 and welcome back. Today I'm going to talk about synthetic substitution. This is a process, this is kind of an additional process, kind of something you can add to your repertoire of your mathematical processes that we know. This is something that is a little bit extra that we can do. It's an interesting example for substitution. But anyway, to get down to what it actually means. Okay, synthetic substitution is for evaluating numbers. Okay, so for example, if you have a function, in this case my function is p of x equals 2x to the third plus 5x squared minus x plus 7. If I want to evaluate this number, so I want to figure out what this is when x equals 2. Now, some of you are thinking, well, don't you just plug it in? Absolutely, yes, you just take this number and plug it in. So example would be p of 2 equals, let's just plug it in, 2 times, shoot, oh, plugging in 2, there we go, 2 times, 2 cubed plus 5 times 2 squared, I'm going to run out of room maybe, times minus 2 plus 7, oh, just barely got there. Okay, and then so this is going to be 2 cubed, which is 8, 8 times 2 is 16, plus 20, minus 2 plus 7. So this is going to be, what is that going to be? That's going to be 36, plus 7 is 43, I just lost track of my numbers, good job, yep, 43, and then 43 minus 2 is 41. Okay, so in this case p of 2 is equal to 41. Okay, now that's what you would normally do, okay, so there's my chicken scratch off there to the right side of what you would normally do. Okay, now what I'm about to show you is this is an additional process that you can use. Okay, now over here what I did is I did a lot of mental math, okay, I cubed the numbers and multiplied times 2, I squared and multiplied times 5 to get all this, then I added numbers, I added these two to get 36, then I added the 7 over here for easy addition, what kind of easy, can I just stall there for a second, and then minus 2 to get 41. Okay, now that's what you would normally do, and there's nothing wrong with that process. It's easy, it's straightforward, you plug in the number, you get an answer, fantastic. But, synthetic substitution is an additional process that you can use to get this same answer. Okay, so showed you that first, now let's go back over here to what the video is actually about, synthetic substitution. Okay, now the setup for synthetic substitution is very, very similar to synthetic division, it's actually almost exactly what it is. Okay, we are plugging in x equals 2, which means over here in this little box is that exact number. Now that is different, that is different from your synthetic division example that I've done. In synthetic division, we would change the sign here, but in this case we're not changing any signs, we're not doing any of that kind of stuff, we're just using the number straight up. Okay, but then over here these numbers were the coefficients, okay, these numbers right here are the coefficients, so the 2 here, the 5 here, the negative 1 and the 7, no gaps, so I got no zeros. Okay, so those numbers actually are the exact same for, or excuse me, from synthetic division. Okay, and now I just go through the process, okay, take this 2, bring this down, multiply here, 2 times 2 is 4, bring that up here, bring this down to get, add down to get 9, multiply to get 18, add down to get 17, multiply to get 34, add down to get 41, there we are. Okay, very, very quickly, very quickly I was able to get that answer and notice, I know what the answer is over here, p of 2 is equal to 41 and notice where my answer is, ding, ding, ding, right here. Normally this is the remainder spot, if we were dividing, that would be the remainder spot for my synthetic division. So in this case, if I'm doing synthetic substitution, if I'm evaluating numbers, this isn't the remainder spot, this is my answer spot. This is my answer spot, so in this case p of 2 is equal to 41, there we go. Okay, now as you can see, this is a process for those who might not be very good at substituting in numbers. If you're not very good at, okay, cubing first, then multiplying times 2 and keeping track of all these numbers and that kind of stuff, you know what, substitution might not be for you. Synthetic substitution here, this process might be a little bit easier for you. Okay, now this is actually also a process that we're going to be doing for solving polynomial equations that are a little bit difficult to solve. Okay, so this is actually a process that we're going to be using in the future, which is really going to be helpful. But anyway, that is synthetic substitution. Thank you for watching the video and we'll see you next time.