 We realize that all matter is merely energy condensed to a slow vibration. That we are all one consciousness experiencing itself subjectively. There is no such thing as death, life is only a dream, and we're the imagination of ourselves. So let's say they've given you the following polynomial. It's a function, it's h of x is equal to this huge thing here. Now, we can't use any of the other polynomial techniques that we've used before to factor this right now because that's x to the power of four. It's not in form of a trinomial. It's not in form of a, you know, where we can use the quadratic formula or simple trinomial factor or complex trinomial factor. And it's definitely not the difference of two things, right? So we're stuck. We can't factor it unless we decide to factor it with synthetic division which is what we're going to have to do. Now, as we stated before, the possible factors, or the possible factors that we can do manually, right? There could be other factors. It's just the factors that we can do manually using synthetic division. The other factors, you know, we're going to have to use computer graphing technique or some kind of algorithm trying to figure out what the factors are if they're not part of, you know, natural numbers. Natural numbers are the real numbers that we're talking about, right? Now, the first numbers we're going to try out are the smallest possible factors, right? Now, the possible factors of 60, there's a whole bunch of them. It starts, you know, plus or minus one, plus or minus two, plus or minus four and it continues on from there all the way up to plus or minus 60, right? For two, the possible factors of two are plus or minus one and plus or minus two, right? So if we can't find any factors using the possible factors of 60 straight out then we're going to have to start dividing the possible factors of 60 by two. Now, there's a whole bunch of numbers there. We're not going to start off with plus or minus 60, right? Because that's going to be crazy. You know, you always start off with the smallest number. You always start off with plus or minus one or plus or minus two. Sometimes, you know, the more of this you do, you can sort of see through the coefficient what's going to work. If all the coefficients, there's tricks to this, right? If all the coefficients are positive, then a positive factor is not going to work because it's not going to kill any of the coefficients. So you're going to try the negative factors first, right? So let's try one of these things out, try one of the factors out first and we're going to lay down the, you know, synthetic long division right here and just do it right here, okay? And keep in mind, one thing we did mention, if you had any gaps in the descending powers, you would have to put a place marker there, zero there, right? Right now, we don't have any gaps. All the x's are descending and there's no skips from, you know, x to the power of four to x to the power of two. It's all descending in order. So we don't have to put any zero place markers, okay? So let's lay down the synthetic division form right now and just go ahead and do it and see what we end up with. So the first number that we're going to try out to see if it's a possible factor of this thing is x is equal to one, which means the factor is x minus one, right? Remember, the one always comes over when we're going to finally write it in the factor form. So we're going to try out one. So what's going to happen is the five is going to come, or the two is going to come down, right? The two is going to come down. We're going to multiply by one. It's going to come up here and then add those two guys and then come up this way and continue this way. Now, right off looking at this, I know this doesn't work. One reason is because I set up the problem. The other reason is I'm multiplying by one. That's a negative. That's a negative, right? So that's negative 29 minus five. That's going to be negative 34. And that's not going to add up to 60 for us to take it out, right? So what we can do right now is do this or change the number to two and see if two works out in this. Let's do the one first where we're going to see that it doesn't end up being zero. And then we're going to go ahead and try the other numbers. And the odds are after this one, we're not going to continue to try all random ones, right? We're going to try one that is a factor, right? Now, I picked one that was really large and the factors didn't start off with the first numbers down here, right? And that becomes a real pain on a test or an exam if you end up writing because you're going to have to try these smaller guys first before you get to a possible factor of the bigger guys, right? Now, in general, if you're writing an exam, if they give you questions like this, they're going to give you ones where the possible factors are the first couple of terms and that reduces down to a level where, you know, you can use the other factoring techniques, okay? So keep this in mind. They're not all going to be, you know, crazy things like this, okay? So let's go ahead and do the synthetic long division so you know how it works out. And then we'll go ahead and actually do the factors and then reduce it down from there. So the two comes down. You multiply by the one. One times two is two. And you add these guys, you get four. Four, one times four. That goes up there and you add those guys and whatever the result is comes down here. And then from here, you go across, come down again, go across, come down again. So I'm just going to go ahead and do all those, okay? One multiplies four, four comes up here. Out of them together, you get negative 25. Negative 25 goes up there, multiplies by one. It's just negative 25. Negative 25 plus negative 5 is negative 30. Negative 30 times one is negative 30. Out of them together, you get 30. So the remainder is not zero. So we know X minus one is not a factor of the top polynomial. What this does mean is if you put in X is equal to one in the polynomial, right? So it becomes two, one to the power of four, plus two, one to the power of three. If you do all this, your H of one, your Y when X is equal to one is going to be 30. So this is really a coordinate that we just found for the polynomial. So what we just found out was H of one is equal to 30. When X is one, the Y is 30. It's a coordinate on this graph. It's not what we're looking for. We're looking for factors of this guy, right? So what we're going to do, what we have to do now is go back to the beginning, go back to these guys again, the coefficients, write them out again, right? And then try another possible factor of 60. And we're going to do that right down here, okay? So we're just going to move down and try another number to see if it goes into this, okay? Now one thing you should be careful of is make sure you copy down the coefficients correctly when you transform over, okay? Some people end up making a mistake here. And, you know, again, always remember the sign in front of the number always goes with the number. The other thing you should be careful of is don't draw your line there. Make sure you leave enough room for your workspace, okay? So it comes down and just leave enough room to be able to transfer the numbers when you multiply them up and, you know, add them together. Now what we could do is try one of the other factors of 60 and, you know, see if that works out. So we could try x is equal to negative one. We could try x is equal to, you know, positive two, negative two, those don't work out. And then continue from there and see which ones work out. Just sequentially eliminating the ones that don't. And as soon as we hit one, our life becomes easier because all of a sudden, this polynomial is no longer x to the power of four. The bottom answer is x to the power of three. So we've reduced it down one level, right? Now, we're not going to sequentially go, you know, start from the bottom and start eliminating the ones that aren't factors of this because I set up this problem and I already know what the factors of this are. So what we're going to do is do x is equal to three. So x minus three being a possible factor of this guy. We know that one's going to work out. So we're just going to do that one and then reduce it down one level and then use synthetic division for this guy. So the two comes straight down. Two times three is going to be six. Six comes up here. Add these two guys. Two plus six is going to be eight and then so on and so forth. We got down to here. Two comes down. Six, six plus two is eight. Eight times three is going to be 24. So 24 goes up here. My negative 29 plus 24 is going to be negative five. So we got negative 29 plus 24 is going to be negative five. Negative five times three is going to be negative 50. Negative 15 plus negative five is going to be negative 20. Negative 20 times three is going to be negative 60. Negative 60 plus 60 is going to be zero. This guy is a factor of our original polynomial. So x minus three is a factor of the original polynomial. And our original polynomial started off with x to the power of four. So we just divided out an x term from a polynomial that's x to the power of four. That means our answer down here is now x to the power of three. So we can write that out. You know, put in the variables, put in the x's just so you see what it looks like now. So our polynomial would be two x cubed, eight x to the power of two minus five x minus 20. So x minus three times this guy would give us our original polynomial. And what is x minus three being a factor of me? It means when x is equal to three in the original polynomial, the y is zero. So it crosses the x-axis as x equals three. So on the Cartesian coordinate system, we just found one point where the polynomial, our original polynomial crosses the x-axis. Now what we've done is, you know, reduced our quotient. This is basically our quotient, right? That's our divisor. That's our dividend. This is our quotient. And that's our remainder, right? We talked a lot about this before. And what we've found right now, what we've reduced it down to is a quotient that's to the power of three. We can't use any of the other factoring techniques to continue to factor this, right? So what we're going to have to do is use synthetic division on this guy as well. To kick it down one level, which brings us down to, you know, if we're able to. You know, sometimes you might not be able to divide this out, right? I mean, maybe this doesn't factor, right? That means from here on, it doesn't cross the x-axis, right? It would mean that this polynomial or original polynomial only had one x-intercept if we can't factor this, right? But I know we can factor this because I set up the problem. So what we're going to do is factor this guy using the same technique, synthetic division, to be able to take out one more x from this. That means we would end up with x to the power of two. That means we can go, you know, back to some more simpler factoring techniques. And, you know, continue to factor it and find all the x-intercepts. So if you're going to think about it in, you know, polynomial terminology, it's just going to be h of three is equal to zero. And we're going to start using a lot, you know, terminology like this a lot more later on when we start analyzing functions, right? So this basically means when x is three, y is zero. It's, you know, a coordinate, which is basically three and zero. Three and zero is a coordinate of the original function. We're the original function characters. So there's multiple ways, multiple ways to express things in mathematics. It's just like there are multiple words that mean the same thing in different languages, right? There's multiple ways to say the same thing in the language of mathematics, okay? So what we're going to do is copy these coefficients down right now on another wall and use the possible factors of negative 20 divided by possible factors of two and, you know, we're not going to need to go into fractions right now. We will do one later on where it is fractions where we're going to have to, you know, multiply out and add out fractions to be able to solve for it. But right now, you know, most likely, most likely across your fingers, I know this, it's going to be a possible factor of 20, okay? So let's lay out the coefficients and, you know, continue synthetic division and see where we end up.