 In this video I want to do one final example of synthetic division. We're going to do a big one this time, right? 2x to the fifth, we have this degree five polynomial, a quintic. Oh no, it doesn't actually change the calculation too much for us here, because we're just going to write down all the coefficients of the numerator in descending orders. We get 2x to the fifth, we get a 5x to the fourth minus 2x cubed plus 2x squared minus 2x plus 3. All right, that's great. Now we have to divide by x plus 3. What does that mean? Well with synthetic division, we actually prefer an x minus c, x minus a constant there. And so we need to think of this as x minus a negative 3, for which case that's the number we're going to record right here. The first step is always the same, drop down the number, so you're going to get a 2 right here. And then just go through this step by step by step. 2 times negative 3 is negative 6, plus 5 is a negative 1. Negative 1 times 3 is a 3. 3 minus 2 is a 1. 1 times negative 3 is going to be a negative 3. 2 minus 3 is going to give us a negative 1. Negative 1 times negative 3 is going to be a positive 3. Subtract from that 2, we get a 1. And then 1 times negative 3 gives us a negative 3, which then adds with 3 to give us a 0. And so this situation we see our remainder is going to be 0. So if we were to write this thing out, we're going to get 2x to the fifth plus 5x to the fourth minus 2x cubed plus 2x squared minus 2x plus 3. If we divide this by x plus 3, this will give us our quotient, which the quotient will always have 1 degree smaller than what we started with. So we get 2x to the fourth minus x cubed plus x squared minus x plus 1. And then you add the remainder of 0, which there's nothing to add here. So you notice that in this situation, when you took the numerator divided by x plus 3, we actually got a polynomial, right? This actually suggests we have a factorization. We could write this in the following way. Give myself a little bit more space. Turns out we actually found a factorization. Because when the remainder is 0, that actually means we have a factor. x plus 3 is a factor. And then the other factor is 2x to the fourth minus x cubed plus x squared minus x plus 1. So wow, we took this big old polynomial x something to the fifth power and we actually factored it. Who would have thought you would have ever been on the factor of polynomial that's so big? And so that was actually pretty nice. That was great that we tried x plus 3. So if only there was a way we could know to predict that x plus 3 would be a factor, then by division you could figure out the other factor, which is degree 4 is as well. And that's what we're going to be talking about in the next lecture, which is another video of course, that synthetic division will be a tool we use to factor really big polynomials. That if we know how to predict who we're going to be the right things to divide by, x plus 3, x minus 3, maybe we do like an x plus 1 or something. If we know how to predict what those are going to be, we can then run synthetic division and boom, boom, boom, boom. We can check very quickly whether we have a factor or not. And so synthetic division is very much going to be the key for us as we learn to factor large polynomials. And I will do this in the next video. See you everyone.