 All right, welcome back. We are going to talk about synthetic division Okay, so in a previous video I talked about polynomial long division and this is going to be polynomial synthetic division this is a This is a process that it's basically a quick way of dividing polynomials It doesn't really look like the polynomial long division that I did a previous video looks much much different But it is a much faster way of doing division. You'll see that here in just a moment now the process itself Does it looks very confusing it really really is but the one thing you got to do just stick with me to the end of the episode Stick with me to the end of the video and we'll get through this and you'll see actually how much easier Synthetic division is then your normal polynomial long division Okay, so the first thing that we want to do identify kind of what we're divide and what we're dividing here in this case We're taking 3x squared Minus x to the third plus 5x minus one this polynomial this Cortic four-term polynomial divided by this linear binomial x plus two Okay, so we want this we want to see we kind of see what? What we get when we divide okay now if you saw from the polynomial long division video I can get a little bit hairy about what all you're multiplying and that kind of stuff It's actually this synthetic division is actually a simpler process. There's a little bit of setup to it though So here's the setup okay now on this on this side what we have this is your setup right here You have you have a number over here in this kind of little half box Then you have these numbers that enter row right here. You have a gap right here We're gonna put some more numbers right down here. You have this line and then you have this box right here We're gonna have numbers down here also. There's gonna be numbers all over the place Okay, but this is kind of the setup you have your box here row of numbers We have a gap right here. We have our line and then we have this little kind of three-quarters box I guess you could call it down here Okay, now where do all these numbers come from? Okay, now we're dividing by x plus two and so the number that's going to go in this box is always going to be the opposite Opposite of this number Okay, so notice it's x plus two so the number that we actually put inside this box is going to be a negative two Okay, that's gonna happen every single time you just use the opposite of that number. Okay, all right then these other numbers here What are these numbers come from? We'll notice I have this up here. These numbers are our coefficients co e coefficients Okay, now these are the coefficients of our of our dividend the numbers have to be on top right here Okay, but notice that there's actually something extra. We have this zero x squared notice that there's a gap right here Okay, there's an x squared variable that is missing there. We're not necessarily missing. We just don't have one there But for the synthetic division process, we have to have all of the coefficients even the ones that are not there Sounds kind of weird. We have to have all the coefficients even the ones that are not there So if a number is not there, we assume it to be zero. So this is actually zero x squared So then my coefficients I got from up here and from this little gap So these numbers are just the coefficients of three negative one zero positive five and that negative one right there Okay, so that's where those numbers come from And now now we actually get that that's the setup. Okay, that's the setup now We get into the actual synthetic division process So now what we're going to do the your first step is actually we're going to start here on the left side We're going to start here working on the left. This three is what we're going to start with your first coefficient I'm going to take this number and I'm going to just bring it down Okay, I'm just going to bring this number down Okay, and then this right here this this row of numbers that I'm going to get are going to be my answers These are going to be my answers Okay, well that answers plural my answer singular. Anyway, so The numbers that I get here are going to be part of my answer now Actually writing the answer is going to be a little bit a little bit different But we'll get to that here in a minute Okay, so take this three and move it on down and then what we're going to do is we're going to take this number That's inside the box and we are going to multiply it times any number that we bring down here Okay, we're going to multiply times any number that we bring down here So in this case for this number I'm going to take negative two Times three to get a negative six and that negative six We are going to put up here in this gap. I mentioned this earlier. We have a gap right here Okay, so again negative we take the three brought it down. We're going to use negative two We're going to multiply that times three to get a negative six. So that's what I have up here. All right And then we just continue and then we just kind of repeat this process I'm going to bring these numbers down This is a negative one and a negative six which are going to combine to get a negative seven You can also think add down You can also think just add these numbers down to get negative seven And then just continue with the process I'm going to take negative two and I'm going to multiply that times negative seven to get uh positive 14 forgot the number there we go five positive 14 And then you just continue the process continue the process. So I'm going to add down to get 14 Multiply up to get negative 28 Add down to get negative 23 and then multiply But oh this is okay negative two times 20 negative 23 is a positive 46 Okay, and then I'm going to take negative two. I'm going to take negative two Oh shoot. Nope. I'm done. Oh, I'm done. I just have to add this down. So negative one Plus 46 is a 45. There we go. That's better All right. So that's the synthetic division process I didn't do my arrows for each one of these but you can't we kind of get the idea So let's recap we we're not going to write the answer quite yet. So let's recap a little bit So I take I start right here. I start with this three I bring it down and then from there I take negative two times this three to get negative six Add down to get negative seven then I take negative two times negative seven to get a positive 14 Then I add these numbers down. Okay, take negative two times 14 to get negative 28 Okay, then add these numbers down to get negative 23 Then I take negative two times negative 23 for a positive 46 that I add those numbers down to get 45 Now I've done the entire process, and that's a relatively straight forward process. Add down, multiply up. Once you have everything written out, it's add down, multiply up, which actually is kind of simple. But now from here, we need to rewrite our answer. This number in the box over here, this one is unique. This is the remainder, so if you remember from our long division example, our polynomial remainder gets written like this, I'm going to write this down here. We write it as 45 over what we're dividing by. Now in this case, we are dividing by x plus 2. We're dividing by x plus 2. So that's what our remainder is. Now what I'm going to do is I'm going to actually write these numbers in reverse. Now you'll see why I write them in the reverse here in just a moment. I'm going to take this 23, this is my constant number. So I don't have a variable with this. This is going to be negative or minus 23. Then I'm going to move on. Now in my answer down here, the numbers are simply just going to increase, the variables are going to increase as I go up to the left here. So this 14 is my linear term, so this is going to be plus 14x. Next is negative 7, this is going to be my quadratic term, which is negative 7 squared. Next is 3, this is going to be my cubic term of 3x to the third. Now notice the progression, x to the third, x squared, x, no x's remainder. Notice the progression down the line. And that right there, there's our answer. So you can kind of realize why I did this backwards. I started with the remainder and then worked my way up from having no x's to 1x to 2x's to 3x's. That is synthetic division. That's the whole process right there. Yeah, and the next video that I do will be synthetic substitution, which actually uses this synthetic division process to do some substitution stuff, which is actually kind of neat. Anyway, that's it for this video. Hope you enjoyed it and thank you for watching.