 Synthetic division, synthetic division. Now I've been mentioning this technique for the last few videos but we still haven't got around talking to what it's really about, right, which is what we're about to do right now. Now the terminology for synthetic division, you know when we say synthetic division, that's a little misleading because we're not really going to be using synthetic division to divide polynomials together. We will for when we have a linear function, the denominator, and we will talk about that but what we're really going to use synthetic division for is to factor large polynomials, polynomials that are higher degree than degree of two, polynomials that are have more terms than three terms, so polynomials that aren't just trinomials but you know they have four terms or five terms because as soon as you go beyond three terms of a polynomial you have to go to a degree of three or a degree of four more than more than a degree of two, right, because if you have a trinomial it has to be degree two, degree one and a constant, right, degree zero. As soon as you go to four terms your first, the largest term has to be degree three and then degree two and then degree one and then degree zero. If you go to five terms and you start off with a degree four, right, because that's you know you set your for polynomials, you set your degree starting up with the highest degree first and going down, right. To learn this technique what we're going to do is talk about long division. We're going to do a couple of long divisions of you know using synthetic division and then we're going to put the long division of polynomials aside and just attack polynomials, large polynomials and try to factor them using synthetic division. So when we talked about you know dividing polynomials and we did polynomial long division basically we had the following types of equations, right, following types of expressions that we have to do. For synthetic division what we're going to specifically state, you know, work with is the top could be any degree, okay, but the bottom is going to be degree one. Now we can factor, we can use or we can do division, synthetic division for polynomials. You know there is a technique for it to do synthetic division for polynomials where the degree in the bottom is higher than degree of one but we're not going to go there, right, because I personally never ended up using it and if you do get some expression like this when you have to divide two polynomials together then you're just going to use long division if your degree in the bottom is higher than the degree of one. So what we're going to focus on for synthetic division is division of the following form where the denominator is just x minus a and a is a constant, okay. So this is what we're going to use synthetic division for and if you recall all this means if you know if you have x minus a in the denominator what you can do is just set that equal to zero and what you do you bring the a over so what you're doing is checking to see what your function is equal to when x is equal to a so what you're doing is solving for f of a and then finding out what your what your answer is which becomes your you know when you do your division it becomes your remainder right and we talked a lot about this in doing polynomial long division. One thing to keep in mind and this is the first thing we're going to start talking about is what you should be really careful of because in most things you look at most of the time that we're going to do this is it's going to be x minus a it's going to be the coefficient in front of the x is just going to be a one however you can do synthetic division where the coefficient in front of the x is not an a it could be it's not one but it could be another another number it could be two it could be three it could be any other number so we can do synthetic division for expressions where we have another number in front of the x instead of a one so we can do synthetic division for the following if we have something like this bx minus c in the denominator here we can still do synthetic division but there's a trick to it okay now let's just lay this out the way it works is all you do you set this equal to zero and then bring the c over divide by b and all it ends up being is your a becomes a fraction right so what you end up having is i think i had to do it this way because i went off the camera but what you end up having is x is equal to c over b right and your a term here would just be a fraction it would be just x minus c over b in general we're going to do synthetic division when it's just x minus a in the denominator okay if we get anything else a higher degree down here we're going to use polynomial long division and in general if we get a get a expression like this where we have this thing in the denominator we're still going to use polynomial long division where we are going to use synthetic division for is not necessarily for polynomial long division we're going to use it for factoring large polynomials polynomials that are higher degree than degree of two so let's go take a look at the special case here where you know we have this type of expression the denominator and you know make it a point to show you what it is that you have to watch out for and how things change when you write it you know move things over and you write it as a fraction where your a is really just your c over b okay