Dividing polynomials of degree one

0:00- [Instructor] What we're going to do in this video is get some practice dividing expressions. So what do I mean by that? So let's say that I have the expressions 6x + 12, and I wanna figure out what that divided by, maybe I'll write this in a different color, divided by 2x + 4 is.

0:19And if you are so inspired, I encourage you to pause this video and try to work through this yourself before we do this together. All right, so first of all, we could think that there's many ways to write this same thing. We could have also written this as 6x + 12, let me put a parentheses around it, divided by, divided by 2x + 4, 2x + 4.

0:49And when you divide something by something else, and let's say we get a question mark right over here, or we could say we do this division, and we get a question mark over here, another way to think about it, let's imagine if we multiplied both sides by 2x + 4.

1:05Another way to think about, an equivalent statement is that 6x + 12 is going to be equal to this question mark, question mark times 2x, 2x + 4. So another way to think about is what can I multiply by 2x + 4 to get to 6x + 12? Well here you might say, all right, to go from 2x to 6x, I would have to multiply by 3.

1:36So that's multiplying by 3. And then to go from 4 to 12, I'm also multiplying by 3. So if my question mark is 3, it feels like it's going to work. If I take 3 times, let me write that yellow, 2x + 4, we can try it out. But we can see that, when we distribute that 3, 3 times 2x is going to be equal to 6x, and then 3 times 4 is going to be equal to 12.

2:05So the answer to our original question, what is 6x + 12 divided by 2x + 4? well that's going to be 3. Our question mark is 3 in this situation. Now the example that I just gave you, that's a situation where we don't have a remainder. But now let me give you an example that does have a remainder.

2:24So let's think about, if I have, and I'll try to get the colors again, 6a - 11 divided by, let's go with purple, that's always a little fun, divided by 2a - 5. And actually I didn't have to write the parentheses the way I wrote it right over here. But one way to think about it, just like before, we could say that this is equal to question mark.

2:50Or another way to think about it, if we multiplied both sides by 2a - 5, or you could say, if I have question mark times 2a - 5, 2a - 5, I am going to get 6a - 11. That is going to be equal to 6a, I don't have to write parentheses at this point, 6a - 11. So we could do the same idea.

3:15We say, all right, to go from 2a to 6a, I would multiply by 3. I'm multiplying by three. Now we might be tempted to say, well, what happens if I multiply 5 by 3? Well that's not going to get us the exact same thing. That's actually going to be 15 or -15, I have this negative sign out here, not -11.

3:37And so this isn't going to be exact, but we can write it with a remainder. Let's say that this is going to be roughly 3. Let's figure out what we get when we multiply 3 times 2a - 5, and then from that maybe we can figure out the remainder. So if I took 3 times, times 2a - 5, that, and we just talked about it, is going to be equal to 3 times 2a is 6a, and then we could say, and then we're subtracting 3 times 5, which is 15.

4:13So 6a - 15. That is obviously different than what we saw up here, but what if we could rewrite what we have up here? So we have 6a - 15 and then plus or minus something else. So let's try to do that. So I'm gonna rewrite what we have up here. So I have 6a - 11.

4:36Well how could I write that with a -15 and maybe a plus something else? Well -11 is the same thing as -15 and then plus 4. Remember, I'm just rewriting, looking for colors here, I'm just rewriting this -11, or the subtracting 11, as the same thing as subtracting 15 and then adding 4.

4:56This is the same thing as a -11 there. And if we write it that way, something very interesting happens because I could put the parentheses here. I could also say, well, let's just subtract the 15 first. And then all of that, of course, is divided by 2a - 5, 2a - 5.

5:19Well we could also rewrite this as 6a - 15 over 2a - 5, over 2a - 5, plus 4, 4 over 2a - 5. All I did, think about it, if I was going in the other direction, I have the same denominator here, so I could just add the numerators. Now what we actually did in this situation is we started with the numerators being added, and we say, all right, this the same thing as 6a - 15 over 2a - 5 plus 4 over 2a - 5.

6:02But this is really useful because we just figured out that this whole thing right over here is the same thing as 3. So we could say that our original question, what is 6a - 11 divided by 2a - 5? we could say that this is the same thing as 3 plus 4 over 2a - 5.

6:25And so you could view this right over here as the remainder. It goes 3 times with the remainder of this. And I know this is a little involved. You might have to watch this a couple of times to get exactly the intuition of what just happened. But in the future, we're gonna find more systematic ways of doing this that maybe will require a little, it'll be a little bit easier to do, but it's really good to understand what we just did.