 Summarizing what we learned about with the fundamental theorem of algebra, every polynomial function with real coefficients can be uniquely factored over the real numbers into a product of linears and irreducible quadratic factors where those quadratic factors will be ones which have non real roots and their roots will come in conjugate pairs. So if we wanna factor polynomials over the real number system, we're gonna get a bunch of linears when you have a real root and you'll get quadratics when you have non real roots. Now, if you wanna factored over the complex number system, the fundamental theorem of algebra tells us that we can always factored into linear factors where you might have non real roots. Let's try that right here. Let's factor the polynomial f of x equals three x to the fourth plus five x cubed plus 25x squared plus 45x minus 18. I'm gonna first look for the rational roots and so what are some things I can see here? So notice there's one variation of signs. So that tells me there's gonna be one positive root. If we look for the negative variations, it's gonna be three or one. So we're gonna have either one real, we're gonna have definitely one positive root and we either have one negative root or three negative roots. Now, the reason why we subtract two is why not two? Why can't we have two negative roots? It's because if we're missing a real root, then we're actually missing two real roots because a non real root must come with its conjugate pair. And so if you're missing one real root, you're actually missing two real roots. So you would have to have three or one negative real roots. Okay, so let's, we could look for the one positive or we could try to look for the negatives, whichever you prefer. In terms of the rational roots theorem, the rational roots theorem tells us that the potential rational roots are gonna be factors of 18 divided by factors of three. So that's gonna be plus or minus one, plus or minus two, plus or minus three, plus or minus six, plus or minus nine, plus or minus 18. Ooh, those are just the factors of 18 that then also divide those by three. Those which are divisible by, already divide three won't get anything new. So we're gonna get plus or minus one third, plus or minus two thirds, three's divisible by three, so is six, nine and 18. So those are, that's our list. That's a pretty big list, yikes. And since the leading coefficient is three, we should anticipate there's gonna be some non integer root here. It could be a fraction, but it also could be something non real. It could be like lots of, it could be a irrational number, it could be imaginary number, we don't know. So we're gonna have to look for them. So what happens if we just try one? I like to start with one. And so if we do one right here, we're gonna get the following. We're gonna get, we're starting the coefficients, three, five, 25, 45 and negative 18. If we try the number one, bring down to three. Three times one is three, plus five is eight, times one is eight, plus 25 is 33, times one is 33, plus 45 is 78, times one is 78, minus 18 is 60. You'll notice that every number along the bottom was positive. And so this suggests to us that we are too big. The upper bound theorem applies in this situation. So what I'm gonna do is I'm gonna erase these numbers. And I'm gonna take out basically all these numbers bigger than one. So one didn't work, neither will two, neither will three, neither will six, neither will nine, neither will 18. None of those positive numbers are gonna work. Now there is one third and two thirds. Those are smaller than one. They might still work. But if you're afraid of fractions, then you're not gonna wanna try them. So it's like, hey, JK, I was looking for a positive. Let's try negative numbers. Let's see what happens there. If we tried say like negative three, what do you see there? Bring down the three, three times negative three is negative nine plus five is negative four times negative three is 12, plus 25 is 37, times negative three is negative 111, plus 45 is negative 66, times negative three is 198, and then subtract 18, you get 180. So you see in this situation, it's like, oh, this is alternating signs, positive, negative, positive, negative, positive, like so. So this is actually too small. The lower bound theorem applies to this situation. So that would tell us that yikes, I can't use anything less than negative three. So let's erase that. So even though we had failed attempts here, we learned a lot of information with using the lower and upper bounds theorems here. So everything less than negative three fails. So there's no negative three, no negative six, no negative nine, no negative 18. You don't really wanna try the really big numbers because they oftentimes aren't gonna work. So this is such a mess right here. We have a negative one, a negative two, plus or minus one third and a plus or minus two thirds, all right. So let's try something else. If you still are avoiding the fractions, we might try like negative two. So bring down the three, three times negative two is a negative six, plus five is negative one, times negative two is two, plus 25 is 27, times negative two is negative 54, plus 45, that's gonna be a negative nine, times negative two is a positive 18, when you found a root finally. I can understand that this process can get a little bit frustrating at times because there's a lot of guessing and checking, but if you use these techniques, you can speed through the process much, much faster. So let's look at the factorization we've now found. F of x can factor as x plus two because we know two negative two is a root. We also have three x cubed, drop the power by one, negative x squared, plus 27x minus nine. That's the next term right there. And so we have that, that's great. Let's reevaluate this thing right here. What can we say? Well, in terms of rational roots, we would have the numbers plus or minus one, plus or minus three, plus or minus nine, plus or minus one third, but many of those numbers we already know won't work. Anything, we know that one, three, nine don't work. We also know that negative three and negative nine don't work. So basically we're left with negative one and one, plus or minus one third. So if we summarize our list here, at the moment we have negative one and plus or minus one third is what remains, okay? Also look at the variation of signs here. This tells us that we vary once, right? So there's one variation of signs right there. Then there's a second variation of signs and then there's a third variation of signs. So this thing varies three times, that's interesting. And if you look at the negative variation, so the variation, since it varies three times, that tells us we have three or one real roots. If you look at the negative variation, you're gonna see a negative, a negative, negative. There's no negative variation. The negative variation means zero means that negative one's not gonna work and negative one third is not gonna work. And so our rational roots, the only one left to try is one third. That's really kind of interesting here. In which case, but also look what we had earlier. So right now we're seeing that the variation, we had to have three or one real roots. But if we look at what we saw before, we only have one positive root, right? Positive real root. And so that actually tells us that it's not gonna be three, it's gonna be one, right? And so that's, we already know, I already know at this point that I'm gonna have two non real roots. How are we gonna find them? Well, if we go forward with what we said, right? It looks like negative one third was also out. I need to get rid of that. So one third is what we have to do. We were avoiding it, but it's kind of unavoidable at this moment. Three, negative one, 27, negative nine. If you tried one third using synthetic division, you're gonna end up with, in this situation, bring down the three, one third times three is a one minus one is zero times a third is zero. So we get a 27. 27 times a third is nine, negative nine plus nine is zero. So just like we predicted, we got that negative one third was a root. So we see that f of x equals x plus two times x minus a third. And then you're gonna get three x squared plus 27, but notice three and 27 are both, there's a little bit three. You could factor out a three right here and this would leave behind an x squared plus nine like so. But hey, you could redistribute this three right here. And so we actually could write this as three x minus one. And so you get a factorization just using integers. I also wanna point out that alternatively we could have factored this by groups. So notice you could have done three x cubed minus x squared and then you have a 27x minus nine. The first group you take out an x squared leaving behind three x minus one. And then the second group you can take out a nine leaving three x minus one. And so then you get the same factorization of three x minus one and x squared plus nine. If you can use the more elementary techniques like factored by groups, I would recommend doing that instead of synthetic division. But if you proceeded with synthetic division, you would have found the same factorization as the same. And so in terms of real polynomials, this is actually the factorization x plus two, three x minus one and x squared plus nine. This polynomial does not factor with real numbers any farther but we can't factor it using complex numbers. It would factor instead as x minus three i and x plus three i. They come in conjugate pairs right here. So this is the complex factorization. And how do I know that? Where did the three i and negative three i come from? Well, the idea is the factor theorem says the roots of the factors go hand in hand. So if you had the equation x squared plus nine equals zero you could solve this by the quadratic formula. But we can also just, we could treat it as difference of squares, right? That's one way of doing it. x squared, you're gonna have, because you could treat it as x squared minus a negative nine. And in which case then this would factor as x minus the square root of negative nine and x plus the square root of negative nine, hence three i. That's one way of doing it. Like I said, you could use the quadratic formula or if you just wanna solve this equation, you get x squared equals negative nine, which it says that x equals plus or minus the square root of negative nine, which will give us plus or minus three i. And if the roots are plus or minus three i, that means the factors are gonna be x minus three i and x minus a negative three i, which is what we get right here. So finding the roots is the same thing as finding the factors. And so we found the roots of this polynomial are gonna be x equals negative two, one third and three i and its conjugate negative three i. So we factored this polynomial and found its roots.