Intro to polynomial synthetic division

0:00- [Narrator] In this expression, we're dividing this third degree polynomial by this first degree polynomial. And we could simplify this by using traditional algebraic long division. But we're gonna cover in this video is a slightly different technique and we call it synthetic division.

0:16My personal tastes are not to like synthetic division because it is very, very, very algorithmic. I prefer to do traditional algebraic long division. But I think you'll see that this has some advantages. It can be faster, and it also uses a lot less space on your paper.

0:31So let's actually perform this synthetic division. Let's actually simplify this expression. Before we start, there's two important things to keep in mind. We're doing kind of the most basic form of synthetic division. And to do this most basic algorithm, this most basic process, you have to look for two things in this bottom expression.

0:50The first is that it has to be a first degree. It has to be a polynomial of degree one. So you have just an X here. You don't have an X squared, an extra to the third, an X to the fourth or anything like that. The other thing is, is that the coefficient here is a 1.

1:05There are ways to do it if the coefficient was different, but then our synthetic division will have to, we'll have to add a little bit of bells and whistles to it. So in general, what I'm gonna show you now will work if you have something of the form X plus or minus something else.

1:20So with that said, let's actually perform the synthetic division. So the first thing I'm gonna do is write all the coefficients for this polynomial that's in the numerator. So let's write all of them. So we have a 3, we have a 4, it's a positive 4. We have a -2, -2, and a -1.

1:43And a -1. Then, and you'll see different people draw different types of signs here depending on how they're doing synthetic division, but this is the most traditional. And you wanna leave some space right here for another row of numbers. So that's why I've gone all the way down here.

1:57And then we look at the denominator and in particular we're gonna look whatever X plus or minus is right over here. So we'll look at right over here, we have a positive 4. Instead of writing a positive 4, we write the negative of that. So we write the negative, we write the negative of that, which would be -4.

2:17We write -4. And now we are all set up and we are ready to perform our synthetic division. So first, this first coefficient, we literally just bring it straight down. And so you put the 3 there. Then you multiply what you have here times the -4. So you multiply it times the -4, three times -4 is -12.

2:41Then you add the 4 to the -12, four plus -12 is -8. Then you multiply -8 times the -4. I think you see the pattern, -8 times -4 is positive 32. Now we add -2 plus positive 32. That gives us positive 30. Then you multiply the positive 30 times the -4, and that gives you -120,

3:14- 120. And then you add the -1 plus the -120 and you end up with a -121. And now the last thing you do is you say, "Well, I have one term here." And in this plain vanilla, simple version of synthetic division, where you're only dealing actually when you have X plus or minus something.

3:33So you're only going to have one term there. So you separate out one term from the right, just like that. And we essentially have our answer. So to simplify this, you get, you get, and you can have a drum roll right over here. This is going to be, this right over here, this is going to be the constant term.

3:53You could think of it a degree zero term. This is going to be an X term, and this is going to be an X squared term. You can kind of just build up from here saying this first one's going to be constant, then this is going to be an X term, then an X squared. If we had more, you'd have an X to the third, an X to the fourth, so on and so forth.

4:08So this is going to be equal to, this is going to be equal to 3x squared, 3x squared minus 8x minus 8x plus 30, plus 30. And this right over here, you can view as the remainder. So minus 121 over the X plus 4. This didn't divide perfectly. So over the X, over the X plus 4.

4:40Another way you could have done it, you could said this is the remainder. So I'm gonna have a -121 over X plus 4, and it's gonna be plus 30 minus 8x plus 3x squared. So hopefully that makes some sense. I'll do another example in the next video and then we'll think about why this actually works.