Synthetic division of polynomials
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Let's do another synthetic division example. And in another video, we actually have the why this works relative to algebraic long division. But here it's going to be another just, let's go through the process of it just so that you get comfortable with it. And now is a good chance to give it a shot, to actually try to simplify this rational expression. So let's think about this step by step. So the first thing I want to do is write all of the coefficients of the numerator. So I have a 2. Oh, I have to be careful here. Because the 2 is the coefficient for x to the fifth, I have no x to the fourth term. Let me start over. So I have the 2 from 2x to the fifth. And then I have no x to the fourth. So it's really 0x to the fourth. So I'll put a 0 as the coefficient for the x to the fourth term. And then I have a negative 1 times x to the third. And then I have a positive 3 times x squared. Negative 2 times x. And then I have a constant term, or zero degree term, of 7. I just have a positive 7. And now let me just draw my little funky synthetic division operator-looking symbol. And remember, the type of synthetic division we're doing, it only applies when we are dividing by an x plus or minus something. There's a slightly different process you would have to do if it was 3x or if was negative 1x or if it was 5x squared. This only works when we have x plus or minus something. In this case we have x minus 3. So we have the negative 3 here. And the process we show-- there's other ways of doing it-- is you take the negative of this. So the negative of negative 3 is positive 3. And now we're ready to perform our synthetic division. So we'll bring down this 2 and then multiply the 2 times the 3. 2 time 3 gives us 6. 0 plus 6 is 6. And then we multiply that times the 3, and we get positive 18. Negative 1 plus 18 is 17. Multiply that times the 3. 17 times 3 is 51. 3 plus 51 is 54. Multiply that times 3. The numbers are getting kind of large now. So that's going to be what? 50 times 3 is 150. 4 times 3 is 12. So this is going to be 162. Negative 2 plus 162 is 160. And then finally, 160 times 3 is going to be 480. And you add 480 to 7, and you get 487. And you can think of it, I only have one term or one number to the left-hand side of this bar here. Or I'm just doing the standard, traditional x plus or minus something version of synthetic division, I should say. So I can separate this out, and now I've essentially gotten my answer. And it looks like voodoo, and it kind of is voodoo. And that's why I don't like to do it, because you're just memorizing an algorithm. But there are other videos why we explain why. And it can be fast and convenient and paper saving very often, like you see right here. But then we have our final answer. It's going to be-- and let me work backwards. So I'll start with our remainder. So our remainder is 487. And it's going to be 487 over x minus 3. And so this is our constant term. And so you're going to have plus 160 plus 487 over x minus 3. Now this is our x term. So it's going to be 54x plus all of this. This is going to be our x squared term. So this is going to be 17x squared plus 54x plus 160 and all of that. Then this is going to be x to the third term. So this is going to be 6x to the third plus all of that. And then finally, this is our x to the fourth term-- 2x to the fourth. And let me erase this. So then I have my x to the fourth term. So it is 2x to the fourth. And we are done. This thing simplifies to this right over here. And I encourage you to verify it with traditional algebraic long division.