Polynomial subtraction & set closure

- [Voiceover] We're asked to subtract negative two x squared plus four x minus one from six x squared plus three x minus nine, and like always I encourage you to pause the video and see if you can give it a go. All right, now let's work through this together. So I could rewrite this as six x squared plus three x minus nine minus, minus this expression right over here, so I'll put that in parentheses, minus negative two x squared, negative two x squared plus four x minus one. Now what can we do from here? Well, we can distribute this negative side, we can distribute this negative side, and then if we did that, we would get the six x squared plus three x minus nine won't change so we still have that. Six x squared plus three x minus nine, but if I distribute the negative side, the negative of negative two x squared is positive two x squared. So that's going to be positive two... Get a little more space. Positive two x squared, and then subtract, and then the negative of positive four x is... I'm going to subtract four x now, and then the negative of negative one, or the opposite of negative one is going to be positive one. So I've just distributed the negative side, and now I can add terms that have the same degree on our x... The same degree terms, I guess you could say, so I have an x squared term, here it's six x squared, here I have a two x squared term, so I can add those two together, six x squared plus two x squared. If I have six x squareds and then I have another two x squareds, how many x squareds am I now going to have? I'm now going to have eight x squareds, eight x squareds, or six x squared plus two x squared. We add the coefficients, the six and the two to get eight, eight x squared. Then we can add the x terms. You could view these as the first-degree terms, three x... We have three x and then we have minus four x, so three x minus four x, if I have three of something and I take away four of them, I'm now going to have negative one of that thing, or you could say that the coefficients, three minus four would be negative one. So I now have negative one x. I could write it as negative one x, but I might as well just write it as negative x. That's the same thing as negative one x. And then finally, I can deal with our constant terms. I'm subtracting a nine and then I'm adding a one. So you could say, "Well, what's a negative nine plus one?" Well, that's going to be negative eight. That's going to be negative eight and we are all done. And one thing that you might find interesting is I had a polynomial here and from that I subtracted another polynomial, and notice, I got a polynomial, and this is actually always going to be the case. If you think about the set of all polynomials, if you just think about the set... Let me do this in a neutral color. So if we think about the set of all polynomials right over here, and if you take one polynomial, which you could imagine this magenta polynomial. So this is a polynomial right over here, let's call this p of x. So this is p of x right over there, p of x, and then you have another polynomial, this one right over here, let's call this, I don't know, we can call this q of x, q of x, just for kicks. So that's q of x, just like that. And if you apply the... In this case, we applied the subtraction operator. If we apply subtraction... So you took these two, you took these two... Let me see how I could depict this well. So we took p of x and you subtracted from that q of x. We still get a polynomial, so that's going to give us... We stay in the set of polynomials. At any time you have a set of things, and you might be more used to talking about this in terms of integers, or number sets, but you can talk about this in general. Here we're talking about the set of polynomials, and we just saw that if we start with two polynomials, two members of the set of polynomials... Let me be clear, this is polynomials, polynomials, right over here. You take two members of the set and you perform the subtraction operation, you're still going to get a member of the set. And when you have a situation like this, I could call this one, I don't know, I'm running out of letters. Well let me just call this one f of x, so we've got f of x here. When you have this situation where you take two members of a set, you apply an operator on them, or you take a certain number of members of a set, you apply an operator on them, and you still get a member of the set, we would say that this set is closed under that operation. So we could say that the set of polynomials, set of polynomials, polynomials closed, closed... I won't even put it in quotes. Closed under, under subtraction. And I didn't prove it here, I just did one example where I subtracted two polynomials and I got another one, and there's clearly more rigorous proofs that you can do. But this is actually the case, as long as you have two polynomials, you apply subtraction, you're going to get another polynomial. And the fancy way of saying that is the set of polynomials is closed under subtraction. This notion of closure sometimes seems like this very fancy mathematical idea, but it's not too fancy. It's just you take two members of a set, you apply an operation, if you still get a member of the set after that operation, then that set is closed under that operation.