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Hi, I am Dr. Shah. I was the National Lecturer Competition winner in 1989, and I am the maths master at Mathscool. Now, ready for newer things in math. Synthetic division - if you have watched the video on polynomial division, this is exactly the same topic, but a speed up way of performing that same calculation. So, I have given myself the same example as we did, when we did the polynomial division, the same cubic. I am going to divide it by the same linear divisor. I am going to start exactly the same way. Put my cubic inside the division, and my linear outside the division. Now, when we are doing synthetic division, what we are largely doing is ignoring the Xes in the equation. What I am going to do is just put a 1 in front of this x cube, and I am going to just rub out the x cube, rub out the x square, rub out the 4 x and rub the x there. Okay, so what we are doing is just looking at the numbers, and that's why this is the speeded up version of the same calculation. Step one - we have to change the sign of this number in the front. So that's the very first step. Change the sign of that, at the moment, it is +2, I am going to change it to -2. Okay, so it's important in the first step that we must change the sign of that number. Next thing I am going to do is that this number here, the very first number, and then just push it straight up at the top. I need a number in the top before I can start synthetic division, so I just take that number and push it up. And then multiply 1 by -2, which is -2. Write underneath the second, and then add these together. Adding them this time, so +5 add -2 gives me +2. And then do the same thing again. Multiply that into that. That's 3 times -2, that's -6, and again adding these two together, will be -2. And then again, the same step. Multiplying into that, -2 times -2 is going to be +4. Now, there are no more digits at the end here. So I am going to just stop here, by adding these at the bottom and put my 5 here. Now I'm going to put back in all those Xes. So, I am going to put that back in again. x, I am going to put that back in again, x cube, put that x square back in, put that x back in. And also at the top here, I am going to put a x square and x and the last one is going to be a unit. I know the last one is going to be a unit, so that's going to be an x, and that, an x square, and we have the same answers we got before. That is our quotient, and that is our remainder. And that is synthetic division.
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