Notes 2 5 Zeroes of Poly Functions

Added: 2026-05-12

[00:00:06]in this section 2.5 we're going to be talking about the zero zeros of polynomial functions uh we have the first thing we have is the rational root theorem or the rational zeros theorem i call it the rational group theorem it says you have a polynomial where a sub zero does not equal zero so basically that you always have a constant here okay then you can find the nonzero rational numbers p divided by q will help you find the zeros of the polynomial p must be a factor of the constant term which is this one right here and q must be a factor of the leading coefficient so i call them your p's over your q's now notice the word possible these are possible rational zeros so your p's are factors of 12.

[00:01:22]so your p's are factors of positive 12 so to be 1 and 12 2 and 6 3 and 4 they could also be negative 1 and negative 12. so basically all the factors of 12 are positive and negative 1 positive and negative 2 positive negative 3 4 6 and 12. those are all the factors of p now all the factors of q this is your p and this is your q all the factors of q all the factors of one are one times one or negative one times negative one so plus a minus one so if you take your p's divided by each of your q's you get all the possible rational zeros now if i take one and i divide it by one i get positive one right if i take one and divide it by negative one i get negative one if i take a negative one and i divide it by a positive one i get negative one if i take a negative one divided by negative one i get positive one so the only combinations out of dividing these as plus and minus one and that's the same thing for this plus and minus two plus and minus three plus and minus four plus and minus six and plus and minus twelve so that's 1 2 3 4 5 6 7 8 9 10 11 12.

[00:03:12]12 possible zeros rational zeros rational means fraction these numbers can be written as a fraction so they're considered rational numbers so those are the possibilities of your zeros when you graph this so i just want to know what are the possible rational zeros we'll work on finding them later okay all right so let's do the possible rational zeros of b factors of fifteen well one times fifteen and three times five so that would be plus and minus one plus and minus three plus and minus 5 plus or minus 15.

[00:04:03]those are all the factors of p now all the factors of q all the factors of 6 1 two three and six so it'll be positive and negative one positive negative two positive negative three and positive and negative six those are all the factors of six so what are our possible rational zeros of this polynomial p divided by q all the p so this divided by each one of these and this divided by each one so we're gonna have a big long list here so one divided by one is plus and minus one plus and minus one half plus and minus one third plus and minus one sixth now three divided by each one of these plus and minus three plus minus three over two now three divided by three is plus and minus one we already have that so we have to write it again and three divided by six is one half so we already have positive and negative one half so we don't have to list that again okay five divided by one plus and minus five over one or just five five over two we don't have that yet plus and minus five halves five divided by three plus and minus five thirds and plus and minus five six now fifteen fifteen divided by one plus or minus fifteen fifteen divided by two plus or minus fifteen over two fifteen divided by three is plus and minus five we already have those and fifteen divided by six fifteen divided by six that reduces by three so that would be five over two all right five over two if you divide this by three you get a five if you divide this by three you get a two so 15 over six is five over two which we already have those so we don't need to put those 15 over 6.

[00:06:37]so those are all the possible so we've narrowed it down from infinite number of possibilities and now we have all of these possibilities that could be zeros of this polynomial now sometimes if you if and again a lot of this was before graphing calculators so if you were to plug in a positive one what would you get if it was a zero you should get zero because that's the remainder theorem if you plug in one and you get something other than zero then it's not a rational zero so these are we've narrowed down all the possibilities the infinite number of possibilities on the number line you know you're like well that's a lot of numbers but in reality we've narrowed it down pretty well so we could plug each one of these in and see which one gives us a zero back if you get a zero that means that your remainder is zero for that a value and therefore it's a rational zero or an x-intercept on the graph okay so right now we're just listing them nice thing is we're gonna let you use a calculator okay or they'll be given to you i'll give you one and you can find the other so all right so what kind of calcul what kind of polynomial equations can we solve easily okay we can solve linear equations we can solve quadratic equations okay but these are cubic and fourth power equations so having to solve them plugging in zero for y solve for x that's going to be a lot harder to do so we're going to use our possible rational zeros and we are going to use the calculator to find out which ones of those will will work for us so what we're going to do is um grab a graphing calculator if we don't have a graphing calculator you should really get one i think santa will bring you one they're cheaper than your cell phones um there are some apps that you can download i might have directions i might send you to to get it on your computer so um okay so if i take this first equation 7 x to the third plus 18 x squared minus 97 x minus 60.

[00:09:40]okay and i were to graph that and i'm gonna go zoom six which is the standard window when that's done graphing and i'm going to see if i can find a zero zoom six okay now we assume that it goes through here or here but let's look at the table let's look at the table all right so three zero is on the graph so that's one of our zeros so i know one of our roots is three from my graphing calculator so i know i have a root at x equals three okay now i could give you that root if it's a non-calculator and i could say find the other ones okay well we know what all the possibilities would be but um i just want to find all of the uh zeros if i gave you one you can use synthetic division which we talked about the other day so let's go ahead and take our synthetic division uh problem 7x cubed 18x squared negative 97 and negative sixty and let's do synthetic division bring down the seven seven times three is twenty-one and we add so we get thirty-nine okay so then what 39 times 3 multiply and now we're going to add get 20 and get a remainder of zero did i know i was going to get a remainder of zero yes because i knew that three was a root so the remaining polynomial that's left here is 7 x squared plus 39 x plus twenty and that is a quadratic so now we can solve this quadratic by factoring or using the quadratic formula so i want you to factor that see if you can factor it okay so the factored form again 7 and 1x those are your only options there now this is a little bit harder guessing and checking again some of you may have learned how to use the box but you really want to just use guests in check i have a 2 in a 10 a 1 and a 20 um what else four and a five i need to get my outside plus my inside to total 39 okay so i'm just going to kind of guess and check i need my outside outside and my inside whoops my inside here to total 39. so i can't put a 20 over here because on the outside it would be 140.

[00:13:36]so let's guess um [Music] positive 4 and a positive five i think i made a good guess now again five and four two and ten ten and two i don't really know i'm just guessing i know i i need factors of twenty so i chose four and 5. now i'm going to check 7x times x is 7x squared i know that 4 times 5 is 20.

[00:14:10]that's where i got those numbers from but i need to make sure that my outside which is 35 and my inside which is 4 total to be 39 and they do so that is the factored form of that quadratic if it doesn't factor you use the quadratic formula okay so the question is is find the exact values of the zeros well one i got off my calculator or one i will give you so why don't i just get them all off the calculator well yeah you could but i'm not teaching you how to type things into your calculator i'm trying to get you to understand how synthetic division works and where the roots come from okay so your final answers are x equals 3 which we got from the calculator then if we're going to use our factors set your factors equal to 0 and solve and that will give you your other two which is negative four over seven and negative five so those are the exact values of your zeros okay all right so if i give you anything higher than a quadratic you have to know or be able to find the roots that's going to help you reduce it so this is a fourth degree polynomial so i would need to know at least two zeros okay i would need to know at least two zeros well i'm gonna give them to you you have roots at x equals negative three and negative one if they're not given to you you can pull them off of the graphing calculator okay so that's why it says use your calculator to get started all right so what we're going to do is we're going to take our first one it doesn't matter which one you start with and you get 22 65 negative 20 negative 45 and 18. do our synthetic division bring down the 22 so we're going to multiply and add multiply [Music] and keep going okay again we know that that is a root we know the remainder is zero now we are going to take if if negative one is a root of this polynomial it's also a root of this polynomial if it's if x plus 1 is a factor of the original x plus 1 is also a factor of this so what we're going to do is we're just going to do synthetic division again but we're going to use our third degree polynomial so we're just going to steal these numbers and we're going to use synthetic division one more time so so we're going to multiply and we're going to add multiply gives us a positive 23.

[00:18:01]multiply and add so again we get a remainder of zero but again we knew that because i told you that negative one was a root so you have now reduced this equation to a quadratic you've reduced it to 22 x squared minus 23 x plus six so now again you can easily solve this quadratic um or factor or use the quadratic formula i don't know if i want to use quadratic formula see if you can factor that let's see if you can guess and check and factor that go ahead and stop the video for a second because it's going to take you a little bit of time guess and check stop the video okay what you should come up with is 2x here and 11x here you could have tried 1 and 22 but it won't work so how do you know well you don't you just guess and check you can use now if this is negative and this is positive i know that these both are going to be negative they have to be okay they have to be that's the only way you're gonna get a negative second term and a positive third term so i know they're gonna be so i can be it can be one and six it could be six and one now i know it's not six i know it's not six because there's a gcf with the two and the six so i know six can't go here i know two can't go here because they have a common factor and there was no gcf in the original so there's not going to be any gcf here gcf meaning grades common factor okay so i know a 2 can't go here i know a 6 can't go here okay so i could do a 1 and a 6 i could do a 3 and a two that's pretty much what it narrows down to and i'm gonna do a one and a six and let's check 2x times 11x is 22x squared i know a negative 1 times a negative 6 is a positive 6. now let's check my outside is negative 12 my inside is negative 11. that totals my negative 23. okay so i was given when they say what are all the real zeros don't forget the lit the one that they gave you don't that's a very common mistake what are all the real zeros well negative three negative one what would this be one half and over here is six elevenths so x equals all of these oops would be the real zeros of that polynomial okay so um you know you could use your possible rational zeros and i could make you list all of these and then plug them in to see which ones are actual roots so we could do that for these um i'm not going to be that mean and make you list all the possible rationals and then check to see which one i'll usually um give you one um or let's use the calculator to find one to start the problem okay all right all right so fundamental theorem of algebra every polynomial function of positive degree with complex coefficients has at least one complex zero the zero may be a real number since any real number r can be expressed as the complex number r plus 0i so if you remember complex numbers a plus b i remember this a plus b i that was called a complex number its conjugate was a minus b bi if if a plus bi is a root then a minus bi has to also be a root they come in pairs same thing with this if m plus the square root of n so if 3 plus the square root of 2 is a root you need to know that its conjugate is also a root so the conju conjugate pair theorem says just that they come in pairs so if i give you one you have to know that the other one is included okay i skipped the number of roots theorem if f of x is a polynomial of degree n is greater than or equal to one so n is greater than so n is equal to two three four and then f of x equals zero has exactly n roots let's go back here this was a third degree polynomial and if i set it equal to zero i get my roots how many roots did i have one two three okay how about this one this is a fourth degree polynomial if i set it equal to zero i get all of my roots how many do i have one two three four all right so you have four that's basic that's the fundamental theorem of algebra okay so this is part of the fundamental theorem of algebra okay so determine the zeros of each polynomial you're going to have four i'm gonna have five i'm gonna have four i'm gonna have four okay all right so i going to give you the two roots that you need this polynomial has roots at x equals negative one and two those are given or you can pull them off the graphing calculator if i didn't give them to you okay so negative one i'm going to start with negative one one x to the fourth negative one x cubed one x squared negative three x and negative six go ahead and do your synthetic division now we're going to do synthetic division again on the new this is going to be your cubic so we're going to use our cubic equation so again this is a root we should get zero what's left over this is x to the fourth this is x cubed this is x squared we have 1x squared plus 0x you don't have to write that down plus three plus no remainder this is our final factor okay from this one this tells us that x plus one is a factor of the original polynomial this zero tells us that x plus two oh x minus two sorry x minus two is a zero of the given polynomial so this is a factor this is a factor this is the last factor so this times this times this equals that so find the zeros well we take the factors this is a quadratic is it well let's factor that x squared plus three well that doesn't factor let's use the quadratic formula no let's not use quadratic formula let's solve by taking square roots okay we only have one x here so let's solve by taking square roots so we get x squared equals a negative three square root of both sides is plus and minus the square root of negative three which is plus and minus i root 3.

[00:27:11]so that is a complex number 0 plus or minus i root 3. sometimes they ask you to write your answers in a plus bi form so you put a zero there so determine the zeros well we knew there were going to be four right you knew they were going to be four so negative one two and positive i root three and negative i root three there are your four zeros two of them happen to be real and two of them happen to be imaginary okay but there's four all together all right let's look at at b here okay let's look at the problem with x to the fifth now remember we couldn't even use p over q's on this one we could do it on this one we could have done checked our p over q's and found that these worked but again that's a lot of work this one we can't use our p's over q's even if i asked you to but i could factor out an x squared and that would give me x to the third plus 5 x to the second minus eight x plus forty so now i know that this is a double root zero is a double root and i have three more to find zero is a double root and i have three more to find so this tells me that x equals zero with a multiplicity of two right double multiplicity comes from there so that's that counts for two zeros okay so i need to find three more well i'm going to tell you that um you could take this cubic and graph in the calculator and find one of the roots but i'm also going to tell you that there is a roots at x equals negative five so that's the one that's given okay so we're going to take that negative five and we put it into here but that's only going to give us an x to the fourth so i want to use this one okay if it's a factor of of this it's a factor of that as well so one x cubed five x squared negative eight and oh this is a negative forty now has negative forty negative forty so we have multiply zero again zero we get negative eight we get positive forty again this tells us that x minus five is a factor which means that negative five is a zero so what's left over again this is a cubic so it's going to be x squared plus zero x minus eight is another factor so if i set that factor equal to zero i get x squared equals eight i get x equals plus and minus the square root of eight which is four times two or plus and minus two square roots of two so what are my five zeros well i have zero with a multiplicity of two that's two there's three there's four and there's five five answers all together for fifth degree polynomial now let's look these are real these are not imaginary these are real these are this is real and this is real so all five of them are real and again this is these are conjugates this is zero plus two square roots of two and zero minus two square roots of two so i'm gonna reiterate if i told you that two square root of two was a zero then you need to know that negative two square roots of two is a zero and vice versa okay all right let's try this one let's not look let's not worry about this one this one's a little little too big let's try this one okay um fourth degree polynomial i'm going to tell you that it's given that you have a root at x equals um [Music] negative four so go ahead and see if you can find all zeros go ahead and stop the video and try it on your own try on your own okay now try it on your own all right all right check your answer now if you didn't stop the video then hopefully you can do it on your own you need to practice so here you should end up with negative 8 negative 12 40 and 100. if you did it correctly this is what your synthetic division would look like again i know that i'm to have a 0 because i told you that that was a root so if you don't get a 0 you did something wrong and just again to reiterate if negative 4 is a 0 then x plus 4 is a factor because they might ask you to put it in factored form okay be able to put it all together okay so what does that give us that gives us a third degree polynomial right so that's 2x cubes so i need another zero i need another zero i have to find it on a calculator or i have to use my rational root theorem my p's over my q's and plug them in to see which one actually works but i'll just give you another one this is going to be given 5 halves so we're going to use our new reduced polynomial and we're gonna do something so multiply the twos cancel add it's gonna be four times five add that's going to be 5 times 5 and again i knew i was now what factor comes from five halves just in case they ask you 2 x minus 5. now it's not very i'm just reiterating the fact that if you know a root or you know a zero you should be able to give me a factor and if i give you a factor you should be able to give me a zero okay right now we're just trying to find all of the zeros roots zeros x-intercepts all the same answer so then i get this quadratic two x squared plus eight x plus ten but we know that we have to divide that by two right so it's actually x squared plus eight oops plus four x plus five would be the remaining factor so if i asked you so far we have x squared plus four x plus five so this times this times this gives you the original polynomial okay that is a true statement all right so setting this equal to zero is the same thing as setting this one equal to zero this almost looks like factors to find the two remaining roots it looks like it factors but it doesn't multiples of five that total four not not going to happen okay so how do you solve the quadratic yeah quadratic formula good thing that we didn't use this space over here because i'm going to put it over here so x equals the opposite of b plus or minus the square root of b squared minus 4 times a times c all divided by two times a so that's 16 minus 20 that's a negative four so that's a negative four plus or minus uh square root of negative four two i now they can all be reduced by 2. this is a common denominator for both terms so you can reduce both terms and get negative 2 plus or minus i so here's your conjugate again negative 2 plus i is a root so negative 2 minus i is a root they come in pairs so that's two roots that's three that's four there's your total of four roots now again two are imaginary two are real okay imaginary roots are never singletons they're they're they always come in pairs always okay continuing on determine the remaining zeros of the polynomial given one of the zeros okay so this is kind of what i've been giving you um but they're giving you an imaginary zero but we need at least two zeros because i need to get this down to a quadratic which i know that i can solve so are we gonna pull out our graphing calculators to find one of them no we're not give me another zero 3 plus 2i comes with 3 minus 2i okay so those are the two zeros based on the given one so what are what are the other two well we're not going to use synthetic division because that just wouldn't be fun at all and [Music] you can't even do it with the the uh with the imaginaries so the only other way to get the other two is to use the graphing calculator okay this one it comes from this conjugate the other two from the calculator let's take a look go over finding zeros on your calculator not just not using the trace feature minus 13 [Music] x to the third plus 61 x squared minus 127 x plus 78 i hope i type that in right okay so we have two more here and here now i'm just going to show you that on a table you can find them but they're not always on the table okay they're right here six and one from the calculator but what happens if they're not on the table you have to know how to do um whoops you have to know how to be able to find these because they're only on the table if they're integers but if they're decimals of some sort they're not going to be in the table so let's go over finding zeros on the calculator we're going to graph it and we're going to calculate number two we're going to calculate a zero and it's going to say left bound well i need to find my little cursor where did that go where am i moving my table calculate a zero i'm at x equals negative two so i gotta come back down here now there we go there it is okay my little cursor here okay now if you look at if you think of the graph you think of the x-intercept this point is on the left side of the x-intercept this one right here so that's as close as i can get unless i zoom in so i'm going to press enter and it's going to ask me to be on the right side i'm doing this 0. so then i'm going to move my cursor so if it's below if you think about what that is going to look like again i'll draw it here for you might be if i'm up here i'm on the left side if i'm down here on the graph i'm on the right side that's what left bound right bound means so then it asks me well guess where you think the zero is that doesn't matter just press enter again and it gives me a 0 at x equals 1. so i can only do 1 0 at a time so then i'm going to do the other calculate the other zero i'm going to go to the other side where are you there it is okay now it asks me for the left bound so i need my again if i think my zero is somewhere around here anything on the graph down here is falling on the left side of the zero so i'm on the left side of the zero and then anything up here is falling on the right side of the zero that's why it says right bounds and i need to be up here which i can't see but i know i'm up there yes just press enter and there and it gives you this answer right here okay so this one was a one and this one was a six now again you have you have to do that if it's not in the table do not use your trace feature there's a trace on here somewhere i don't even know where but i don't use it because it's not accurate okay so from the calculator we get one and six from the conjugate we get this one okay so the zeros are three plus and minus two i one and six two real to imaginary all right says use your calculator to answer the following questions about the given polynomial function what are the possible rational roots okay well that's not a calculator question so again let's go over our p's oh boy factors of 88 uh one two three doesn't go in there how about four yes five six seven eight goes in there um nine ten eleven goes into eighty-eight okay um what would be the next one so it would be one i can do the start from the outside it'd be 1 and 88 it would be 2 and 44 right it would be 4 times 22 and it would be 8 times eleven okay so these two go together and there will be four times twenty-two two times forty-four and one times eighty-eight so there are all your factors of 88 now we're gonna do all of our factors of q which is oh just a one so we like that so what are our possible rational roots it's our p's divided by our q's so possible rational roots are basically all of these divided by one which gives you the same thing plus or minus one plus and minus two plus or minus four eight eleven twenty-two forty-four and eighty-eight okay what are the actual rational roots the actual rational roots so how many of these that's one two three four five six seven eight times two that's 16 actual rational roots possible so plug in one plug in negative one plug in two plug in negative two plug in four plug in negative four plug them all in and see which one gives you a zero no i'm not going to make you do that now i might make you do that later but i'm not going to make you plug in that many but know that if you plug in a rational root into the polynomial and you get zero as your y value it's an actual root how do you think we're going to get which one of these work how about we just use the calculator so once you go ahead and try and find them on your calculator stop the video if you have to coming back you should have seen that four and 11 from your graphing calculator again all this rational root stuff was before the graphing calculator and they'd actually have to plug every single one of those back in to see which gave them a zero so you guys are lucky but how many real roots are we supposed to have i have two actual rational roots but how many real roots am i supposed to have well i'm supposed to have a total of four roots they don't necessarily have to be real though i have to have to have a total of four i don't like that that says real you go back here n groups not necessarily real roots so let's take that off how many roots they don't all have to be real but they have to total four so that's the type of hair okay so how many irrational roots do you think you're going to have how many complex roots will you have in this problem well i have two rational roots so i have two reals here and i have need a total of four so that means that there must be two irrational okay okay now they could be imaginary they don't have to be irrational but maybe that's why it says use your calculator because use your calculator so when we graph this oh okay when we graph this on our calculator you're gonna see that this graph crosses the x-axis four times okay that's why it says four real roots but just so you know the only reason that you know that there's four real roots is because there were four zeros on the graph okay but later on i don't want you to be confused because there has to be four total but they could be um complex we know that real roots we can actually see in imaginary we can't so go ahead and graph this it'll cross the x-axis four times but we only got two of them from here how are we going to find the irrational roots well look at that use the zero feature on your calculator that's what i just showed you how to do just showed you how to find the zero so why don't you practice it on this one pause the video and come back okay so if you rounded to the second decimal two decimal places that would be the hundredth you should get a negative one point four and a positive one point four one okay so i know that there's four real from the graphing calculator because it crossed four times on the graph now if you couldn't see all four um maybe you need to check your window so you can see all four if you do the standard window it's a 10 by 10 so you might not be able to see this one so you might have to zoom out okay i saw across the x-axis four times so i know there's four reals two of them are rational two of them came from this list so that means the other two have to be irrational and there's zero imaginary well how do i know they're zero imaginary because i already have four one two three four so i already have four roots i can't have any imaginary now let's say that you saw this graph cross the x-axis twice and that's it then you would have two real roots and there'd have to be two imaginary roots because remember it has to total four okay rational irrational are both reals okay so let's look at this last example number six sketch a possible graph with the following conditions you have fourth degree polynomial with a positive leading coefficient two distinct negative real zeros greater than negative five and one positive real zero that's less than four with a multiplicity of two all right why don't you pause the video and come back after you sketch these [Music] all right are you sure you did it okay a multiplicity uh of two one positive real zero less than four with a multiplicity of so anywhere in here it's gonna bounce okay two distinct negative distinct means two different ones well greater than negative five is this way and they're negative so one here and one here doesn't really matter where okay fourth degree polynomial positive leading coefficient we know it's going to bounce here and we know that the end has to open up so it's going to do this and then it's going to go here and up that way again these are just estimates we don't know just nice smooth curves okay i don't care how high you go just make sure it's a nice smooth symmetric curve all right and then this one hopefully you tried it when i told you to you're gonna have one less than three you're gonna have two over here um the left side should be up this is a triple so you have a cubic behavior here okay i'm coming back up through here and back down for your odd power your fifth degree polynomial with a negative leading coefficient so that's kind of a review from before okay that's two five two five which is covering all of the zeros of a polynomial function they could be rational they could be irrational those are both reals and they could also be complex or imaginary totaling the degree is the fundamental theorem of algebra