? Well, remember, coefficients was a nonvariable 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 it is, well, xy is the same thing as 1 times xy. So some people would say that, hey, you have a coefficient of 1 here on the xy. Or it's implicitly there. 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, it's clearly made up of three terms. The first term is xyz. The second term is x plus 1, that 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, and 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 y. The first factor is x plus 1. And the second one is y. It's multiplying this expression. This smaller expression itself is one of the factors. And the other one is y. And then this third one also has two factors, a 4 and an x. And if someone said, hey, what's the coefficient on this term? You would say, hey, look, 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, 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 part of the nesting you're thinking about. If you're talking about the terms of this whole expression, there's one, two, three. But then you could look at this subexpression, 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 terms? Well, once again, there's clearly three. Actually, let me add one more, because I'm tired of expressions with three terms. So I'm just going to add a 1 here. 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, 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 and 1/y. If you multiplied 3 times x times 1/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. So it would be very tempting to say that you have four factors. But the convention, the tradition that most people use, is that they consider the exponent with x as a base as just one factor, this as 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 5. 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? Well, it's just 1. It's just got a 1 sitting there. It's not being multiplied by anything.