foreign welcome back students we're in section 1.5 and this and the Plex because it's talking about complex zeros and also there's a lot of stuff so we made two separate sections for 1.5 we're going to call them A and B we're going to get started in 1.58 today as I said we're going to be talking about complex zeros remember what zeros are right okay zeros are where a function output is zero now complex if you recall means that we have real numbers which are those numbers you worked with all the way up through algebra two they're the regular old numbers that we have but also the non-real numbers those imaginary numbers that show up like the square root of negative one and you've talked about that in algebra two but if we have a function let's suppose we plug a value a into it let's look at this right here if you have some some function p and you plug in a into it and the resulting output is a zero then we have a word for a we call it a zero you're right or a root we use both words so we'll say that a is a zero or a root of P furthermore because we know the output is zero then we know it's an x-intercept so whatever that value of a is it's an x-intercept we can make a coordinate point a comma zero now lastly if a is a real number then we know there's a factor that um you know that real solution came from and that's x minus a that is a linear Factor because if I were to graph this it would be a straight line now we're going to have to remember how to factor some polynomial functions we're not going to get into it too much but we will review in fact here's our first example right here find all the x-intercepts of the function X cubed minus 2x squared minus 8. so a quick review of some factoring I hope you remember you look at all these and we want to pull out a greatest common factor so hopefully when you factor out that GCF you look at all these terms there's no whole number that comes out of all of them but I can pull an X out and that's going to leave a trinomial of x squared minus 2x minus 8. now as I said we're not going to review too much how to factor you should know how to factor by now but I know the way that a lot of teachers teach it is they say okay when this is a trinomial and we have uh a is equal to one right that's that first number right there a b and c then we can we can look at this and say well what multiplies to C and it'll give you B when you add them together so in this case we get a negative 4 x minus 4 and then a positive 2 which will be X plus two and I always advise my students to multiply this back out and to make sure you get that trinomial because you know what factoring is hard now at this point we want to find all of the x-intercepts remember the x-intercepts that's where the function equals zero so I'm going to set this function equal to zero because where this function equals zero that will be an x-intercept and there will be three places where this function will equal zero okay how do we make this equation equal to zero well this is called the zero product property one of these has has to be a zero that's the only way you get a zero when you multiply so either the first factor which is X either that's going to be zero or we're going to say that x minus 4 equals zero and if you remember that I'll give us x equals positive four or lastly we get X plus 2 will equal zero and then X will equal negative two so I'm going to graph all of these on our axes right here especially the x-axis is where we're looking we have 0 negative two and four why don't we put a point right here negative two zero and four one two three four and I also remember from algebra two this is X to the third power so I know the general shape of it is kind of something like this right we have n behavior that starts kind of low and it goes up high and if we're looking at this we're just making a rough sketch here we have we're going to go down at that point and come through here and we have a nice little sketch but the question really just asks us to find all the x-intercepts so to answer the question x equals zero and the output would be zero x equals four and the output would be zero and x equals negative two and the output would be zero those are all of the X intercepts of this function okay there's the first type of question it's just simple factor and find where the zeros are let's look at the next one which is a little more complicated because we have an inequality it says given the polynomial function h of x equals x squared minus two x minus three what are all of the intervals on which the function h of X is greater than or equal to zero well if if you can guess it yeah we're going to have to factor it again so let's look at this this function h of X factored here is H of X I'm just going to write it down here to give me a little more room and I want to factor it so you know h of X is going to equal what this is the same situation as the other one a is equal to one so I really just need to look at this last number and see what multiplies to give you a negative 3. you got two options it's either one and negative 3 or negative 1 and positive 3. well how are we going to get a negative 2 when we add right they have to add to that middle term so the only way to do that is x minus 3 and then X plus one now as I look at this as I said go ahead and in your head you can multiply that out to make sure you get a negative 2x in the middle and you do so we're good to go but from this we can find the x-intercepts right that's where the this function would equal zero so zero equals x minus minus 3 times X Plus 1 and just like above we're going to use the zero product property and we're going to say that either x minus 3 is going to equal 0 or X Plus 1 will equal 0. one of them has to be a zero because they're being multiplied and the product equals zero it's the only way you get a zero when you have a product right one of these has to be a zero so we set them both equal to zero and we get an x equal to three and we get an x equal to negative one so now you can go over to our axis and we know that there's an x-intercept at negative one so it crosses right here and it also crosses at x equals three all right so far so good now there's a couple ways to get the general shape of this if we wanted to graph it the first way I recommend you memorize what a quadratic equation looks like this is x squared that is a u right now there's two ways this can go can either be a u a regular old U or it can be an upside down U that depends on the value of a here this one's positive so it's going to be a regular old U it's going to look something like this and and we're just doing a rough sketch right now we're not trying to get too accurate but there's a different way we can do this let's pretend like I didn't remember this rule about the quadratic let me undo that for a second go away you the other way I could do this problem and figure out the graph is I could divide it into different sections we basically have a section from here all the way to here we have that point we have this middle section we have another point and then we have this section out here so I'm going to create a table let's call uh let's see let's call the top part we'll just call that like X and then I want to look at specifically the sine all right so let's write The Sign of H of x and I'm going to go through all these intervals one at a time and maybe it's not an interval it's just a single point but as we work left to right we have this interval of negative infinity and it goes all the way up to this value right here which is negative one so I'll say negative Infinity all the way to negative one and remember this is not a coordinate point this is an interval of X values and when we have that I'm going to plug in let's plug in a value we'll say this value right here but you can pick any value we're going to plug it into each of these factors uh individually and we're gonna see what sign we get okay so let's plug in a negative three that'll be like right here we're looking at this interval from negative Infinity up to negative one I'm going to plug in a negative 3. when I plug in negative 3 into this first Factor right here I'm going to get negative 3 minus three that's negative so the sine would be a negative times whatever I get here I plug a negative 3 in it's another negative right that's this Factor when I plug in that negative 3. so I get a negative times a negative guess what that is that is positive all right so I've just learned that then this interval the function is going to be positive because I picked a I don't say a random point but I picked a point here that's in my interval and I plugged it into each factor I found the sign of it and I know the resulting sign of the polynomial should be positive then I'm going to look specifically at this point we know that at negative 1 that's already a zero right so the sign is it's just going to be equal zero so I'll just put a zero here there's no sign right it's positive or negative it's zero and then I have this interval from negative 1 all the way until three so again I'm going to write it using interval notation and from negative 1 to 3 what I'll do is I'll pick an easy number in this interval I'll pick zero right because that's easy I love zero so plug a 0 in here if I plug a 0 in the first factor I get 0 minus three that's negative so I'll have a negative number times if I plug a 0 into here we'll get a positive what is a negative times a positive you're right that's going to be negative so I know for this part of the interval right here the function is going to be negative right and then we continue the process we know that at 3 here we get 0 and if you were to follow through and do it on the other side here you would get positive okay so it's going to be positive 0 negative 0 positive and we know that at each zero it changes sign right so what do we get we get this resulting function where that's positive and then it's negative and then it's positive again and that gives us the same U shape that we had before so you're like hey why do we just go through all of that well these polynomial functions are going to get complicated this one's pretty simple so hopefully you can understand analyzing each of the X intervals that we have that are created by these x-intercepts we are not done with this problem yet because the question actually asks us to give all the intervals on which h of X is greater than or equal to zero greater than or equal to well here's the greater than here's the equal to and there's the equal to and there's the greater than so the way I would write this I'm going to use interval notation from negative Infinity all the way up to including negative one right so I'll use the square bracket there easy enough along with we're going to start at three including three square bracket closed and we're going to go all the way to hey I didn't finish my table oh well we all go to Infinity easy enough so that was complex a little bit but now we're going to talk about the multiplicity of these different factors if we have a linear factor and it's repeated n times and we use an exponent for that normally right so if we have X Plus 1 and it's squared then we'd say it's repeated two times but if it's repeated n times we're going to say that that factor has a multiplicity of n well what does that mean I can look at each one of these factors and I know the multiplicity of X plus one is two the multiplicity of x minus 3 is 3 and the multiplicity of X plus four is one remember when we don't have a number there there's really a one if a real zeroa has even multiplicity then the graph will bounce off of the x-axis at that value of x equal to a let me try to show you something here I hope it works out you don't have to write this down just watch just watch what I'm doing so suppose I have a wonderful axis here and of the polynomial function so maybe I don't know it looks like in the end but maybe we'll just make it look like this okay so we have some polynomial function and we have two zeros or two Roots one right here one right here and maybe there are others like back here but I want you to focus on these and I want you to imagine what would happen if these two Roots became closer to each other and closer to each other if we move these closer and closer the function is actually going to probably look like this right as those roots get closer and closer until there's a point where these two roots are right on top of each other and what happens is that is a repeated root that means that we would have one value here that has two roots and this this polynomial function is going to bounce off it there'd be no width between it where it's negative right it would just bounce off it so hopefully that demonstration helped us understand why when we have a factor that has even multiplicity like this one even multiplicity that there is a bounce going on okay there's a bounce going on so let's look at our factors for this polynomial I think we could just look at it can we just look at this and figure it out a little bit we know that x equals negative 1 is going to be a factor right we can do this in our head X Plus 1 equals zero move it over you get x equals negative one but that has even multiplicity so we won't forget about that we also have x equals three that has odd multiplicity because there's a 3 here right and then x equals negative 4 will be our last root or zero and we can plot all of these right here one two three four we're going to go negative four first then negative one and then three and I know this polynomial function P of X is going to have these roots or these zeros I just need to figure out what it looks like now way back in Algebra 2 you learned that if the degree of a polynomial is even that means it has the same end behavior it goes up okay or it goes down on each side but it doesn't go up and down it's either both up or both down so one of the things we could do is knowing that we can look at the roots and here at negative four I know the multiplicity is a one right so I know that the function actually crosses at negative four and then I can analyze x equals negative one that's even multiplicity so I know that at this root it's going to bounce and then looking at x equals three that's odd so we're going to cross there now that's one way of graphing this the other way is to set up intervals like we did on the last problem the last example we could go negative Infinity all the way to negative four we could do at negative 4 and then negative 4 to negative 1 and then so on and so forth you set up a table and you can find the sign of the polynomial that was quite a lot I told you complex right another way of thinking about this is to look at output values for the input values near x equals a the output values will all have the same sign positive or negative what that means is if we have a even multiplicity like we do at x equals negative one that's right here around negative one they have the same sign these are both negative it doesn't switch and that's what this sentence right here means let's do that table example remember I said you could build a table let's do that right here for number three given the polynomial function P of X is equal to x minus 2 uh remember that's to the first Power if there's no exponent there and then x minus 5 to the fourth I got some bells ringing right there that is even multiplicity and then we have X plus seven that is also to the first Power what are all the intervals on which P of X is less than or equal to zero so looking at all the factors right now it's pretty easy to come up with the zeros and The Roots right because a linear factor is easy x minus two you get x equals two x equals five that has a multiplicity of four though we're going to keep our eye on that and then we have x equals negative seven so if I were to sketch these let's be a little bit sketchy all right put up your thing here what are you doing all right so we need x equals negative seven we'll put that here we need x equals two and x equals five now I wanna sketch this polynomial function let's build that table that we built back two examples ago let's build that out so my table is going to have X and then the sine of P of X now I'm gonna look at my x-axis here that's an arrow I want to divide this into different intervals so negative Infinity up to this point which is negative seven and then from negative seven to two and two to five and then 5 Infinity so I'm gonna I'm gonna spell that all out here in my table so here is my table and as I promised you I was going to save you work but look at that table we got to figure out now as I said what we could do is in each section we could plug a value in so let's this first section here goes from negative Infinity all the way up to negative seven so maybe like negative 10 I plug that in to each factor and I find the signs so a negative 10 here to give me a negative and negative 10 here would give me a negative but it's raised to the fourth power so a negative raised to the fourth power is positive and a negative 10 here would give me another negative so a negative and positive negative we could do all that kind of stuff or we could just be easy and smart about it so we know that at each one of the zeros the polynomial will equal zero so we can fill that out in the chart that's pretty easy zero zero and zero okay now what else do we know we know about the multiplicity right here at x minus 5 where x equals five the multiplicity is four so there's a bounce that happens right here and the other uh two routes here they don't bounce which means that the polynomial crosses so it's going to cross a two it's going to cross at negative seven but it will bounce at five I really only need to find one point in this polynomial and we can build the rest of it so let's pick an easy value you know what the easiest value is to me x equals zero let's find x equals zero and what it looks like at x equals zero if I plug a 0 into all of these factors I'm going to get 0 minus two so that's going to give me negative 2 times 0 minus 5 to the fourth power so that's going to be negative 5 to the fourth power right and then lastly we get zero plus seven that's just going to be seven all right that's a four there I don't want to confuse anybody but what does this end up being in terms of the sign I don't care about the big number you can plug in a calculator if you want to but I have negative 5 to the fourth power that's positive 7 is positive so it's going to be a positive times this negative 2. the resultant value will be negative so I know that at zero the function is negative that's all I need to know so I'm going to go over to my table right here 0 occurs between negative seven and two I know the function is negative now I can use my investigative skills here I know that at 2 it crosses so if it's negative here that means this value must be positive I also know that at 5 it bounces right because that's the even multiplicity so if it bounces if it's positive and it's coming down to five two it crosses it's coming down to five and it's going to bounce right then I know it's going to be positive on this side furthermore I can go all the way back to the left hand side here at negative 7 and it's zero I know it crosses there because the multiplicity is odd so it's going to be negative it's going to end up being positive on this side so we're negative and we're going to end up being I was so close I did very well to that last part but we get this polynomial function we could sketch it it's a rough sketch we get the idea of what it looks like and I can just use one value in this table and I can determine the rest of the values if I know that even multiplicities mean that the sign does not change when you hit the root it bounces right there so now we did all that work the last step is to answer the question what are all the intervals on which P of X is less than or equal to zero well look at your little table here it's less than or equal to zero it could be either one less than or equal to so we're talking about all the way from here to here right and this value right there so how do we write that let's use a closed bracket from negative seven to two a and X has to equal 5. and if you use both of those hey check that out you have all the values where it is less than or equal to zero complex as we promised complex zeros now remember complex zero or zeros they're both real and non-real so a polynomial degree n has exactly n complex zeros When you're counting multiplicities remember again complex reverse are both real and non-real zeros if there are any non-real zeros they always come in conjugate Pairs and we'll talk about that below this means there are either no non-real zeros or an even amount of non-real zeros but there there can't be one or three or five let's look at this quadratic all right here's a nice little quadratic what could happen with a quadratic which has a degree of two right so back here the polynomial would have degree two that's even how many real zeros does this quadratic equation have I can look at the graph and I know there's a zero right here I know there's a zero right here so we would say there are two real zeros which means because the quadratic has a degree of two these two numbers have to add up to the degree of the polynomial so I know that there are zero non-real zeros okay let's look at this possibility uh what do we have we have a root right here looks like it's at one but they don't really say the scale but let's pretend it's a one uh it looks like a repeated root right so how many real zeros are there there are still two but it's repeated you have the same value twice so we're going to call that two zeros it's one repeated root that means we still have zero non-real uh zeros now let's look at this last example how many zeros well zero real zeros and I know that because it doesn't touch anywhere right there's no x-intercept so there are no real zeros but these two numbers have to add up to the degree of the polynomial so I know in this situation there must be two non-real zeros all right let's try cubics which has degree of three I can look at this polynomial I know there are three real zeros therefore there have to be zero non-real zeros now if I look at the suppose we have this cubic right here oh my goodness and it got moved up and I can look and I see one right so there's one real zero I know for this polynomial there are two non-real zeros because these two numbers have to add up to the degree lastly this one's a little bit tricky but I got a zero right here and did you notice I have a repeated root of repeated zero right here that means we have three which means there are no non-real zeros easy enough right so number four says the degree of a polynomial is eight all right that's important so I'm going to underline that and we have real zeros at x equals negative 10 x equals five and x equals 16. so there are three points where we have real zeros and x equals five has a multiplicity of two so really we have one zero here we have two zeros at x equals five and one zero at x equals sixteen how many non-real zeros does a polynomial have well we know if the degree is eight of the polynomial right all of these zeros have to add up to that if you take the real and the non-real there are four right here so there must be four non-real zeros and again we just look at the degree of the polynomial and we count the real zeros and the total has to equal whatever the degree is easy enough okay so now let's talk about non-real zeros if a plus bi is one of the non-real zeros then it's conjugate a minus bi is also a zero so these imaginary zeros or these non-real zeros they come in pairs and each of the pairs has a plus and a minus now a lot of students they forget which one uh you know it says given one non-real zero they forget which one to change do I change the negative three or do I change the positive six to find the other zero and so I'm going to give you a little clue over here remember your quadratic equations we've got x equals negative B plus or minus square root B squared minus and then the 4 8C all over 2A anyways this part right here is where the I comes from right and so when you were studying quadratic equations you would actually figure this out and sometimes you get an i in there right and that would give you it actually reduced down to one of these type of answers this non-real answer the plus and minus is in front of the square root so that means that that part that gives you the I it's going to give you a positive and a negative I value so if they give you one of the non real zeros or the polynomial you know that the other one must include a negative that I value if it's positive and if it's right here negative then we know we're we're going to have a positive in front of that I because there's always going to be plus or minus that's how I I tell students to remember it but this is more general for any polynomial function that's just for quadratics but that's how I remember it hopefully that works for you that's non-real zeros we have one more part right now successive differences to find the degree of a polynomial this is crazy if you have input and output values of a polynomial function so here we go we have a nice little table this is a polynomial function it's possible to figure out the degree of this function just by using a little trick of differences this technique only works if the input values are equal intervals which means notice how it's going up by ones every time and that's how both examples are but it has to be the same difference it doesn't have to be one but it has to be the same length of interval every time and if we do that this rule will work every time the degree of the polynomial function is equal to the least value end for which the successive nth differences are constant that made my head hurt so let's just go down and do an example and see what we're talking about all right so the first difference we're going to find right here we're just going to say what is the difference between zero and one all right how far do you have to go from zero to one and that is a one and how far do you go from one to four the difference is three and then we get five seven nine right and we can keep going for as long as we need to go these are the odd numbers and then we're going to find the second differences so these are the first differences so doing the same thing finding the differences from one to three how far is that well that's two from three to five two two can you believe it all right so how many different layers do we have here how many times do we have to do it we had to do it twice we found the first difference the second difference is and then they're all the same so we can say the degree of this polynomial is two let's write that out the degree equals two and that's because we have two layers here we had to do the first difference and then the second difference why don't you try it with the next one you pause the video do number eight all by yourself go okay so hopefully you figure this out wasn't too difficult I found the first differences I found the second differences and then the third difference is you gotta be careful because these are negatives oh look what I lost I lost the negative all right so then the third difference is is when I finally get a constant value so that means that this polynomial has a degree of three fantastic wow that's it that was a long video but I promise the next one will be short this is Mr Kelly remember it's nice to be important it's more important to be nice good luck on those Master Jets