 stay one more time, that's the one more time. Hi everyone, this is Chichou, and I'd like to welcome you to Series 3B of the Language of Mathematics. Now, this series is a continuation of what we did last year in Series 3A, and since it's a continuation that I'd like to do a little recap of what we did last year, which was basically we started off, it was supposed to be a series about the equal sign and units, and we started off fine. I started talking about the equal sign and the different symbols that have branched off the equal sign, and what the equal sign means, and basically it's just equivalence, trying to compare one thing to another thing, seeing if this side of the scale equals this side of the scale. And again, there's other symbols that branch off from there. Some of them are inequality symbols, saying that one side of the equation is larger than the other side of the equation, or one side is smaller than the other side of the equation. There's congruency when you come to geometry trigonometry. You can have the element of number theory, set theory, basically, where you can say these numbers belong to this set, and we talked about some of the stuff in Series 1 as well with the real number set. When you say one, two, a whole number is going up, or a whole number is zero, one, two, three, all the way up to infinity, they would belong to a whole number set. They're the element of the whole number set. So I sort of touched on the equal sign at the beginning when I made a couple of videos about comparing two things, and I took it to the extremes to try to make a point. I went to the smallest thing that we've been able to measure in the real world, and the largest thing. Some would say the largest thing that we've been able to measure in the real world, and that's black holes and elementary particles. And I put a couple of videos together just at the beginning to try to emphasize the point that when you break it down, when you're talking about two things, you can always compare two things. A lot of teachers say when they're initially trying to introduce units to people, and I've done this myself too, is they always say you can't add apples and oranges, right? If you go apple plus an apple, or two apples plus two apples is four apples, right? And your units there would be apples. And they say, trying to make sure that students don't add the wrong units together, they say you can't add apples and oranges together, right? And I've done this myself and guilty of it myself. However, that's not really true. You can't add apples and oranges together as long as you break them down to a specific unit, right? So two apples plus two apples is four apples. Two apples plus two oranges is four pieces of fruit. So what you're doing there is finding a common unit between two things that you're comparing. And that's where the importance of units comes in. That's where the importance of how it relates to the equal sign comes in. So there was a little bit of talk about that at the beginning of Series 3A. So we sort of messed around with that a little bit and brought it down to the point that if you're talking about something, it's really important to know what property of that thing it is that you're talking about, which is units. Now from there, I was going to expand our units and talk more about the equal sign units. But what I did is sort of take a tangent, I went off on a tangent, and that occurred when we started talking about the equal sign and how to move around the equal sign, right? How you can take numbers and move them over or solve for equations or solve for variables. And we dealt with addition and subtraction, how you move around and have addition and subtraction. Talked about how you move around with the multiplication division, exponents and radicals, which is basically, if you're trying to get rid of something on one side of the equation, you do the opposite to it. And if you do something on one side of the equation, you've got to do the other side of the equation. That's what the equal sign is, right? It's a scale, it's a balance. If you try to keep things balanced, if you put something here, you've got to put something here. If you remove something here, you've got to remove something here, right? And that's how you move around the equal sign. We also talked about one tool I like to use, and I teach this to a lot of my students, is cross multiplication. I know most people have encountered hate fractions. So what I like to do is get rid of fractions, initially anyway, with simple equations, get rid of fractions right away. And if you have one fraction equal to another fraction, all you do is take the denominator here, the denominator here, and cross multiply up, right? Across the equal sign, and that gets rid of fractions. And we talked about that a little bit. And, you know, sort of solve equations. And we solved some simple equations. From there, we kicked it into quadratic functions, quadratic equations, which are basically anything in the form of A x squared plus B x plus C. And these are just polynomials, quadratic equations are polynomials of degree two. And to solve quadratic equations, what we talked about was the different factoring techniques that we have to solve quadratic equations, which are basically four manual one using a formula. And four manual are the graze common factor GCF, simple trinomial factoring, the difference of squares and complex trinomial factoring. And the fifth one that we use a formula for, when we can't easily, you know, factor things out, which is a quadratic formula. Now out of those five, we've already talked about GCF, and GCF you use anywhere for any type of polynomial equation, or non-polynomial equations. It's basically the greatest common factor. Anything similar between them, you can take them out and put them in front, right? So we talked about GCF, we talked about the simple trinomial factoring. And what we still have to talk about is the difference of squares, complex trinomial factoring, and the quadratic formula. We're also going to talk about another form of factoring that we use to factor larger degree polynomials, larger than quadratic function, larger than degree two, which is synthetic division. And that is used for anything larger than degree two. And we're going to talk about that, right? So what we ended up doing was, you know, talk about quadratic equation specifically, and some other some higher degree functions as well, where, you know, talk about what it means when we're factoring a quadratic equation or when we're factoring a polynomial equation, which is basically factors of anything, factors of polynomial equation or quadratic equation, are the x-intercepts. And there's multiple terms that mean the same thing when we're talking about factors, right? When we factor a quadratic equation, what we're getting, the solutions are basically the x-intercepts, where the function crosses the x-axis. And you also refer to that as, you know, the zeros, the solutions, the roots, the factors, and, you know, the x-intercepts. So whenever you're factoring for something, when you get x equals something, you know, multiple things or, you know, x only equals, you know, one number, what you're really getting is solving for, what the solution gives you is where the function crosses the x-axis. And we talked a lot about this. Basically, we're exploiting the property where if you have multiple things, multiple things together to give you zero, the way you solve for any type of equation like that is take each, you know, what you do is exploit the property of zero where the only way you can have multiple things multiplied together to give you zero is if at least one of them is equal to zero. And since you don't know which one is equal to zero, you solve for all of them equaling to zero, right? And sometimes all of them give you solutions, sometimes none of them give you solutions, sometimes you get, you know, some give you solutions, some don't give you solutions, right? And we talked a lot about this stuff. One of the things we did when we're solving equations, we talked about, you know, some of the things that you cannot do when you solve an equation, which is eliminating solutions, dividing by zero, dividing by zero basically gives you asymptotes and polynomial equations and that checking solutions, right? So checking solutions is quite important. I, you know, initially when I started teaching these techniques to my students, I get them to check their solutions because I don't want people to take for granted that the answer they get at the end really solves the equation. It's a valid solution. Sometimes, you know, you might have done everything correctly, but once you substitute it back into the polynomial equation, the solution doesn't make sense. So you have to eliminate the solution, right? So checking solutions, checking answers is important. And, you know, we did a fair bit of different types of equations. Some polynomial, some not polynomial. We did a fair bit of polynomial. Some of them were not polynomial equations. And we'll touch a little bit further on those non-polynomials. So from, you know, factoring and graphing quadratic equations, you know, we've got our hands really dirty with this stuff, right? And from there, at the end of series 3A, what we ended up doing, what I did was take a step back and went, you know, in depth into the terminology of what a polynomial equation is. And, you know, I put up three videos towards the end talking about polynomial equations, talking about, you know, what they can give you and, you know, what the x-intercepts means. And we sort of did a summary and defined a lot of terms and explained what it is that we're talking about when we say polynomial equations, right? And that's where we left off series 3A. In this series, in series 3B, we're gonna wrap up the factoring techniques, talk about the remaining four techniques, right? Which are, what's left is the difference of squares, complex trinomial factoring, quadratic formulas, effect division. So we're gonna start solving more quadratic functions and more polynomial functions. And we're gonna talk a lot more about polynomial functions in general and start graphing polynomial functions and, you know, maybe see where we can apply them and some of the polynomial functions or quadratic functions that we have in the real world, right? That's where we are. That's where we left off series 3A. This is where we're going with series 3B. I'm also going to do, you know, put in as the additional stuff in there as well. Wherever I think, you know, something is needed. If we need to start discussing something else, specifically related to polynomial equations, we will start talking about it, okay? So this was just a quick little recap. Just give you a, you know, rapid explanation of what we did last year. And I forget how many, you know, how many videos there are. It's two to three hours worth of material on there. Four hours maybe. Worth of material on there that we're sort of recapping. And, you know, if you're not familiar with that series, you should take a look at it because what I'm going to do is probably just jump, you know, continue that right away. So I'm not going to do too much recap and the video's coming up. I will do a little bit just to, you know, just to remind everyone what it is, what's going on and what we're talking about. So, you know, people aren't lost because there has been, you know, a few months gap a year basically since we, I did the last video. And that's about it. That's what we're going to do for series 3B. And one thing that has changed for this year is for the previous three years, what I did was, you know, I had summers off. So what I would do is do as many videos as I could for the summer before my work kicked in in the fall. And in the fall I would take a break. One reason I did that is because I had work coming up in the fall. And the other reason was that the weather wasn't nice so I would have to go outside and do my videos, you know, on walls outside, right? That's changed. My schedule has changed because math, these videos, this is the project I'm going to be focused on for the foreseeable future. So I'm going to continue series 3B until I'm satisfied that we can, you know, end it and then start the next series. And the next series I'm not sure if it's going to start right away after the series is done or not. I might take a little break but back at work to hopefully get the websites up. I'm still going to do that on the side. That's one of the things I'm going to do while I make these videos. But so this thing is not going to end, you know, in a couple of months or whatnot. I'm going to continue this. I'll probably shoot into the winter. So it's going to be, I'm going to go into the cold. So you're going to see smoke coming out of my mouth. I'm going to be dressing a lot warmer. And one thing I'm going to do, you know, as soon as the weather starts getting a little colder is I do miss my beard. So I'm going back bearded style. So hopefully for this series, you're going to see a beard slowly growing on me. Okay. Anyway, that's it. I hope that reminds people if you followed it from series 3B, I hope that reminds you of what we did. And I hope you're looking forward to this year because I am seriously looking forward to it. And I got a few lessons already organized to shoot. And we'll take it from there. We'll see where we go. And welcome back. Glad to be back again. And I guess I'll see you guys in the next videos.