Hi there. And this problem we're going to be analyzing some graphical traits of polynomial functions to higher degrees. All right. So when we're talking about degree we're talking about is the total number of exponents. Right? So we actually have here is what we call 1/5 degree polynomial. Right? Because if we wouldn't multiply this out, we did distribution on this guy. We'd end up with a term that had X to the fifth power. All right. We can see that by looking at these two terms knowing that these two expressions are being multiplied. Alright, when we multiply exponents, right? Or multiply variables with exponents. Remember that we add those exponents to get the final result. First thing we're gonna analyze here is we're gonna try to come up with the zeros. Remember zeros have different names? They're also known as X intercepts. Sometimes you hear teachers or books say routes. They're also called the solutions. Right. And really what it just as graphically as where does this function cross the X axis? If we think of that as a coordinate, anytime we cross the X axis. Right. We're seeing what value It's going to give us an output or Y. of zero. All right. So, in this particular problem, our first step is where do we think this is what values of X would create an output of zero? Right. We can just look at each of these little expressions here. Right. On their own. So the expos foursquare but let's just focus on the X plus for what value of X would make this become zero? Right. Well, we got to think of the opposite what's the inverse or opposite of positive for that's negative for if you put negative for into here, you'd end up getting negative. Four plus four is zero. Zero square is still zero. So, we found one of our points on hearing. I know my graph here. We could even put that on there. Here's negative four comma zero. And then if we go look at the second set of brackets, change colors for that. What would make this second term become zero? Well, this guy would be five if I put a five into here. 5 -50, zero to the third power. It just means zero times zero times zero. Yeah. So either five or 0 would end up giving us our X intercepts. Right? Yeah. Yeah. And there's probably only two values that will will make that true. So there's only two X intercepts here. They occur at -4 and four clean my function up so we can look at it again. Mhm. All right. So those are identifying where zeros are located. All right. Now, when we grab this guy are zeros can do one of two things. They can either cross through the X axis. Right? They can go through it like this or they could touch it and bounce off and kiss that access. Right? What determines whether or not our function does this uh is whether or not this X opponent is even or odd. We have what we call repeating zeros. All right. And when that repeating zero is and has an even degree or even exponents like to for example this right here is just going to touch the X axis. Yeah. For our other zero. Right? Since it's an odd degree or odd X opponent, it's actually going across the X axis. Right? So those are second part of this. All right. Next thing. We're gonna kind of start making a picture of this guy, right? And in order to do that, we need to know the number of turning points. What we call turning points. All right. So if I just if I gave you a linear equation, for example, a linear equation has zero turning points. It doesn't change direction. A quadratic parabola has one turning point. It changes direction once. Right. As we start getting into other functions polynomial. They can have multiple turning points where they change direction. Right. Yeah. And our rule for this to figure out how many turning points they are, is to take the degree of our function. Okay. Yeah. And subtract one. So, the degree of our function here was five. If we subtract one from that, we're gonna end up with four turning points on our graph. All right. We're gonna start showing you how this looks in just a second last piece of information. We need to know uh Is the end behavior alright? In the end behavior is just concerned. It what's happening way over here on the left and what's happening way over here? On the right. And behavior doesn't tell us anything in the middle here. Just what's happening at the left side and on the right side. Okay, So again, I like to use examples, simple examples that you've worked a lot with to get us started. The end behavior of a linear equation could be two different ways. Right? The left side could be going down the right side, could be going up or if we change the soap, your left side could be going up the right side. Could be going down When we look at quadratic six and notice something different happens. We can either have both the left and right going up or both the left and right. Going down right. The degree of linear equations, The highest x point in a linear equation is one. The degree in quadratic six is to All right. So this is going to be the first piece of and behaviour is the degree of your function even or odd. Okay. And we said that the degree of our function right back up here at the top. The degree of our function here was five. So we had an odd degree on this guy. Okay, We have a non degree. It means one of our end is going to be going up and one is going to be going down right. When when we have functions where the degree is even they both go up and they both go down. So how do I know Which one it is? Right. Well, with a linear equation and we had a positive slope or a positive coefficient. The left went down the right went up and we had a negative slope or a negative coefficient. The left side went up, the right side went down. So we got to go up here and look at do we have any coefficients? Right? It looks like maybe we don't but there's a one out here. Right? We don't show that one in math. And we also don't show that this is positive. It's a positive ones. We have a positive coefficient in here and our degree is odd. So, if I was looking back down here, we're having this scenario happened right here. All right. Where the left side is going to be going down on my graph on the right side, It's going to be going up right. And now we can start to kind of rough sketch this. We said that when we hit -4 since it's a repeating zero with an even degree, it's just gonna kiss that line. It's gonna come down. All right. Said we might have four turning points. So this might be an approximation of our graph. Yeah. And blue. I'm gonna label are turning points there's one 2, 3, 4. Turning points right? R N behavior is going down on the left, up on the right. And the reason that's happening is this is an odd degree of fifth degree polynomial. It has a positive coefficient. So that's why it's down on the left up on the right. We only touch the X axis over here at negative four because that value or that zero has an even degree even ex opponent. We cross through five because we have an odd exponent. All right. So a lot going on with polynomial is and a lot of things to pay attention to her. We're going to focus on the exponents. We're looking at the zero values in here. We're paying attention to whether or not it's positive or negative, right? And in person can definitely show you some call a polynomial. Dance moves to help you remember this. Um But I hope this gives you a good starting point.