 All right. Welcome back. Today, what we're going to be doing in this video, we're going to talk about how to use factoring to solve polynomial equations. At this level of mathematics, you have solved linear equations. You've solved quadratic equations. You've maybe even solved rational equations. Your fractional equations, all that kind of stuff. Now what we're going to do is we're going to solve polynomial equations. Equations to your higher powers here. To do this, we need to factor. Equations are very basic process that we've been using for a while. So that's what we're going to use, that very basic process to help us to find the different factors of these polynomial equations and to help us solve. So I've got two separate examples here to go through. Here we go. First thing we've got to do is whenever you look at an equation like this, you want to try to figure out if anything is in common. Looking at this equation, I see a couple of things in common. I see a 4. I see a 4. I see a negative 24. So what that tells me is all those numbers are divisible by 4. So that actually tells me that I'm going to factor out a 4 first. And also I see all these x's here. I see that every single one of them has at least x to the fourth. And so what I'm going to do is I'm first going to factor out that common factor. I'm going to factor out a 4 and factor out an x to the fourth. So that's going to leave me x squared plus x minus 6 equals 0. There we go. Now again, when we're factoring, remember that we're dividing this. We're taking each one of these three terms divided by 4x to the fourth. So again, 4 divided by 4 is 1. 6x divided by x to the fourth is x squared. And so on and so forth for the rest of them. All right, now after we factor once, we need to continue to factor if we can. Now if you notice inside here, this trinomial right here can be factored. So what I'm going to do is I'm going to factor using two parentheses. I'm going to put x's in the front of my parentheses. And the numbers that multiply to negative 6 and add to a positive 1 are going to be plus 3 and minus 2. So that right there is the correct factorization. Now what that does, that just gives me multiple parentheses to use. Because when we're solving polynomial equations, you can well imagine there's going to be a lot of x's that we can put in here to get 0. And as we see here, there's at least three of them. There's 1x here, there's an x right here, and then there's an x right here. These three factors are going to tell us what x can equal. Anyway, now after this step, once we've factored everything down, as far as we can go, there's no more factoring I can do. My next step is to take each one of these factors, each one of these, and set them equal to 0. So 4x to the fourth equals 0, x plus 3 equals 0, and x minus 2 equals 0. Now with those three, set them all equal to 0, and now take all of those and solve them. So in this case, this one is actually easy to solve. The only number that's going to work in here to get 0 is 0. Subtract 3 over to the other side, x is equal to negative 3. Over here, add 2 over to the other side, x is equal to 2. So when I'm trying to solve these polynomial equations, these are my three answers for this one. So these are the three x's that will work. If I plug in those three x's up into that equation, every single last one of them should be 0. Now, I don't have time to check them, but you can just take these numbers, plug them in, if you want to double check your answer. This 0, that one's easy to check, because 0, 0, and 0 is equal to 0. That one's easy to check. These ones you might have to plug into a calculator, or maybe use a really quick synthetic substitution to check to see if you did that correctly. All right, moving on to the next one. Now, if you ever want to solve a polynomial equation, if we look at this next one, this one's kind of out of order. What we want to do is we want everything to be set equal to 0 like the first one. So the first thing I'm going to do is I'm going to take this 26x squared, subtract it over to the other side so that I have everything on one side. So I have everything on one side. Okay, now, this is going to be a little bit different factoring than the first one, that's why I chose this example. Now, for this type of factoring, we have x to the fourth, x squared, and 25. This is actually going to be, it's kind of commonly called quadratic graphing. All right, quadratic graphing. Quadratic factoring is because it usually only works with quadratics, but it does work with your higher powers. In this case, this factoring is actually going to be the exact same factoring as what I did right here over here for this one, okay? When I factored and I put the x's in the front, then I found numbers that multiply to negative six and add it to one. Actually, it's going to be the exact same thing over here. And now, notice that it only works if it's x to the fourth and x squared. Okay, notice the jump, notice the jump there. We don't have any x to the thirds. We don't have any x's, notice the jumps there, okay? Notice over here, okay, x squared and then x and then six, and there's really no jumps, but it just kind of goes down two to one to zero. This one is four to zero, okay? So there's a little bit of similarity there, but again, it can only work if we have this type of factoring, or if we have these type of exponents over here where it's four, two, and zero, okay? So now, let me factor it and show you, okay? So in this case, instead of just x's being at the front, there are going to be x squareds at the front, because again, I need numbers, I need x's that multiply x squared times x squared that gives me back to x to the fourth, okay? And then I need numbers that multiply to a positive 25 and add to a negative 26. Well, it's going to be negative 25 and negative one, okay? Those numbers multiply to a positive 25 and add to a negative 26, okay? Now, you can very quickly, if you really want to, you could actually foil this and get back to where you originally were. That's a good way to check to see if you did things correctly, okay? All right, now that is the correct factorization. Now, one thing you also got to think is that, if I have these x squareds, I might be able to factor those one more time. And in fact, I can. In fact, these are, this is a difference of two squares type of factoring. So this one's going to be x minus five, x plus five. And this one is the same deal. This one is a difference of two squares. Now, one, sometimes you don't see that as a square number, but one is a square number. Just like five times five gives me 25. One times one is gonna give me one. So this is gonna be x minus one, x plus one, okay? Equals zero. All right, so there's the different levels affecting, lots of different types of factoring that we did for this one. And then just like the last problem, what we're going to do is we're gonna take every single one of these parentheses and set them equal to zero. Now, here's a little step. Here's a little shortcut here. If you have these very simple, if you have these very simple parentheses where it's just x plus five, x minus five, x minus one, x plus one, just do very simple ones with just x's. The trick here is, is that all of your solutions are just gonna be the opposite of the numbers that you have inside the parentheses. You may have already caught onto that with your other lessons. So in this case, I have x equals five, x equals negative five, x equals one, and x equals negative one, okay? Those are my four solutions that I have for this equation, for this one up here, okay? Those are the four numbers that'll work. In this case, for the first example, I only had three numbers that worked. For the second example, I had four numbers work. With your polynomial equations, with your equations with these higher powers up here, you're gonna get lots of different answers. Sometimes you'll have two or three or four, five or six answers. I mean, there's a lot of different ones that you can have. All right, so anyway, that's factoring to solve polynomial equations. Hopefully, these two examples will help you out. All right, thank you for watching, and we'll see you next time.