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# Intro to end behavior of polynomials

## Video transcript

what I want to do with this video is talk a little bit about polynomial end behavior and this is really just talking about what happens to a polynomial if as as X becomes really large or really really really negative for example we know we're familiar with quadratic polynomials where Y is equal to ax squared plus BX plus C we know that if a is greater than zero this is going to be an upward-opening parabola of some kind so it's going to look something like that the graph of this of this equation or of this function you could say and if a is less than zero if a is less than zero it's going to be a downward-opening parabola it's going to be a downward-opening parabola we spent less time with third degree polynomials but we've also seen those a little bit so for example if you have the third degree polynomial y is equal to a x to the third plus b-- x squared plus C X plus D if a is greater than zero if a is I don't want to use that brown color if a if a is greater than zero when X is really really really negative this whole thing is going to be really really really negative and then it's going to increase as X becomes less negative less negative it's going to do something that might do a little bit of funky stuff in between but then as X becomes more and more and more positive it will become more and more and more positive as well so it might look it might look something like this when a is greater than zero what about when a is less than zero well then just like here we would flip it we would flip it so that so let me write this so if a is less than zero if a is less than zero when X is really negative you're going to multiply that times a negative a and you're gonna get a positive value so it's going to look something like this and then it's going to go like this it might do a little bit of this type of business in between but then its end behavior it starts decreasing again it starts decreasing so when we talk about end behavior we're talking about the idea of what is what is this function what does this polynomial do as X becomes really really really really positive and as becomes really really really really negative and kind of you know fully recognizing that some weird things might be happening in the middle but we just want to think about what happens as as extreme values of X now obviously for the second degree polynomial nothing really weird happens in the middle but for a third degree polynomial we see that some interesting things can start can start happening in the middle but the end behavior for third degree polynomial is that if a is greater than zero we're starting really small really low values and as a becomes positive we get two really high values if a is less than zero we have the opposite and these are kind of the two prototypes for polynomials because from there we can start thinking about any degree polynomial so let's just think about the situation let's just think about the situation of a fourth degree polynomial so let's say Y is equal to a X to the fourth power plus B X to the third plus C x squared plus D X plus I don't want to write e because he has other meanings in mathematics I'll say plus well I'm really running out of letters here plus even F I don't want to I'll just use em all though this isn't the function f this is just a constant F right over here so let's just think about what this might look like let's think about its end behavior and we can think about it relative to a second-degree polynomial so its end behavior if X is really really really really negative X to the fourth is still going to be positive we're going to be if a is greater than zero if a is greater than zero when X is really really really negative we're going to have really really positive values just like a second-degree and when X is really positive same thing x to the fourth is going to be positive x a still going to be positive so it's end behavior its end behavior is going to look very similar to a second-degree polynomial now it might do it in fact it probably will do some funky stuff in between it might do something that looks might do something that looks kind of kind of like that in between but we care about the end behavior so this is I guess you could call the stuff that I've dotted line in the middle this is called the non it behavior the middle behavior this will be obviously be different than a sec agree polynomial but what happens at the ends will be the same and the reason why when you square something or you raise something to the fourth power you raise anything to any even power for very large as long as a is greater than zero for our very large positive values you're going to get positive values and for very large negative values you're going to get very large positive values you take a negative number raise it to the fourth power or the second power you're just going to get a positive value likewise if a is less than zero you're going to have very similar end behavior to this case for at a polynomial where the highest degree where the highest degree term is even so this is a is less than zero your end behavior when a is really really really really negative this thing is going to be really really really positive we're going to be multiplying it times the negative so it's going to be really really really really negative so it'll look like this and likewise when X is x is really really really positive you get the same thing because our going to be multiplying a positive times a which is negative and in between it might be doing something it might do something like that but its end behavior you see is very similar to a second-degree polynomial so if you ignore this its end behavior is very similar now the same is true for a fifth degree if you were to compare it to a third degree and the overall idea here is what happens to what happens to this value and we get really large axes or really small X's now we're taking it to an even power in which case for either really negative values or really positive values we're going to get positive values and then it depends what our coefficient a is or are we taking it to a odd are we taking it to an odd power so the general idea and actually let me just do a fifth degree just to to make the clear so if I had something of the form Y is equal to a X to the fifth plus B X to the fourth plus and it just went all the way I want to have to even write it this thing if a is greater than zero if a is greater than zero would look something like this it's end behavior is very similar or it is similar to is similar to a third-degree polynomial where a is greater than zero at the end it would do this now it could it might do it might do kind of some craziness might do some craziness like this let me might do some I might have to get this rights as one two three it might do some craziness like this in between but then for really large X's it will look the same as X to the a X to the 3rd when a is greater than zero so once again very very similar and behavior when a is greater than zero and very similar and behavior when a is less than zero it would look like this at the ends a negative value it will be positive because this part is going to be really negative but then it's going to be multiplied by negative to get a positive and for really positive values of X it will be negative because once again this a term is going to be negative and then what it does in between where at least it for the sake of this video we're not really thinking about so the big takeaway here and this is kind of a little bit of a drumroll here when we're talking about end behavior if you're looking at an even degree polynomial if you're looking at an even degree polynomial it's going to have end behavior like a second degree polynomial if you ignore what happens in the middle what happens at really negative values of X and really positive values of X is going to be very similar to a second degree polynomial and if your degree is odd you're going to have very similar end behavior to a third degree polynomial you might do all sorts of craziness in the middle but given for given a whether it's greater than 0 or less than 0 you will have n behavior like this or end behavior like that