Welcome to episode 10 of Math 1050, College Algebra. I'm Dennis Allison, and I teach in the math department here at UVSC. I have a bit of a cold today, so I was a tenor. I may be a baripuntone today, and tomorrow, if I'm a base, we're in big trouble. Let's look at the objectives for the course today. We want to look at, first of all, what's called the division algorithm. Then we'll talk about long division and a shortcut called synthetic division. Actually, those, I think, were covered in intermediate algebra, so that's probably just a review for you. Then we'll look at the remainder theorem and the factor theorem, and finally, we'll talk about constructing a polynomial from its roots. Now, you might say, what's the purpose of all these topics? Where are we going with this? Well, if you remember in the last episode, just at the end of the episode, we were discussing how you graph a polynomial if it's factored using its x-intercepts. Let me just give an example.  Now, what I'm doing here in this example is I'm purposely putting in a factor with multiplicity 1, a factor with multiplicity 2, because there are actually two factors of x plus 2, and a factor of multiplicity 3. And I want to demonstrate what is the effect of those powers, those exponents, on the graph. Now, the first thing I would do would be to locate the x intercepts. And if you remember, the x intercepts are the places where the function value, p of x is 0. Well, now, p of x is 0 whenever one of these factors is 0. So that could be when x is negative 2, when x is 1, or when x is 4. So the x intercepts would be, and I'll put them over here, x equals negative 2, x equals 1, and x equals 4. Now, this helps me draw the graph, because if I locate those points on the x-axis, I'll just put a heavy dot on each one of those points. So I'll put a heavy dot here at 4, I'll put a heavy dot here at 1, and a heavy dot here at negative 2. Now, because when I multiply this out, I think I'm going to get x to the sixth power, it looks like. Because you see, I'm going to get an x squared there. I have an x here. And when I cube this factor, I'm going to get an x cube. So when I multiply those lead terms, x squared, x, and x cubed, this is going to end up being a positive x to the sixth plus, and then a whole bunch of other terms. Now, what's significant about this is if I have a polynomial of even degree, then you remember both sides of the graph, the in behavior, has it either both sides going up or both sides going down. In this case, do you think both sides will go up, or will both sides go down, based on that lead term? Well, if the coefficient's positive, both sides go up. If the coefficient's negative, the graph has been inverted and both sides go down. So I'm expecting to see a graph that goes up over here, and when I finish, I should have the graph going up over there too. OK, well, let's see. If it's going up on that side, then if I go backwards, if I come down from up above, I guess the question is, how does it approach this x-intercept? And if you remember, in the last episode, we said that what the way we determined the behavior at the x-intercept is we look at the multiplicity of that factor, and the multiplicity there is 3. And therefore, it behaves something like a cubic function. That is, it comes in, it turns, levels off, and then it goes down again. So I get this little bit of a ripple in the graph. Now, we don't know how far it goes down, because we're not making a graph that's that accurate. I just am looking for the general shape, but I know I have an appointment to come back and touch the x-axis when I get here to 1. Now, what happens at 1? Will it have the same ripple effect there when I get to 1? Well, what I do is I go to the factor that produced the 1, and that factor has multiplicity 1. And therefore, it passes through just about like a straight line. So when my curve's going to come up and it's just going to pass right on through, there's nothing special that happens in the behavior there.  But eventually, I have to come back to the negative 2. So I'm going to turn right about here and come back to the negative 2. Now, what happens at negative 2? Well, let's see. If I look at the factor where the negative 2 came from over here, what is its multiplicity? Multiplicity 2. And Steven, do you remember what happens when the multiplicity is 2? What happens at that x-intercept? It touches the x-axis and turns back. Right. It just turns around sort of like a parabola, sort of a little mini parabola right there. But actually, in this case, it turns very quickly. It turns and it goes back up. And you notice we have both sides of the graph going up just like we predicted in the beginning. One thing to point out is when you draw these graphs, you don't want to make them bounce in like a bouncing ball.  OK, well we've talked about this idea in the last episode. But this raises the question, what do you do if the polynomial hasn't been factored? Now this is a sixth degree polynomial. What if this had been multiplied out and we had not known what the factors were? It seems like we'd really be stuck in that case. For example, what if we were wanting to graph this polynomial function, and in fact in just a little bit we will, what if I graph p of x equals 2x cube plus 7x squared plus 4x minus 4? Now you see this time it's a whole new ball game because we don't know what the x-intercepts are because this hasn't been factored. And generally we don't have experience factoring cubic polynomials with the one exception of factoring by grouping. And occasionally factoring by grouping will help. But factoring by grouping won't allow us to factor this one. If you remember in factoring by grouping what you do is you pair up the terms two and two. Sometimes you group them in threes and ones, and you try to factor this, you try to factor this, and then overall you look for a common factor between the two. That will sometimes allow us to factor cubic polynomials, but I don't think it would in this case. So we're stuck in the very beginning because we don't know how to factor that. Well the lesson today, and the lesson that follows this, episode 11, will be about information for factoring higher degree polynomials. I remember when I was a student back in high school and so forth, I thought you could only factor quadratics. And I thought that it was just sort of serendipity whether you could factor higher degree polynomials. But in fact there's a whole theory for this. It's not very complicated. And that's what we'll be talking about today in an episode 11. OK, so to begin this, let me remind you of something that you've seen maybe back in middle school. It's called the division algorithm. And this is the division algorithm for arithmetic. And then we'll see what is the division algorithm for polynomials. Now the word algorithm means it's a procedure or a method. So if you solve, for example, a quadratic equation using the quadratic formula, that would be an algorithm for solving a quadratic equation. If you solve a quadratic equation by factoring, that would be a different algorithm for factoring quadratics. Well, there is an algorithm for division that goes like this. It says if you have two numbers, a and b. And let's say a is bigger than b. So let me put an inequality sign in there. Suppose a is bigger than b. Then I can write a as a multiple of b plus a remainder. I'll just put spaces in there to represent those numbers. So I can take a times a multiple of b and then there's some remainder left over. For example, what if I said 11 is equal to some multiple of 4 plus a remainder? This is multiplication right here. Well, I could put a 1 there, 1, 4. And then what would be the remainder in that case? Seven. Seven. Yeah, exactly. Another possibility is I could say 11 is equal to 2 times 4. But in that case, what would be the remainder? Three. Three. Now, this is the preferable way to express this relationship because I have a remainder that's smaller than the number 4. So rather than using the equality there, I'll use the equality here. This is referred to as the algorithm for dividing 4 into 11. Basically, 4 goes into 11 two times with a remainder of 3. We don't usually think of division as being, whoops, that should be a plus right there. We don't usually think of division in terms of writing products and sums. But the idea is I would like to find a number that is a coefficient of 4 so that my remainder is smaller than 4 but non-negative. Let's take one more example of that and then we'll look at this for polynomials. What if we take a number like 37 equals some multiple of 5 plus a remainder that goes right here? Well, I could put in a 1, a 2, a 3, a 4. But what's the biggest integer I could put in there so that I get a remainder that's either 0 or larger but smaller than 5? 7. Yeah, I think I could put a 7 here. 7 times 5 and the remainder would be what? 2. Would be 2. OK. The number that goes here is referred to as the quotient. And the number that goes here, just as I've been referring to it, is referred to as the remainder. And the way you normally write this is you say 5 is divided into 37 and it goes 7 times and then I subtract off 35 and I get a remainder of 2. And that remainder is normally expressed as a fraction 2 over 5. So what we do is just write that as 7 and 2 as 2 fifth. So this remainder comes up as the numerator of a fraction. OK, now back in middle school, you may have seen some expressions like this, but I want to now write this division algorithm for polynomials. And let's go to the graphic that's on the screen to express this. It says for two polynomials, p of x and d of x, I'm calling it d of x because that would be the one that I would want to divide by, for p of x and d of x, where d of x is not 0, p of x can be expressed uniquely in the form, p of x equals q of x times d of x plus r of x, where q and r are polynomials and either r is 0 or its degree is less than the degree of d of x. Now that looks pretty complicated. Let me just give you an example of what that refers to. Suppose I were to have the polynomial x squared plus 2. And I want to put a multiple of a divisor, let's say the divisor this time is x minus 1, plus a remainder. Now the polynomial that goes here in that graphic I call q of x, what do you think I called it q of x? Quotient. Because it's the quotient. Yeah, it's called the quotient. And the one that goes over here, the polynomial that goes over here I call r of x because it's the remainder. So the question is, can I find a polynomial that goes here so that my remainder will be a polynomial that's either 0 or it will have a degree smaller than my divisor? This is my divisor, by the way I call that one d of x in that relationship.  Well I think what I should put in here is x plus 1. Now what's the product of x plus 1 and x minus 1? x squared minus 1. x squared minus 1, right, because that's the difference in two squares. Let's just see what the remainder would be in that case. x squared plus 2 equals this product, well that product is x squared minus 1 plus my remainder. Now can you see what I should add on to this to get x squared plus 2? 3. 3, exactly. Now I don't get a remainder that's 0, but I do get a polynomial you might say whose degree is smaller than the divisor. Because this divisor has degree 1 and constant polynomials other than 0 are said to have degree 0. So this has degree 0, its degree is smaller than that divisor. OK, as a general rule in the division algorithm, if you have a polynomial p of x and if you choose a divisor polynomial d of x, like in this example I chose d of x to be x minus 1, then according to this algorithm there should be a unique polynomial that I can put here so that I get a unique remainder over here. And this remainder will be either 0 or have degrees smaller than d of x. So this polynomial I call q of x. This polynomial I call r of x. OK, now that's fairly abstract, but this leads us into what we call long division. So let me do an example of long division and see how we would actually divide polynomials and express them in the form of this division algorithm. Suppose I wanted to carry out this division, 4x cubed minus 2x plus 3 divided by x plus 1. Now I'm not sure that x plus 1 will divide that exactly. I may have some remainder, that is some rational expression I'll have to add on at the end, but I'd like to carry out that division. So the way I do it is I put x plus 1 outside the division sign, and inside I put my polynomial 4x cubed. I better leave a space for the second degree term. I'll move the minus 2x and the plus 3 over here. At the moment I don't have any x squared terms, but I might have some in the course of the division, so I have to allow for that. Now when I divide these two polynomials, what I do is I divide the lead term here into the lead term there. I divide x into 4x cubed, and x goes into 4x cubed, 4x squared times. I'll put that in the x squared column. Then I multiply 4x squared times x and 4x squared times 1, and I get 4x cubed plus 4x squared. Now let me ask you, do you think I should add these or should I subtract these? Subtract them. Yeah, I think we should subtract them, so I'm going to put parentheses on there and put a minus in front of that. When I subtract 4x cubed minus 4x cubed, of course I get a zero here, so I won't write down anything. And over here I'm subtracting zero minus 4x squared is minus 4x squared, since I'm subtracting that from a zero that we don't see. Then I bring down the next term, the minus 2x, and I'll divide x plus 1 into minus 4x squared minus 2x. Now the way I do that is I actually divide the two lead terms. I divide x into negative 4x squared. How many times will that go? David, what would you say? I'll divide x into negative 4x squared. I want to say four of it. Let's see now. Well, I want to divide x into negative 4x squared. I'll just come over here on the side. Sorry, just once. Sorry about that. Yeah, so what would that ratio be? Two to one or one to two? No, let's see. If I divide by x, it's going to be a negative 4x. That reduces to be negative 4x if I cancel that off. So right here I'll put a negative 4x. So when I divide this guy into that guy, what I'm thinking is negative 4x squared over an x. OK, now I multiply negative 4x times x plus 1. And I get negative 4x squared and I get negative 4x. And I'll subtract those off. You see my purpose in this is to always get these first terms to cancel off here. I got them to cancel off there. And as that cancels, then I can bring down the next term over here, and that forces me to move across the problem from left to right. Well, when I subtract negative 4x squared from a negative 4x squared, I get a zero. And if I take negative 2x and I subtract a negative 4x, let's see, negative 2x, I'm going to subtract a negative 4x. That's going to give me, who can say how much will that be? 2x. 2x. That'll be a 2x. And then I'll bring down the 3. OK, now here we go again. I want to divide x plus 1 into 2x plus 3. How many times will my divisor go into this expression? Twice. Two times. Right, very good. So I'll put a 2 here. When I multiply, I get a 2x plus 2. And you see, once again, I'm wanting those two lead terms to cancel off. And when I subtract here, I get a 1. That's my remainder. I have this quotient and I have this remainder. So the answer to this problem up here of what is the cubic polynomial divided by the linear polynomial, the answer is 4x squared minus 4x plus 2 plus. Now, the way I express the remainder is in the form of that rational expression, 1 over x plus 1, sort of like I did when I divided 5 into 37 a little while ago. If the remainder is 0, then I don't have to put a fraction there. I don't have to put a rational expression there. And I get a nice polynomial answer. And what that means is x plus 1 is a factor of 4x cubed minus 2x plus 3.  And this is one way of finding factors is to see if they will divide evenly. OK, I'm going to do one more division problem, long division problem. And then I want to show you a shortcut for this process. In the next case, suppose I want to divide x to the third power plus 3x squared minus 7x plus 1. And let's say I'm going to divide by x squared plus x minus 3. OK? What makes this problem a little different is I'm not dividing by such a simple divisor like I did before. What was it x plus 1? Now I'm dividing by a quadratic expression. But the process is still the same. So if I come down here to carry out my division, outside I put my divisor. And inside I put my numerator. And I don't need to leave any spaces because every term is represented. The cubic, the quadratic term, the linear term, and the constant term. OK, so when I do this division, I compare lead terms. x squared goes into x cubed. How many times? x. x times. I'm going to put an x over here in the x column. Now you might say, Dennis, is it OK if we put the x right up there in front? Well, yes it is. Where you place the x really doesn't matter. But to me, it seems reasonable to put the x's in the same column with all the other x's. So when I multiply, I get x cubed plus x squared minus 3x. And I have to subtract that off. And I've done this in such a way that my x cubed terms will cancel. We always want those to cancel, which allows me to move over and bring down the plus 1. When I subtract in the quadratic column, what will the difference be? 2x squared. 2x squared. And be careful. When you subtract here, what do you get? Negative 6x. Not negative 6x. Negative 4x. Negative 4x. Yeah, negative 4x. Because you see, we're subtracting a negative 3, so that really makes that a plus 3. So I have negative 7 plus 3 is negative 4. So I'll bring down the plus 1 now. And starting over again, I divide my divisor into 2x squared minus 4x plus 1. I think that goes into it two times. So I'll put a plus 2 over here. And I take two times my divisor.  And when I subtract, the quadratic terms cancel off. When I subtract here, what will I get? That is the negative 4x. And I'm subtracting off the 2x. Negative 6x. Negative 6x. You know, the reason I'm asking you these questions is because when people make mistakes in division, it's generally with the subtraction. And when I'm balancing my checkbook, if I make a mistake, it's usually with when I'm trying to subtract off a check balance, I make a mistake in subtraction. So I'm purposely asking you about those subtractions. And if you get them wrong, that's fine. Because I think some of the people at home are probably wondering the same thing, how we're getting those. So we want to sort of go over that slowly. When I subtract over here, what will I get? 7 plus 7. Yeah, plus 7, exactly, plus 7. And you see, I'm no longer able to divide my quadratic into this expression because this has a degree too small. So that's my remainder. And that tells me that the answer to this problem is x plus 2 plus a negative 6x plus 7 over the divisor x squared plus x minus 3. OK, now what does this have to do with that division algorithm that I wrote on the screen here a little while ago? Well, you see, what this tells me is that my p of x, which is the numerator, and my d of x, which is my divisor, can be expressed in this form. We said p of x was equal to q of x times the divisor d of x multiplied together plus the remainder, r of x. And can anyone refresh my memory? What was the restriction on the remainder? How was it related to the divisor? It has to have a lesser degree. It has to have a smaller degree than the divisor, or it's 0. You might say, well, if it's 0, it doesn't have a smaller degree. Well, if this is equal to 0, it doesn't have a degree officially. A constant other than 0 is said to have degree 0, but the polynomial r of x equals 0 technically has no degree to it. So if I put x cubed plus 3x squared minus 7x plus 1 here, I'll need to put a quotient times the divisor. A quotient times the divisor. Well, there's the divisor. It's x squared plus x minus 3 plus a remainder. Well, the quotient is? x plus 2. Exactly, x plus 2. That's what goes there. That's the quotient. That's what was up above my division when I carried out the division. And the remainder is what? It's that, remember, the negative 6x plus 7. That's what came out at the end of the division. So I'll put a negative 6x plus 7. And by golly, it does have degree smaller than the divisor. The divisor has degree 2. The remainder has degree 1. Now, if you multiply this out, add on the remainder. Collect all those terms. You will get the original polynomial. Now, if it had turned out that the remainder had been 0, then you see I could have just erased that portion of the problem. And that would tell me that this quadratic is a factor of the original cubic. It would also say that the quotient is a divisor of the cubic. But we didn't get a 0 there, so we don't have any factors here. OK, now, we would certainly like to shorten this painful process of long division. And let me now show you a shortcut that applies in many cases. It doesn't apply in every case, but we'll explain the idiosyncrasies of that. Suppose I wanted to multiply, let's take x to the fourth power minus 2x cubed plus x squared minus 3x plus 2. And I want to divide this by x minus 2. Now, the process I'm going to show you here is referred to as synthetic division. What I was just using is referred to as long division. Long division works all the time, regardless of whatever you're dividing by. Synthetic division only works if you're dividing by a linear polynomial. That is a first degree polynomial. And the coefficient of x has to be a 1. So this could be x minus 2. It could be x plus 3. It could be x minus 7. But it has to just be a single x and then plus or minus a constant. OK, I'm going to do the division on the left. And then we're going to see how we could shorten this. x minus 2. And here I'll put x to the fourth minus 2x cubed plus x squared minus 3x plus 2. OK, I'll have to write sort of small so we can squeeze this in. x goes into x to the fourth x cubed times. So I get x cubed minus x to the fourth minus 2x cubed. And when I subtract, what do I get? We get 0 because these two terms cancel. And these two terms cancel. We get nothing other than 0. So rather than bringing down one term, I'll take the opportunity to bring down two terms because I want to divide my divisor into a pair of terms over here. x goes into x squared x times. I'll put that over here. x times. When I multiply, I get x squared minus 2x. And when I subtract those off, who can tell me what will I get when I subtract those off? Negative x. I'll get a negative x right there. Exactly, negative x. Because that's negative 3x plus 2x is negative x plus 2. And x minus 2 goes into negative x plus 2. How many times? Negative 1. Negative 1 time. I'll put a negative 1 right there. And I get a negative x plus 2.  I get a remainder of 0. First thing that tells me is x minus 2 is a factor of this fourth degree polynomial. It's a factor of it. Now, that's going to be very useful information to be able to identify factors. But more importantly, I want to find a way of shortening all this expression.  So right here, I'm going to put a 1. Right here, I'm going to put negative 2. Right here, I'm going to put a 1. Right here, I'll put negative 3. And right there, I'll put a plus 2. I think I'll leave out the plus signs because if I see a 1 or a 2, I know it's a plus 1 or a plus 2. Let me move that over to kind of center it there. Because I'm thinking, you know, when I write these down, it's the coefficients that are important. And I can remember that this is the constant term, the linear term, the square, this is the cubic, this is the fourth power term. I'll also put a 1 over here. And I'll also put a 1 right here. And I'll put negative 2 there. And I'll put a 1 here and a negative 3. Here I'll put a 1. Here I'll put negative 2. You see what I'm doing is trying to leave out any extra symbols that I don't have to write down every time. Here I'll put a 1. I'll leave out the plus. Here I'll put a 1, negative 1. And here I'll leave out the plus. And up above, I'll put a 1 for the x cubed. Let's see to get it in the right column, I'll put it there. I'll put a 1 in place of the 1x. And you see, I know that's 1x because it's in the x's column and then negative 1 over here. So this is sort of a shorter version of what I just did. You know, furthermore, it seems to me that this 1, 1 times 1, that's a duplicate right there, so let's just leave that out. And over here, I knew that I was going to get a 1 below that 1 because I always get these first terms to cancel, so I'll leave that out. And right here, I always get those two to cancel, so I'll leave off the negative 1 because I know that's supposed to cancel off. How can I reduce it further?  What if I don't even bring down the 1? What if I just leave it off and I just look up there to find the number? Same thing here, the negative 3. If I look up there, there's the negative 3. I'll leave that out. And so it looks to me like what I can do is just press this all together. Oh, one more thing. I'm going to leave off the plus 1 right here because you remember we said this only works whenever you're dividing by x plus 2, x minus 3, whatever. There's always a 1 there. So now I'm going to push this all together and make it look very thin and it's going to look like this. I'm going to erase this bottom portion so I can squeeze this in. In synthetic division, I draw my division symbol that way and I put the coefficients in directly across the top. And outside, I put a negative 2. Now I begin by bringing the 1 straight down and I multiply here, negative 2 times 1 is negative 2. And I subtract and when I subtract those, I get a 0. If you remember, when we subtracted right here a moment ago, we got a 0 and I brought down the next two terms, that's the 0. The myth there was a 0 remainder. Now I multiply negative 2 times 0 and I get a 0. And when I subtract, I get a 1.  When I subtract, I get negative 1. And then finally, negative 2 times negative 2 is 2. And when I subtract, I get a 0. This is my remainder right here. And this is my quotient. You might say, how's that the quotient? Well, look up above, look up here. I was getting a 1, a 1, and a negative 1. And I could put a 0 to hold that space open. This is my quotient, but it's represented only with coefficients. So what this tells me is the answer to my division problem is, let's see, constants, linear, quadratic, cubic. That's a 1x cube. There aren't any quadratics. There's a linear term plus x. And there's a constant term, negative 1. And the remainder is 0, so I don't have to add on any term over there on the n. And I have found the quotient of these two polynomials using what's called synthetic division. Now, you don't have to remember why synthetic division works. I just merely want to give you an indication of how it works. Let's go to the next graphic, and we'll do a synthetic division problem that's on the screen there. This says that when you are using synthetic division, when dividing by x minus c, c is a constant, we can use synthetic division as in this next example. Now, let me just copy that down before we take that off the screen. We're dividing 2x cube plus 7x squared minus 4x minus 11. And that's divided by x minus 3. OK, let's try solving that example here on the green board. You notice, first of all, we don't make the regular division sign. I make the squared off arrangement. This is just sort of standard. You'll see this in your textbook. And then across the top row inside the box, I put the coefficients 2, 7, negative 4, negative 11. And my divisor goes out here. And my divisor is x minus 3, but I'm throwing the x away. In fact, I'm throwing the coefficient away. I put a negative 3. And I'm just going to make a note out here that I'm going to be subtracting when I get to these columns, just like I do in regular division. So the process begins this way. I'll always bring the first number straight down, whatever it is. That's a 2 there. And then negative 3 times 2 is negative 6. And I subtract here and get 13. And then negative 3 times 13 is negative 39. And I subtract here. Got to be kind of careful. I subtract here. So that's actually negative 4 plus 39 is 35. And then negative 3 times 35 is negative 105. And I subtract here. And that's where I put my remainder. So I'll put a little box around it to separate that number from the others. When I subtract, let's see, that's negative 11 plus 105 I think is 94. So what's the answer to the problem? Well, this is the remainder. This is the quotient. That says 2x squared plus 13x. Let me make that 2 a little better there. Plus 35 with a remainder of, I'll have to put this right below it. I'll say with a remainder of 94 over x minus 3. So this is much quicker, I think, than the long division process. And we'll do another example or two here in a minute. One thing to point out. You remember this 2 represented 2 cubed, and this 7 represented 7 squared. But when you look at the answer, I'm saying this 2 represents x squares. So you say, wait a minute. You had x cubes. Here you have x squares. How did that happen? Well, if this is the remainder and this is my quotient, I have to count from here constants, linear, quadratic. So those two don't represent the same degree polynomial. This is always going to be one less. Now, how could I make this even faster? I mean, this is fairly fast. Well, most people, if they make a mistake in synthetic division, it's with this subtraction here. Because you always make more mistakes with subtractions than with addition. So I would like to change this to addition. And I can do that if I merely change the sign of that number and make that a plus 3. So when I look in this factor, x minus 3, I throw away the x. I change the sign to a plus 3, and then I can add. Now, I'll just try this right below, 2, 7, negative 4, negative 11. And over here, I'm going to add only for convenience, because I figure I make fewer mistakes when I'm adding than when I'm subtracting. We start off the same way. I bring a 2 straight down. 3 times 2 is 6, and I add. I get 13.  Well, by golly, it's the very same result. What I got here and what I got here are the same. So the actual synthetic division algorithm has you change this sign so that you can add. Another way to look at it is if you look at the divisor, x minus 3, this polynomial, this linear polynomial, has a 0, and the 0 of that polynomial is 3. See, if you're plugging a 3, you'd make that 0. So it's the 0 that you write out here, and then you add. If you don't change the sign, you have to remember to subtract. And if you want to subtract, that's fine with me, but I think you'll make fewer careless errors. I know I will if I do the addition route. OK, let's take another synthetic division problem. Let's see. Let's suppose this time we have 3x to the fourth minus 5x cube plus x minus 11, and I'm going to divide by x plus 2. x plus 2. I have no idea what this answer is. I'm just making it up, but we can carry out the division nonetheless. So if I'm going to divide, I'm going to use synthetic division. You remember, I can only divide if that's a linear factor with a x coefficient of 1. So I'm going to put the 0 of that factor in front. What's the 0 of x plus 2? Negative 2. Negative 2. Or another way of looking at it, I'm writing down a 2 and I'm changing the sign. And then across the top row, I put 3, negative 5, 0, 1, and negative 11. The 0 is holding open the fact that there are no x squared terms. If you leave that out, these numbers move over, and that 3 ends up looking like it's the coefficient of x cube, not x to the fourth. OK, here we go. I bring down the 3. I multiply. I add. And I multiply. And I add. And I multiply. And I add. And I multiply 86. And I add. What am I going to get there? 75. 75. The remainder is 75. What that tells me is x plus 2 is not a factor of the polynomial on top. But furthermore, it tells me that this quotient is 3x cube, let's see, constants x, x squared, x cube, 3x cube, minus 11x squared, plus 22x, minus 43, plus, and then I'll squeeze in here, 75 over x plus 2. OK, that's synthetic division. Now, if I were to write this in the form of the division algorithm, I would write it this way. I would say that 3x to the fourth minus 5x cube plus x minus 11 is equal to the quotient, OK, here's the quotient without the remainder term, times the divisor plus the remainder. And I'll just put the remainder right below there, plus 75. I'll leave this to you to verify. If you multiply this four-term polynomial times this binomial, if you multiply that out, add on 75, you will get this fourth-degree polynomial that I've given over here. OK, now, what if I wanted to carry out this division? Let's say we have 2x cube plus 3x squared minus 17x plus 12, and I want to divide this by something that doesn't have coefficient 1. It's linear, but other than coefficient 1. Let's say it's 2x minus 3. Now, when I see a 2 there, I think, wait a minute, I can't use synthetic division here because I'm only supposed to do that when I have a 1. So what I'm going to do is rewrite this as 2x cube plus 3x squared minus 17x plus 12 over 2 times x minus 3 halves. I'm factoring the 2 out of that. So I'm going to use synthetic division to divide by x minus 3 halves, and afterwards I'll divide by 2. So I'll go back and divide by the 2 separately. So here's how I'll do this. I'm going to put 3 halves out in front. I'll put the coefficients inside 2, 3, negative 17, and 12. There's a 12 there that's kind of hard to read. And when I finish, I'm going to divide my quotient by 2 because I have an extra 2 in the denominator. So I bring the 2 down, 2. I multiply 3 halves times 2 as 3.  And 3 halves times 6, let's see, 3 halves times 6 is 9. So when I add, I get a negative 8. And 3 halves times negative 8, let's see, can someone help me out there? How much is 3 halves times negative 8? Negative 12. And when I add, I get a 0. Whoa, hey, we're all excited now. Because what does the 0 mean? No remainder. It means there's no remainder. But what does it tell me about this factor? It's the visor of the top one. Yeah, actually, what does it tell me about this divisor? It's a factor of the numerator. Yeah, I got the words backwards. But we're not quite finished because I've just divided by x minus 3 halves. I also have to divide by 2. So when I divide by 2, that means I'll divide this by 2. 1, 3, negative 4. The remainder's still 0. And that tells me that the answer to this problem is x squared plus 3x minus 4. That's the, if I erase what's in the middle here, I've now divided by the 2. And this cubic polynomial divided by this linear polynomial is, gives me that answer. So I had to bring the 2 out and treat it separately from this linear factor with the lead term coefficient 1. OK, that's about everything I can tell you about synthetic division. Now let me show you where it leads us. Let's go to the next theorem that's on the graphic. This theorem has a name. It's called the remainder theorem. And this is one of the things I need to point out to you somewhere along the way. When you come across a theorem that has a name, what that means is it has a name probably because you're going to be referring to it a lot. You have a name because people refer to you a lot. Your parents gave you a name. They didn't just say, hey person or whatever, but they gave you a name like Jeff or Stephen. So when a theorem has a name, it must be because it's something you're going to refer to more frequently. So the name of this theorem is the remainder theorem. It says when a polynomial p of x is divided by x minus c, the remainder is p at c. Let me explain what this means before we go to that example. Suppose I have a polynomial. And let's say the polynomial is x squared. Let's say x cubed plus 2x squared minus x minus 4. And suppose I'd like to know what is p at 2, p at 2. Well one way to find p of 2 is just to plug the number in. Let's do that. Right here I'd get 2 cubed or 8. Right here I'd get 2 times 4 is 8 minus 2 minus 4. How much will that be? 10. That'll be 10. So p of 2 is 10. Here's another way you can get that answer. Use synthetic division and divide x minus 2 into this polynomial, x minus that number. So if I divide x minus 2 into that polynomial, I'll put a 1, a 2, a negative 1, and a negative 4. And if I'm dividing by x minus 2, I'll put its 0 or 2 outside. You notice that's the same number that I would like to plug in. And you know what? The remainder here should be 10. The remainder will be the function value. Let's try this.  And then I add these, and I multiply. And then I add these, and I multiply. And I add these, and I get a 10. Now the remainder you get will always be p evaluated at this number right here. Now that seems pretty incredible that that could be true. Why is that the case? Well let me explain, and then I'll work another example like this. This goes back to that division algorithm that we wrote on the screen at the very beginning. You remember that we had p of x is equal to a quotient times the divisor. And in that example, I was just dividing by x minus 2 plus a remainder. Now this remainder would have to be constant because its degree is smaller than the divisor. The divisor has first degree, so this has to be 0 degree. Look what happens if I substitute in a 2 here. If I plug in a 2, I get q evaluated at 2 times 0 plus the remainder. Well it doesn't really matter what q at 2 is. When I multiply, this says p of 2 is the remainder. That's what the remainder theorem says. The remainder upon division by x minus 2 is the same thing as the polynomial value at 2. Let's take an example. I'm going to ask you to give me a polynomial this time. I'd like for someone to make up a polynomial, not terribly complicated, but not exactly trivial either, a polynomial that's, let's say, a fourth degree polynomial. Susan, can you give us a fourth degree polynomial, just whatever? 2x to the fourth. 2x to the fourth. Minus 3x squared. 3x squared. Plus x minus 1. Plus x minus 1. And suppose we'd like to know, what is the value of this polynomial at, let's say, negative 3? Well, here's the most primitive way to do it. You could actually plug in the negative 3 in three different places and multiply that all out and expand it. But you see, taking negative 3 to the fourth power, taking negative 3 squared, and doing all that arithmetic is sort of tedious. We can surely do it. Here's a faster way to do it. I'm going to use synthetic division. And I'm going to put negative 3 outside. And I'll put 2, 0, negative 3, 1, negative 1. And I'm looking to see what the remainder is. Why is it 2, 0? Well, you see, you didn't have a cubic term, so I have to hold that place over. That's a good question. In fact, I'm actually glad you gave us an example that was missing a term, so we could emphasize that, because it's easily overlooked. OK, I bring the 2 down. I get negative 6. And let's see, I'm adding. Because I'm not dividing by the plus 3, but the negative 3. So when I add, I get a negative 6. Multiplying, I get 18. Adding, I get 15. Multiplying, I get negative 45. Adding, I get negative 44. Multiplying, let's see, I'm going to get a positive number. 3 times 4 is 12. 3 times 4 is 12. And one more is 132. And I add, and I get 131. So how much is P at negative 3? 131. 131, that's the answer. Now, you might say, well, Dennis, you know, if it's all the same to you, it seems to me it's just simpler just to plug in the negative 3 and get it over with, even if the arithmetic is a bit more complicated. OK, well, imagine this isn't a fourth-degree polynomial. What if it's a 14th-degree polynomial? So it's really long. And I want to plug a negative 3 into the 14th power, the 13th power, the 12th power. You get the picture. This gets no more complicated, except the numbers can get bigger right here, like I was up to 132. But that's nothing like taking negative 3 to the 14th power. I mean, come on, give us a break. So this is much quicker for big polynomials. When the polynomial has a relatively small degree, like this one does, I don't think there's any clear advantage of one over the other. OK, let's go to the next graphic. And look at a theorem that's called the factor theorem. We make heavy use of this in the next episode. Now, the factor theorem says that x minus c is a factor of the polynomial p of x, if and only if p of c is 0. Now, what lies behind this theorem, as you see, the remainder theorem just told us p of c would be the remainder. But if the remainder is 0, if p of c equals 0, the remainder is 0, and when the remainder is 0, that means that x minus c is a factor. That kind of sums up several ideas that we've mentioned here today. OK, let's take this problem. I'm wondering the answer to this question. Is x minus 1 a factor of this polynomial expression? x cubed minus 3x squared plus 3x minus 1. Is this a factor of that? Well, one way to find out is to divide x minus 1 into the cubic polynomial, see if it divides it evenly. But another way to do it is to say, what's the remainder if you divide x minus 1 into this cubic? If the remainder is 0, this is a factor. So to find out what the remainder is, I'm going to evaluate the polynomial at 0 or the root of x minus 1. If I plug in a 1, I get 1 minus 3 plus 3 minus 1. The 3's cancel, the 1's cancel, I get a 0. So the function value at 1 is 0. And therefore, we can say x minus 1 is a factor of x cubed minus 3x squared plus 3x minus 1. Yeah, it is a factor. So this time, rather than doing the division, I'm calculating a function value. So I can sort of play these off against each other. OK, the last item of business here today is to construct a polynomial when I'm given its roots. And all of the information that we've gone over in this episode comes to life and has purpose in the next episode. Let's go to the next graphic. This is the last graphic of this episode. And it says to find a polynomial of degree 3 that has roots 2, 5, and negative 4, we must first recognize that the roots of the form x equals c leads us to factors x minus c. Now, let me just work that example right here on the green screen. We have three roots, 2, 5, and negative 4. I'm looking for a polynomial that has these three roots. Well, if 2 is a root, then x minus 2 has to be a factor. And if 5 is a root, then x minus 5 has to be a factor. And if minus 4 is a root, what must be a factor? x plus 4. x plus 4. Now, if I multiply those out, I will have a polynomial that has these three roots. Let's see, if I multiply it out, that'll be x plus 2 times, if I multiply these two, that'll be x squared minus x minus 20. And then if I multiply these two out, I'll get x, I'll just multiply here, here, and here. I'll get x cubed minus x squared minus 20x. Then I'll multiply here, here, and here. Minus 2x squared plus 2x plus 40. Going back and collecting like terms, I have x cubed minus 3x squared minus 18x plus 40. This is a polynomial that has three roots, 2, 5, and negative 4, like we asked. However, this isn't the only polynomial that has those roots. For example, I could have put in x plus 4 two times. I could have used that factor twice, and I'd still have the same roots, but this would have ended up being a fourth degree polynomial. I think the problem on the graphic said, find a cubic polynomial, a third degree polynomial. This is the one we'd want. Another way I could come up with a different answer is I could put a coefficient in front here, like a 3. And in that case, what I would do would be to triple this answer, and I would come up with another polynomial, I'll call it p1, 3x cubed minus 9x squared minus 54x plus 120. So when I say find a polynomial that has these three zeroes, the answer is not unique, but this is probably the simplest answer, and any multiple of it, any non-zero multiple of it will be an answer. And I could even, if I was willing to go to higher degrees, I could put exponents on these factors, and I would still have these three roots with yet a higher degree. OK, let me summarize what we've done today. We started off talking about something that we call the division algorithm, and I think we used an example from like middle school. We took 37, and we divided it by 5. We got the quotient, we got the remainder. But we quickly moved on to polynomials, and we looked at the division algorithm for polynomials. From there, we did long division, we did synthetic division, which only works in certain cases. Then we looked at the remainder theorem, and finally, the factor theorem, and we concluded by reconstructing polynomials from the roots, from its roots. Now, if you stay tuned for episode 11, you'll see how we will use all this information to factor big and rather complex looking polynomials. I'll see you next time.


the Ruger GP 100 match champion in 10 millimeter. Will it shoot 40 Smith and Wesson and so that's what we're going to figure out now I've got some loaded up on speed clips we're gonna we're gonna check that out let's load up some 40 and see how it goes with six rounds of 40 Smith and Wesson and the moon clip you need the moon clips case will sink too far into the cylinder 10 yards away we've got 40 let's go so two did not fire from 40 and you can see that all of them have primer strikes the two that did not fire have light primer strikes


YouTube Darius slay day big play slay the best corner on the Eagles the best corner NFL if you ask me The man they dropped honestly the best DB card in Madden this year without a doubt This is the best card Ridiculous card the best slay card ever especially since it's been on the Eagles If you guys remember back when slay was on Alliance He had crazy good cards since he's gone to the Eagles got been kind of weak But this card has seven buckets four of them are discounted.  I mean the rest of it really doesn't matter But he's 99 absolutely everything and when we put him on the Eagles theme scene He'll be 99 speed Which is you know as fast as you can get and right now I'm rocking them with deep zone knockout Deep man knockout short man knockout and pick artists. That's where we're rocking on slay. I'm off of medium knockouts I'm trying that now because I feel like medium knockouts are the easiest to click on and avoid because the balls in the air are long enough for you to click on enough timing and They can avoid them the most and if you can knock out some short passes inside the 10 quick slants quick out routes and bench drags hitches anything you can get people in third and long situations a lot easier For me, I don't know. I'm gonna try this I feel like mediums just haven't been operating the right way for me And I feel like being able to knock out short passes is gonna be clutch with a high blitz game because I just feel like that's That's how you beat the blitz get rid of all fast drags quick flats quick in routes stuff like that You know, so I'll try it out. We'll see how it goes I actually equipped all my corners on the outside with it or all before my DBs I call my shell with Tillman's Surtain blunt Darius slay That's what we're rocking right now My outside corners actually have deep zone knockout as well for cover three and stuff like that And then I have James Bradbury in a slot with deep zone knockout just in case I want to play zone I'll switch him to safety or something like that. It's one AP got to try it boys But this is the new Darius slay best DB all year and definitely the best Darius slay card We've ever seen is he worth the money? Absolutely not dude. Somebody donated him to me shout out to my man DP He gave me the card love it man. Love to see it. He's a lord.  Something's wrong with me Let's get it boys. Super Bowl Wow, I got Broncos jerseys against the Eagles. That's tough But I can't allow this dude Aiden Hutchinson might not be it. It's just tight offset or tight I Tell you what the the next man event I went to I really was like in shambles I went to one in AC where I did with user and like joke that was probably my last For my next one that I went to it was probably like two years after and I had my son He was my son was probably like 10 months or a year or something dog I was in shambles good knock out there. It was different when I had my son Because what made it different was that I thought about spot me and how that was the last time He saw his son and that had me an absolute crying shambles. It was tough tough scene for boys Well, let's go tell them in feet feet feet feet two good kids are just gone You know and you're talking about this scumbag kid. That's why my opinion on guns is so different You know, it's so like it can't be quantum. You know, it can't be compared No, I really have a strong opinion about guns.  That's just got a high ball every time. Oh man Like that's what I mean like he's too tall sometimes I'm kind of like old-school conservative when it comes to like the mental health part I really am if my wife's like career is in mental health and so we are definitely different in that aspect Dog, how can we not throw this past man? Can we come on? Homes that time same defense same play three plays in a row that time somehow my slant wound up in the popcorn vendors I don't know how but Someone said AJ with little man Bro, what's he gonna do when his chair went out of batteries, bro? Wow? I don't Man's that's that's one of the problems man a lot of these Yeah, I do that too if I couldn't score I will do that too. I you guys know that grinds my Three for 11.  How are we just getting left right now, dude The accurate bit right there. Look how bad I didn't think slay would bag Carmichael that bad I had to hold on Carmichael just a little bit too long.  What that's the chat. They know fine hork leg horn That is a practice.  I said I said eating that Like the past like dog like just Insanity are we really just getting bagged right now?  Dog dude, are we trying to block anybody on the line right now?   That's my fault.  I've been throwing that flat route a lot on man coverage and just high ball on it and I Guess I hit it at the perfect time and you know Here we are an absolute hell game against cover four drop How do you execute quick snap you snap the ball partner dog? How did that dude? I guess he's interesting cover four. I cannot fumble this game right now. I'm pulling on conservative It's just I'll be honest boys, and I've said this before bro the four-man rush dog That gets after you I didn't I do oh my god Block boys just please block for me, please Please They just bro.   It is absolute dog Roadish the trash man. Seriously. Look at this slant.   They play the arts out Like it's their last game in the world. They play their hearts out play the game to win mother I don't beat you by 40 points 40 times this year played a game to win. Good job, buddy Jujus though now. They're not giving me a game I just want to get that taste out of my mouth pause and they're not giving me a game They're trolling me. They're trolling me spinner against goal line Moron dude. Oh my god.  I Said I said I said up This is in fact not the same guy he passed the controller to his little brother I said I said I said a lead male loan son. It's tough dude Good run man, let's grow up see one of my beauties of like being a player right a Madden player It's like I'm like really good at like Like kind of getting over right now. I'm just right back to the next game Okay, I got a problem with EA right now I do and this is just a patch man and before I rage about it that time chatters I manned up my D lineman on the running back and what happened was the D lineman then switched on to the tight end tough scene honestly What the hell just happened dude?     I'm different in open field Oh my god in heaven, dude There's just no way.  Just knew that was going to happen. God. This game is just just absolute Like when you see like little like horse when you're going down a horse trail and you see that falls out of the horse's You ever see a horse when it's walking it's like That's what this game is Ah, of course, he was on the deliming right there like just perfect play and he's on the deliming You ask me a lot of questions for men. That's in a seven seven game against cover four drop Y'all ask me a lot of questions.  Tell you the four-man rush is in cracked, bro So my zones are probably on 30 right now.  They are aren't they The amount of times he's done that for this is why salty was saying yesterday That should be a sack. He should definitely not have another chance for that right there ever This has got to be a catch, man The bot rush i'm i've told you guys a million times the bot rush is like the best against in tight The bot rush dude Yeah, they took the they took the announces out this year. They didn't want to pay madden to do it this year It was his last last year on earth. So he couldn't do commentary So they took commentary out of the game this year resting peace john madden Finally finally after ten of them Finally and interceptions like that would make you go off like what like when you get hit Passing the ball because you held it too long and it pops up in the air You should just be able to what just throw the ball away every time you take too long with the ball So come on man. He can't be running away from us like where is hutchinson right there Like that should be hutchinson's spot.    Okay Until you know what happened there. You can't tell me what karma is. All right, chris all day with a k You spell chris with a k. You should just be Your entire opinion on everything should be ignored. Oh my god dog.  This is like actually bagging me worse than anything in the world, dude He's past committing every play. He has two inside stuffs. You can't run the ball.  Do you get our fumble bet?  It really does Thank you.  Good job. Much only for money spenders. Yeah, definitely, dude. Absolutely. What do you do with your free time? This is gonna be a dot. All right boys I'm trying something here Give him a chance. We'll give him a chance for the boys for the content. Why the high ball to the open running back Well, listen, if you've been like me as many times as I have this year You would high ball to the running back to it, dude Did y'all see that guy that was just in the middle of the field and like could have just put his hands up and like These are the people these are the people I struggle with right there. How much was this card? You said how much was this card? How much was he? I mean, did he actually get the strip? Was it him? No, it was peanut tillman Look at him We gotta show some fight here sleigh. Like this is like dog jj jj jj jj jj jj jj


Humanity is in a profound inflection point in human history where we are looking to a new paradigm.  One of the great challenges for me as a journalist is to still be very focused on those facts that are still very important. But finding ways to be able to connect and tell stories better because we're trying to convey some extremely complex things. Telling someone that the planet's going to warm by two degrees doesn't really inspire them to do anything. But telling them that 30 million people in Bangladesh may be displaced in the next 80 years may resonate a little more. In 2010 I embarked on this program called The China Dream. It's really to create a national identity, much like the American Dream. But what is it for China? What technology has brought I think in the last sort of five years really is particularly mobility and more social platforms have emerged. It's sort of rapidly accelerating the opportunity for people who are disconnected who are just pawns in the game of life to actually have a voice. I saw an Al Jazeera once a gentleman who's in Aleppo reporting directly as a victim. What we're seeing is the victim is also the journalist. They're holding the phone right there and they're telling their situation. They're also the photographer recorder. They're also the distribution channels coming straight off Facebook or Snapchat or whatever. So the pain, the energy, the conflict is no longer dissipated. It's literally going from their hand to my hand. So it's much harder for me to like pretend nothing's going on. You need to visualize the social norm that you want at the very beginning for people. And this is why storytelling is so important. What we're doing in the end is creating a platform by which any individual can search through the relationships on this planet. I'm having a really hard time with my breakup. Here's five conversations from around the world of other relationships dealing with a similar issue. I'm excited about bringing Maori voices to thebodiesnotanapology.com. I'm excited to listen to the stories of Maori people in their own voices and their connection to how we create a more just and equitable and compassionate world. We can really learn a lot from our indigenous cultures and see what are some culture and customs and values that they have that we should adopt as we embark on this new level of consciousness in which we all communicate. It allows us to translate addressing a problem and having actionable ways of dealing with it afterwards after being inspired and moved. And in some ways not depressed, happy, joyful, full of content is where we make the most change.  We all have stories to tell. All of us.


Maternal under-nutrition during pregnancy can lead to lifelong health issues in offspring, including increased risk of obesity and insulin resistance. In this study, researchers found that offspring of mothers who had been fed a low protein diet while pregnant were less likely to become overweight when exposed to a high-fat diet as adults compared to offspring of mothers who had been fed a normal protein diet. Additionally, the offspring of older mothers were more likely to show signs of anxiety and reduced physical activity, which could also contribute to their decreased risk of becoming overweight. This article was authored by Swear, JP Voight, and SC Langley Evans.


Ich muss ja sagen aus persönlicher Erfahrung, dass der Kongress für mich gerade deswegen so schön ist, weil er so traumhaft familienfreundlich ist. Wir sind diesmal wieder mit zwei Kindern da und da ist natürlich das Kidspace absolut gold wert. Aber, und das glaubt man gar nicht, dieses Ding hier, das Decktelefon, das macht diesen Kongress auch für mich extrem kinderfreundlich, weil wir eben schon vor vielen Jahren unseren Sohn mit so einem Decktelefon einfach laufen lassen konnten. Okay, so der Speaker ist telling an Anecdote, about how he's giving his kids a Decktelefon from this conference's phone system, that enables the son to walk around on its own, because he can always call his parents. And I'm just a user of the system, but the people that are on stage now will tell us what is possible with this technology. We are greeting civilian and ST, which are parts of the event phone system. And LaForge as a guest hacker will show you this talk that is called hashtag me fail or with Giga set this wouldn't have happened or the subtitle is decked is correct. This talk will be translated by the C3 lingo team. You can leave feedback for the translation on Twitter at C3 lingo. Well, thank you, we are very surprised. We thought this would be a kind of niche translation. We thought this would be like 20 people. Now it's more like 40. Actually, it's probably a couple of hundred. Yeah, so we, the presentation is kind of done. So what will we do? We have to hurry up because it's actually a lot. We'll talk a little bit about the POC. We will talk about the technologies that we are using. And that's why we will show you the hardware live on stage and how it works. So the next thing will be the problem we're having, the security variation that we found. And LaForge will describe that in detail. And we will tell you about how the manufacturer handled it and what we built with it. And how we use the security variability for more features and can do more for our users, in fact. Then we will continue with metadata and how we actually gained that. And then we will show you live on stage how to get unwilling devices to become willing. And then we will show you some plans for the future. So, let's start. What even is the POC and the event phone? So what does the internet say about this? The internet says the phone operation center is an integrated hard and software project, which enables to use phone systems on big events in a big area. That's how Wikipedia puts it. And in fact, that is the case. And we also think it's nice that event phone is a relevant operator of wireless communication for big events. This, of course, makes us very happy. And now we will continue to our one-pager. What are we actually doing? We provide infrastructure for wireless phones. Who of you in the hall has a decked phone? It's probably about half. And then we operate phones for rent for organization and infrastructure teams. We provide them earlier. We ensure that they will work and make them operable. And we also disinfect them, of course. Then we have SIP and premium SIP that we operate. Premium SIP actually does have different functions. For example, the cert, the emergency services here on the congress have more rights and more is more stable connection. Then we operate the EPVPN, which is a phone system that also works outside of the events that we integrate with the events. Then the calling and token handling, which is the connection of phones to the system. Then the guru, which is the generic user registration utility, the web portal where one can register numbers and connect devices, which is also operated by us and is also the link to the GSM team, whose numbers we operate as well. Then the dial in and dial out, which is the connection to the outside phone system. We also offer special services, which is the connection to the chaos operation. There are some games, the time and the automatic reception for noise complaints. Also, we have a new section, which is the education about media competence, which is where we show all of you how to look at everything critically and don't take everything for granted. And for the truth, what you see on the internet and here. Und und und this talk also Eventphone turned 18, which is the age that is required in Germany for doing all kinds of legal things. And I have to say something about this. There is one person that actually is doing this since 18 years and has participated in on every event. Not just the Congress is also every summer who always was there. Wenn er nicht krank war, war er immer engagiert in dem Phonesystem, und das ist Sacha, und wir wollen einen besonderen Dank für die 18 Jahre des Phones für dich sagen. So, okay, das ist es, um es kurz zu sagen, wir sind immer reachbar, auch in der Bedrohung. Wenn ihr das nicht kennt, könnt ihr es sehen unter dem QR-Code. So, was ist das Talk not about? At the 35th C3, we talked a lot about how Eventphone came to be, and how it actually works, and some stories. That's not what we will talk about at the Easter Hag. We explained the new Phonesystem in detail, and why we switched from the old Alcatel system to the new Meteor system. What was part of that, that's also not what this is about. You can see those on the Internet. Today we are only talking about this red line, the connection between the Open Mobility Manager, the OMM, and the RFPs, the radio fixed parts, which is the antennas basically. Then the tech boys will use a lot of abbreviation, and to explain those quickly, OMM, as we saw in the earlier slide, is the Open Mobility Manager, that's the piece of software from METAL, then RFP, that's the radio fixed parts, the antenna base stations, then the PP, which is the portable, that's a decked lingo, it's everything that's mobile, everything that can move the PP's is portable parts, which is normal decked entities, but also for example the small headsets. Okay, now you know what I mean with those. Then the one abbreviation is the E-Pie, you probably know a little bit more about networks, the international portable equipment identifier, you can imagine it's kind of similar to a Mac address, for the example you can see a part of it is written in cursive, and that's the manufacturer part, just like in the Mac address, it's hard coded, you can see what manufacturer produce the device, and then there's just a counter after that. Also similar to a network is the E-Pie UI, which is the international portable user identity, which is kind of similar to an IP address, it's assigned by a system, by the system that operates the phones at the moment, for example the DHCP server on the knock, is the one that hands those out. And then there's the European Telecommunications Standards Institute, the ETSI, which basically creates the standards for Europe, and especially decked. The E-Pie UI as an example has different formats, it doesn't have to look like this exact example, it's not as normed as it is for the IP system. And now let's have a look at SIP decked on MITO. So civilian will talk about that, and he'll watch it by the camera, alright. How to is the title, so we've been asked a lot how we're doing it, can I help with that at all? So what do I need to do for it at all? Naturally you'll need an antenna radio fixed pad, there are different generations of that. We're recommending Generation 3 Plus, they're easy to identify, because for example they have ventilation on the back, and there's also a USB connector. So if you want to buy something like that and don't know unsure about it, we're recommending Generation 3 Plus. You can get them at the manufacturer, for example, but you're able to get them used as well, it's about 100 Euros. Additionally, you'll need a license, that's not an open source software by MITO, it's a commercial product with a special thing about it. So for small decked installations, you don't need a license, but you're limited to five licenses, to five antennas. If you want a call, not just between decked devices, but also to a SIP, or with a dial out, you also need a SIP server. We also recommend YATE, which we use ourselves. In the slides we also have the cookbook by YATE, you can also find it using the QR Code. That's a little how-to, if you have no experience at all about it, and using that you can install yourself pretty easily. There are two possibilities of using the ... you can either run it on the antenna using Linux, but we never tried that, so we don't have any experience about that. Alternatively, you can use anything running sensory. For example ... For example, the one that we run here at the Congress, there is a little server in the knock that has a VM on it, we sent to us. So if you have the antenna running, then the first one is connected to the other ones via IP, and the other ones then need the connection to the YATE. The downside to this setup is you can at most run 512 end devices on this setup, and if you look at the dashboards, you can see that we are vast beyond that, so there is a hard limit on top, so you need to extend it a little bit. If you do it standalone, then you have a lot more radio antennas, so you can have a lot more end devices. This supports up to 5,000 devices, so we already mentioned it a few times, and the question is how do you get it? The official answer of Mitle is that if you get a device, then you get the relevant credentials and you can download it in the portal and just download it. So if you get the antenna from the second hand market, then you obviously don't get the credentials and you don't have the download links, but the release notes are public for the most part, and if you have the right search terminologies, then you can find them easily. So this is why we put them on the slide. You can just use these and you can get the credentials. So the question is, what do you do with it now? And what hackers do is they just look at things in depth and go deeper than the others, and this is what my colleague is going to tell you about, firmware analysis. So for me it's normal to look at firmware, because, yeah, it's kind of obvious for me. Every device I get my hands on, I'm just going to look at. I want to understand how it works. I want to understand what makes it tick. I want to know what it's built off. For me it's natural to do this. So, I just had an antenna and wanted to look at it, and it was 10 years in the past that I for the last looked at this decked. We had a tool that was called detective. We found a vulnerability in it back then, but that was the last time I looked at it. End decked, which is a pun on discovered, but also decked. And there is a specification for this, and I've been doing specifications for the last 10 years, so they're all rather interesting. It's a bit of a matter of taste, but for me it's very interesting. And decked is still very common, so there's also decked ULE for IoT devices, and I heard that since last year, or maybe a little longer, the Pock is very interested in running a new system, the Mitel system. And last year in November I had to move, and the December Congress was too annoying to take your heart or too much effort, so I sat at home for a while, and I looked at those specifications for a while, and if you look at it, you first start with the hardware in the generation 3, and I looked at the board, what kind of pieces are on it, and you can find an ARM system on chip, and if you look at the firmware, then you can see that it's basically a mainline in the next kernel, and it has two Ethernet ports, so if you can't find it, it's usually very often you can find it in network attached devices, like Shiba Plug, it has two of these Ethernet ports in this configuration, one is the one that talks to the ARM, the one that is being led outside, it has a port on the outside, and there's a second Ethernet port, and my question was, if it's inside what is the point of it, and the point is that they actually attach the deck processor over the Ethernet port, so they actually have two of them, one to the outside, one to the deck port. So, it's a Linux, and it has a password that you can use for the root user, and during the configuration it's being set, and you can actually look at it, because it's being set by the ARM, so it's easy to get access to it, you can log in and then have a look around, so you can see the schema on the bottom, it also has a UART here, there's also GPIO and so on, so now if you look at the user space software, that looks interesting enough, it's basically really a very clean Mainline kernel with very few patches, so in the essence there's nothing really deck-specific in the kernel, it's really normal, no extra drivers, there's only just one program that's called IPRFP, that creates the Ethernet connection to the deck processor, and the whole processing of the deck is being done on the processor. So, there's ... it's a very Mainline, very standard vanilla kernel, and there's not a lot you can see from there. So, interesting, however, is ... you can find some other things, which may be interesting, for example, there's a video for Linux in the kernel, and I was wondering what's the point of that in a deck-basis station, is there a configuration option I haven't found yet, but it's actually what's true is you can attach a video camera to the USB, and if you have a phone that actually is video-capable, you can actually view the video that you see there, very interesting function, and then you find, oh wow, there's also Bluetooth, so you can probably also create Bluetooth-Pictures, so maybe to do location services in a museum or something. Very strange, but it makes sense in the end. Yeah, there's actually use cases for these kind of things. So, you also find binary images for the firmware of this deck processor, for example, a bootloader, which is actually connected to this serial interface between the processor and the out. The firmware is installed automatically via Ethernet, it can install the actual firmware. There's a Macmoni.bin, which is a Macmonitor. I'm not quite sure what it actually does. I didn't look into it further, so once you look at it, you have this Ethernet-Device, but what do you need with it? You'll take a look at the decode file. Interestingly, there are just raw Ethernet-Frames. There are different subsystems for the Ethernet-Types, differentiates between which system is actually speaking right now. You take a look at it with everything you know about decked, so you're trying one thing or the other with a telephone that's attached to it, and you try to understand it. Then I was using a minimalistic Wireshark thing to check the protocol, and from a specific point, where you can find the decked standardised packets. But there's no Wireshark-Desactor for that either, so I started something about it, but it's just very minimal just to confirm my results when I was doing it. And then you're getting ahead and you don't find any Wireshark-Desectors either. So you can see the split of functions on the decked protocol stack. I won't go into detail, but whoever is interested into that further, I had a technical presentation about it at the DOSMO, and you can also find it at media.ccc.de. And this is the split in the ARM. There's the DRC-Layer and the NWK-Layer, and what's above that. And the interface between DRC and MAC-Layer, there's the protocol you can see with these raw Ethernet-Frames. So now I have a general idea of what happens. So, I'm further looking into the IP-Protocol. You can see there's a TCP-Connection that's been created. It looks very random, so there's a big Entropy, so it needs to be encrypted in a way. In the RFP and ARM, we can create a lot of Loggings, so we can take a look at the Hackdumps, but it is in no relation to it, it's still encrypted. If you take a look at the Symbol-Table, you see Blowfish. There's some Blowfish-Functions from US OpenSSL. And this is linked with LDP-Load. You can link other libraries. So, there are two of them. There's one LibTracefish, which outputs Hypertext and Printtext. Plaintext und Syphatext. So, you can get the plaintext on the other side to correlate the encrypted with the plaintext. So, many of the raw Ethernet-Frames that you can see are just one-to-one passed. They are getting encapsulated on the TCP-Protocol, but that's mostly it. We'll talk about that more later. Some other people looked at that a little closer. What's important about this is that we can disable this encryption, but this Detachbar that is known from something else, he looked at the encryption a little more with Blowfish and saw that it's pretty clear that there's one CLEE that is globally statically distributed for all devices. And that's, of course, bad, because when one has one identical global key that can maybe be extracted from that firmware, that would mean that one can decrypt the communication from all the other devices, which brings us to this de-decked protocol. So, as LaForge said, it's just one static key that is used for all instances, which is kind of critical, because if somebody is capable of carrying out a man-in-the-middle attack, he can read and potentially manipulate all of the data passing in-between. And then we thought about how could one get to this man-in-the-middle position. So we have a photo here, there's a red circle there, for those of you who can't see.  This antenna is hanging fairly low. So sometimes they're way higher, for example in hardware stores. Then you might even need a ladder.  And then when you screw them open, there's somehow a lot of space in there and a RJ45 plug. Something might be possible there. So, when one doesn't have the motivation to Google about it, there are also devices that are used indoors, for example in hotels. They're hanging from the ceiling in hotels. The problem there is, this photo is from a hotel near Paris, so this is not a problem, only existing in Germany. A lot of these bases have a very wide installation base. They are used in many, many kinds of ways, in domiciles, in events, in hotels and lots of spaces. So this led us to the thought that we can't leave this standing that way. If we can hack this, other people can too. And so we learned that of course we do responsible disclosure, we tell the manufacturer and give them the possibility to solve this. We even got a CVE number for this. So how did the manufacturer respond? We have to make this transparent, a short timeline. First contact tries were in the beginning of October. We called the MITRE support because we only found an e-mail address on the website. That looked kind of generic. So we dialed ourselves through the telephone menu. We got a walker there and told them that we wanted to talk about the security on ZipTech. And then they said, ZipTech, I don't know, I know what that is, but I don't know anything about security. So as an alternative, we contacted a MITRE partner that we have a good contact to. And we told them that the KS Communication Club should tell them before they hold a talk about this. And then we got an e-mail from the communication manager for wireless. And we had a long talk with them. And then this meeting happened a week before this. And we presented to them with the things that we have on this table. That there is a security problem and where it is in our views. And we talked about how to maybe solve this. We explained to them very complicated what Eventphone is. And waited. And a month later we asked again. And then we got the confirmation that yes, they have a solution that they are testing in the lab. And there will be an update for Congress. So you won't be affected. And then one day later we met them at their office and talked about their solution. And the solution was you get a custom firmware that has a different static key. So, of course, we didn't like that. So we thought about it a little. We explained our own solution idea. And we got a yes. And they said, yeah, okay, you'll get a proper update, but only in 2021. And so we said, okay, so that won't solve our problem, but it might be okay. And then on the 20th of December, they said it will be before the Congress. The incident response team from Canada apparently looked into this. And apparently the problem is bigger than expected. So since two days, the update is available. It's available on the download center of Mitel after you enter the credentials for that. There is also a security advisory on the website. And we explained to the manufacturer that we want to give them the possibility to explain things from their view. So we have two slides from Mitel. Wait, did I turn this off? Wait, there's a hack. Use the keyboard. Okay, so from the view of Mitel, the problem only exists when you have a manager middle position that you can gain, which is apparently hard in company networks. And then here again, an update is available, which can be downloaded. And there is a download link there, which, and then there's another link for the security advisory for those of you that don't understand English. Here it is again in German. And if you have any questions, you should contact Mitel and their technical support. So another thing we have to say in sub 90 days from reporting to a bug fix is for such a big company, actually not that bad. We have seen worse things. So in the beginning, ST already talked about RFP proxy. So what is that? That's a transparent two-way encryption machine in the Mitel proxy. So if you're doing it, then do it correctly. It can decrypt and encrypt several types of communication. It takes care of reking in these protocols, exchanging the blowfish keys often. It's able to manipulate selective messages. You can inject messages. You can remove messages and to guarantee the stability. There are several types of processing, for example, at our event. So to take a look at it, that's the way it looks normally. You have an antenna, which has a TCP on port 12,621. In our installation, we also have an RFP proxy on a different port in our installation. So there's a rule for it, which makes it a transparent proxy in between. So if needed, it can get the communication out through a socket. So, for example, you can attach Motorola, a logging tool, which can write in PCAP, which we'll talk about later. You can also do audio, you can process audio with it or control LEDs with it.  Several deck-specific processing of non-deck-specific processing is running in on.  So when you're using this proxy, you still have the problem that there's not just the specific specified deck-centered, but also manufacturer proprietary protocols, for example for LEDs. There's been a lot of reverse engineering efforts and you need a lot of data for that. So, how did we get that data? My colleague Zuckerberg, and I always like to say, so where do you get data from, unless you steal from the users? We took an honest approach. So if we just go to any event and get some data, we might be in danger. So we just did that transparently. Who of you has been on the ESAC? Very few people. So I'll just explain that quickly. We wrote a blog post that we want to use metadata. We want to log all data with all data about all the devices who are communicating, accept the language. If you wanted to use the guru, you needed to confirm that every time. It was not very pleasant, but we wanted everybody to notice. And we really expected a shitstorm for it, for us logging everything with it, so that anyone would say, don't do it. And before we wrote down everything very specifically, we also noted the intent that we wanted to note the data, find mistakes and irregularities, since we didn't know at all what we're looking for. So we just graded a big amount of data to look through it. Using this tactic, it actually worked. So with a SIP, you could create a station with falling from SIP to SIP, which didn't use any metadata off. So all the data we collected extra. We secured them and encrypted them on a secured file system. So very few people had access to it from Pog. It was recommended very precisely. We wanted to do it in a way, which we would expect if somebody's handling our data. So in the end, we thought about how do we get rid of it. And our big maximum was maximum transparency. So maybe somebody saw an event. We also had a video about it, but we don't have. We're not able to show it sadly. So there's a screenshot and a QR code, if you want to watch it. There's the principle for eyes. I'm an actual. I made sure that the developers deleted the data responsibly. And we thought about what we could do about it. And in the end, we wanted to throw it all away. And right now we can't even say how much data it was anymore. And I believe we handled it responsibly. And I wanted to still bank all of the users at EasterHack. And it helped us a lot. It was really informative. And we could actually create new features. And because we were able to analyze them against that test data. And the first new feature, my colleague is going to show now. I already mentioned that these software can control their own LEDs. There's four LEDs that are attached to the base stations that can be turned off and on according to different passwords. And it's being controlled by the OMAM. And what we realize is that if you make one of them a blank red and green, then the third LED, then if you let it blink red and green, then it doesn't just do that. But the fourth one also starts blinking. So there's also functionality that is being controlled via the LED. The fourth LED is usually used for the wireless. But this is not allowed for us to use because that's the domain of the knock. And why is it blinking? Because in the setup decked was a tool that uses Morse code, but we don't use it anymore. Ja, ihr könnt ja mal schauen irgendwo. Yeah, you can have a look. There's probably a few other blinking LEDs. The only problem is that you can only blink with one hertz. So you need to have a bit of time. Do you want to switch back to the slides, please? Genau. Okay, the second interesting feature is the media tone. If you listen to the headset and there's this sound that you hear, this is generated within the antenna. There's a specific message dedicated for that, that he says what frequency to use in which order it is being used. It has up until 256 entries. It can do four frequencies at once. It can do loops. So we were wondering why would we use the boring sounds. So there's a MIDI converter that you can use to generate these dial tones. And it has loop detection and we want to now demonstrate that. This is not mine. This is mine. Please dial 45. Lieber Mann, vom Audio kannst du mich mal muten, bitte. Okay, please mute me now. Ah, ist klingelt. Ah, it's ringing. Hört ihr das? Can you hear this? Sehr schön. Very nice. Habest du wohl zu weit weg? Okay, you're too far.  Ja, the radius or the distance you can use is limited. Please start Tetris. This is a normal telephone connection as you can hear. And I'm going to type Tetris now. Is this missing a tone? Okay, so it's not working right now. Ja, ihr habt das Telefon von ST gerade schon gesehen. Okay, now you just saw the phone from ST. It's a nice Motorola S120X or 1200. And the X is up to the number of phones, headsets in the box. Why this one? Because it's the cheapest one that you can find when you search for decked. It has colors, which is nice. You can have it in Turquoise, in Pink, in other colors. It's important that it has Gap-Compatible. You can get that. This is a feature you can not just attain by writing it on the box. You actually have to implement it. This is, there are a lot of people over the last events that just got that device that we got and they couldn't register it. So this is why we were interested in. We don't have a lot of time, so we still tried it and we could use the proxy to get the traffic from this and we tried to debug the connection request. So when you attach it to the base station, then it sends a message and it gets assigned its identity. That this one ends in DF and when it's trying to ring, then it sends its own identity. It sends the EPUI and it doesn't end in DF. The answer to that is the reject and it gets the reason for it. So in the Etsy-Standard or in the Gap-Standard, it's defined that this is the way the handshake has to work and it refers to the DAG-Standard. There is an access-rise-accept standard message and it has to save it. It probably does it, but it doesn't send it again, we assume. It says that the EPUI is up to 60 bits long, so this is what we saw as an EPUI and it has a 60-bit portable Usernunder. We color-coded it here so you can see it. So the last two bits are the ones that got lost and some of these are special because they're using the EPUI. They use the serial number for the device. And this is still you. And the EPUI has a length of 36 bits. This is defined. And if you look at it now, there are four bits or rather these two zeros that are not being used, at least not at the sub-DAG-Device. And so this was our way of making it compatible to the base station. We basically moved the relevant data to the one byte up. And if it connects, then we move it back and then the Aum is happy. And that's why this year even these phones work on our station. So at the event we realized that... The other way around, sorry. We made sure that it... We wanted to make sure that it only affects these devices, so we moved it back around. So we basically whitelisted for every single device that is affected to make sure that we don't break it for other devices. We also informed the manufacturer, of course. So in September we send it to them via Twitter. They send it to China. And then we asked them again. We heard nothing. Maybe something will happen, we don't know. So at the event we had to realize that this is not the only one that is affected by this problem. So we amended the list of manufacturer prefixes. So that, for example, the Velocomphone has to work, that the Siemens device is working. There's about 20 devices or manufacturers that we know of now, where it doesn't work. Audiovisual marketing. So with marketing we don't really have a thing. But maybe they are good with their phones. AVM. So, last year we had the same problem with AVM phones. The Fritz phones, which is why we always did not recommend them. That they could never show who was calling. They always only showed intern. But ever since we can now not just deliver messages but also read and modify them, we can now actively show who is calling. We also have a positive thing to say about the AVM phones. We also brought Fritzparks. And this Fritzparks can't just do TCP. It can also do D-Trace, the DS4 DECT. And under the given URL we can find this capture interface for all kinds of Fritzparks. And what's interesting about this is that Gigaset phones can actually display the AVM Phonebook. And this does not work on our system yet. But we hope that we can change it at some point. OK. Look into the future. What will we do with this? What are we doing with this? So, people that are interested in DECT in general, people that are interested in the MITER Installation, people that are interested in these kinds of things, we hope we aren't the only ones. And we hope that with these kinds of information you can start getting into it. We hope that the Wireshark Desector will continue to be developed and analyzed more, play with the hardware, play with the software. And there's another thing, right? And, for example, DECT without E. There's another thing. There are EC-Terminals that work with DECT. And our plea for you is that we are still collecting data. And if we also talked about this at Easterheck, we are crowdsourcing. If you know what DECT telephone you have and what model it is and you want to tell us, you can enter it into GURU, which would help us a lot, actually, because we asked the Etsy for the EMC-List, the Equipment Manufacturer Code table. And so the one that was displayed in cursive earlier, which allows us to see what manufacturer makes this device. And the response was that this table is apparently secret. So, it would be great if you entered your model numbers, because then we could actually find out what manufacturer has made the device. This is all that we want to ask of you and thank you. And is there another thing? Yeah, maybe. Oh, yeah, there is another thing about the proxy. It has not been released yet. And we'll tell you why we didn't release it. So, I already explained it in part. We only, on very short notice, got the response that the tool that the update was deployed. We thought it would be irresponsible to release the tool, because it would be very easy to compromise a lot of systems around the world. We do want to release it, and we are talking to the manufacturer, so you can all use it and play with it. But that current plan is that we do that in the first quarter of next year, as soon as the existing installations have the chance to apply the update. Good. This talk was translated by the C3 lingo. You can find us on Twitter at C3 lingo. Please leave feedback for us also under the hashtag C3T. The talk was translated by Moritz, Kaster, and Oscar. And we will next translate the questions and answers coming in. Okay, so, please go to the microphones one, two, and three, and also questions from the internet will be asked. So, you said there was first an update from MITER that was going to do it, and then they said there would be a better solution. So, what's the new solution, and how's the key being generated? Okay, so, the antennas have to be connected to the system at some point, which is called the capture. And so, what we are assuming is that this takes place in a responsible and to be trusted environment, and the keys will be exchanged in there for the coming communications. So, every decked antenna gets its own key now. Okay, a question from the internet. Did the people know from the hardware store know that there would be an outside antenna being modified? That could even be photoshopped. I can't really deny that or say it's true. All right, microphone two, please. Can we use Tetris in our extensions so we could create a me-melody generator by phone? That might actually be difficult because we only have 256 tones, and those only actually work on the 2G antennas. The 3G antennas only can do six tones because they saved on hardware, which is probably not enough. Okay, a question from the internet, maybe. Yes, there's one coming in right now. So, where exactly can you remember, can you notice your telephone device number? Okay, so in the Google system online, the guru3.eventphone.de, if you log in there into your account, you can see a pencil button there, and you can enter it there. There, press the pen, and then you can edit your handset, and then under model you can enter your model number, and the manufacturer, and you can even give it a name, which has the advantage that when you use it the next time, you can just reuse it from right there. You don't have to re-enter the token at the next event, ideally, the phone will just work. Apparently, one of the microphones got turned off. So, after the question from the network came, this is only for people that actually used the event phone and entered their phone there, so then the phone number is connected to the account. If anyone on the Internet that is not on the CCC Congress, they of course can't do this as a further explanation.  Microphone 3, maybe. If I understood correctly, for you stations, there are no downloads, so how are you going to provide updates for that? I can't answer that, we have no idea. So, the contact data is in this advisory. This would of course be told to the people that need it. Okay, Microphone 2. So, you took a look at the communication between the antennas and the EOM. Is anyone trying to replace the EOM through their own antennas? For example, in the Congress. Ja, maybe, it's a question of time and the number of projects, which is of course growing over time. The number of people working on the project is unfortunately not growing. So, yeah, we would like to do that, that would be great. I'd be the first person to scream yes, but of course we would need people to actually get involved there. And this might actually be a good starting point for later open source development and working with DECT. It would be if people would contribute to the Wireshark DECT Dissektor. This is all kind of part of the DECT specification. You would have to read that and maybe understand it anyway, because if you want to work in that open source context. And then you can express that in the Wireshark Dissektor and then we'd have a great baseline to work from. So, yeah, we'd love to have that. Yeah, contributions are what we need. Okay, so I can't see any more questions, maybe on the internet. No, any either. That's it. Thanks for all of the nice discussion from all three of you on stage and everyone from the event forum. Thanks for all of your work. So, for the last time, this talk was translated by the C3 lingo. You can leave feedback for the translation on Twitter at C3lingo.


How do you keep the number of cards down if you are mapping a journey over a number of years? And this was a question by Katrina. Thanks Katrina for asking this question. I think this question is quite interesting because there are two perspectives on this question. One perspective is that the journey covers a number of years. Another perspective is that we have a journey map that we are working with year after year. Let's answer them both. Yeah, I think that the first thing is that if it covers many years, it's again that we have to summarize it. It's again going back to what is then a level 2 in this map, even if it's over a couple of years. I mean, I've been in projects where we were mapped from people getting their first job to people dying, I mean in pension. And it wasn't more than on the level 2, it was 12 cards, I think. And that's a number of years.  Yeah, something like that. That's one of the questions. The other question is, okay, we're going to do this over a couple of years. We're going to work on this map for several years. And I was really happy when I saw this because that's a huge challenge that people do journey maps and then they leave them. They don't keep on working with them. And if this was the interpretation with this question, I'm super happy that we got it because then you just don't just add things to your map over and over again or more and more stuff. It's it's just as important to take away things to I think that people in their mentality and internally in organizations, they think that now we're going to work customer centric now we're going to add even more things. But it's really good to have the mentality that what could we take away? So over the years take away things. I mean look look for instance on on this disruptor industry disruptors. What are they really good at? Narrowing down their customer journeys. They take away simplifying simplifying Yeah, simplifying things simplifying how it is to open a bank account how it is to do buy something on the net how it is to Order a taxi or whatever it could be. So over the years simplify and simplify that should be the That's really interesting like perspective that over the number of years your number of cards in the ideal situation should actually go down. Yes that That's that's a really Really cool take away. I Don't think a lot of companies are ready for that yet but If you see that the number of cards is actually Rowering that could mean that your first attempt to map a journey wasn't complete. It could be but at some point you should sort of reach a maximum Complete less and from there You shouldn't try to add more. No No, because The the competitor you have that makes the most relevant journey it doesn't has to be Short but it will in most industries be shorter But the one who makes the the most relevant journey, they will be the winners. Yeah in your industry So it's a from most organizations. It says it's all about Taking away cards and That's I can I guess we're getting back to really the The essence and the core of journey mapping it helps you to focus on doing What adds value to the customer and eventually to your business? Yes, removing? Removing friction or improving the existing experience And and a journey map just helps you to focus helps you it literally points out what you need to do Yeah, yeah