 This video will talk about synthetic division. Okay so we have synthetic division which is really kind of like a shortcut because all we have to use are the coefficients. If we were to go back and look at our original problem we'd see all these values showing up so we don't have to do that. This here we're talking about the zero with the divisor so that's an important little fact. So let's start there. We have x plus seven and we want to find the zero when is it equal to zero? So x is equal to negative seven. So I need to use negative seven. And I also have to use the coefficients of this one. So we want to use the coefficients. So let's put those in there. We have coefficient of x cubed is one. Coefficient of x squared is five. Leave some space in between because you've got to add and subtract or add and multiply and do good stuff like that. And then it's minus seventeen x and then it's minus six. We're going to write it like this. We're going to put our negative seven out here. And what happens is down here is where remainder starts. Now we've done this problem. This is the very first problem we did in long division so we're going to be able to double check ourselves here. So here's the process. Okay it starts right here. It starts right here. Drop down the first term. So we're going to take this one below the line and then it says multiply the coefficient of the divisor and place it in the next column. So in other words on this diagonal here we want to multiply. So one times negative seven is going to give me negative seven. Now it says add the column. The five plus a negative seven is going to be negative two. And again we're going to multiply when we go this way. Negative seven times negative two is going to give us a positive fourteen. And then we add remember so when we add there we get a negative three. This is a negative two just to remind ourselves. And then we take negative three times negative seven and we get a positive twenty one. And if we add negative twenty six and positive twenty one we get a negative five remainder. So this is our remainder and this is our quotient. It may not look much like a quotient right now but it is. And here's how you do it. You got your remainder and then you've got your constant and then you've got your x term and you've got your x squared term and your x cubed term and however else you want to go. So we have a remainder of negative five and then our quotient is negative three is the constant and negative two x is the x term and then x squared would be our first term. There's your one, your negative two, and your negative three. And then it would be if you really wanted to write it in here as a polynomial you would write it this way. So we think that it's x squared minus two x minus three with a remainder of negative five. Let's go back a few slides and what did we get? x squared minus two x minus three with a remainder of negative five. If we could just see these in the split screen it would be so much nicer. You can see that we actually get all those values just in a much simpler form. Okay so what is the divisor? They sometimes call it c we want to know what the zero is. So subtract the three and we want negative three. Usually whatever your polynomial here is you want the opposite sign you see in the binomial. So we want a negative three on the outside and see if you can set it up. I'm going to wait just a second. See if you can set up the dividend coefficients. You should have gotten one and the zero for the squared term minus seven and six. This will be our remainder down here in this little box. So let's go to work. One. Multiply on the diagonal and then place it inside. Add on the column. This gives us negative three. Negative three times negative three on the diagonal gives us a positive nine and when we add we get two and two times negative three will give us negative six and when we add the column we get zero. So remember this is our constant this is our x and this is our x squared. So we end up with x squared minus three x plus two and we have no remainder and if we were to take that times our x plus three just to check one more time add the remainder which is just zero. X cubed looks good. Three x squared and negative three x squared are going to cancel each other out and we had no x squared term. Minus seven x still looking good. Plus six.