here are the top 10 things you need to know about polinomial [Music] functions must know number one what is a polinomial function let's start with the definition a polinomial function is a sum of terms where each of the terms is a power of X multiplied by a coefficient and the general structure of the equation of any polinomial function looks like this now this structure looks difficult but if I show you a sample equation you'll see that polinomial functions are actually quite simple within each of the terms of our polinomial function there is a power of X where X is our variable which I can see over in this sample equation and then multiplied by each of those powers of X we have these coefficients which are represented by the parameter a and those coefficients can be any constant value they can be any real number or even any complex number and the exponents on the X's that go from n and then keep decreasing until you get to one and then finally an x to the 0o those exponents have to be non- negative integer values and you can see over in our sample function the exponents on the X's are all non- negative integer values and then because this n represents the highest exponent On Any X we know that this term would be the highest degree term it would be degree n making the whole degree of the polinomial degree n so looking at this sample equation the highest exponent is four which means the degree of that whole polinomial is degree 4 and then another important characteristic of any polinomial function is the value of what we call its leading coefficient which is the coefficient of the highest power of X so over at our sample function this is the highest degree term and its coefficient is -2 so the leading coefficient of that function is -2 and those two pieces of information actually tell us a lot about what the graph of this function will look like and we'll look at that through the next few sections and then the last part of a polinomial function we should talk about is the last term which notice does not have a variable x with it which means this last term is constant and graphically that would show up as the Y intercept of the function and one other thing I want to note is that all polinomial functions are continuous they don't have any restrictions on their domain and if I were to show you the graph of this function you'd see that what we have is a continuous function with no restrictions on the domain and now let's analyze a quick table of values for this function I'll choose X values between -3 and 3 and then subbing those X values in for the x's in my f x function I could calculate the Y values and now I want to calculate the finite differences between each pair of consecutive yv values within each pair I want to do the bottom number minus the top number so -10 minus -15 is 105 and then for the next pair 5 - -10 is 15 and so on now these are what are called the first finite differences they're not constant so that means my polinomial function is not degree 1 so I can then calculate the second column of finite differences by finding the difference in each pair of these values still doing the bottom one minus the top one the second finite differences are not constant meaning it's not a degree 2 polinomial function but I'll calculate the third finite differences and finally the fourth finite differences so that I could see for a degree 4 polinomial function the fourth column of finite differences are going to be constant so a general conclusion would be that the nth differences of a polinomial function are [Music] constant must know number two end behavior in this section you're going to learn how to describe how a function behaves at its extremes so basically where does a function start and where does it finish and before we can do that let me remind you about how we number the quadrants of a cartisian grid starting in the top right we have quadrant one and then we rotate counterclockwise to quadrant 2 3 and four I'll shrink this and move this out of the way so we can have it as a reference as we go through this section now there are four types of n behaviors that a polinomial function could exhibit now let's say we have a polinomial function that is an even degree and has a positive leading coefficient I'll short form leading coefficient with the letters LC if the polom function has those two properties I know it's end behavior meaning I know where the function starts it would start up in quadrant number two and it would finish in quadrant number one now there are lots of different shapes that it can make between those end behaviors but if these two criteria are met I know that this is where it starts and this is where it finishes so we could say its end behavior is from Quadrant 2 to quadrant 1 or another way to describe that is using an Infinity notation we could say as X goes to negative Infinity that means as we move forever to the left I see that this function its y value would be going up to positive infinity and as X goes to positive Infinity that means as the function moves to the right what's happening to the Y values well they're going up again they're going up to positive Infinity so these are two different ways of describing a function's end behavior so what would happen if we still have an even degree function but its leading coefficient is now negative that would vertically reflect a polinomial function so that it's now starting down in quadrant 3 and finishing down in quadrant number four so we could communicate that end behavior two different ways we could say the function goes from Quadrant 3 to Quadrant 4 we always State these quadrants in an order so that it goes from where the function starts its leftmost quadrant to where it finishes its rightmost quadrant or we could describe it using Infinity notation by describing what happens to the Y values as X goes to negative infinity and also as X goes to positive Infinity as X goes to negative Infinity that means as we move to the left along the function the Y values are going down to negative infinity and as X goes to positive Infinity so as we move along the function to the right the Y values of the function are going down to negative Infinity so for any even degree polinomial function what you're going to find is that the end behaviors are the same meaning that both sides are pointing up if we have a positive leading coefficient or both sides are pointing down if we have a negative leading coefficient but what happens if we have an odd degree polinomial function let's start with an odd degree and a positive leading coefficient if we have that the function is going to start down in quadrant 3 and finish up in quadrant 1 so as X goes to negative Infinity as we look at the function's behavior as we move to the left the function is going down to negative Infinity but as the function moves to the right we notice that it starts going up towards positive infinity and our last scenario is if we have an odd degree polinomial function but we vertically reflect it by making the leading coefficient negative that would make it go from Quadrant number two down to quadrant number four and I'll quickly write this in Infinity notation as well notice for any odd deegree polinomial function the N behaviors are opposite one side's going up and the other side's going down the Y values are going to negative Infinity on one side and to positive Infinity on the other side and now let me make some room so we can do a couple examples where we test out if you're able to State the end behavior of polinomial functions Let's do an example where we State the end behavior of each of the following part A we have the graph of a function and I can clearly see from the graph that its left side starts in quadrant number two and then its right side finishes down here in quadrant number four so this function's end behavior is from Quadrant 2 to Quadrant 4 and I don't see its equation but because of that end behavior I would actually know a little bit about its equation because it has this n Behavior I know that it is an odd deegree polinomial function with a negative leading coefficient and in Part B we just have the equation we don't have the graph but from the equation if I look at the highest degree term I see that it is degree 5 which is an odd degree and the coefficient of the highest degree term its leading coefficient 3 is positive so because of those two criteria I know it as this scenario where the function starts in quadrant 3 and finishes in quadrant 1 and then for part C here's my highest degree term its degree is four which is an even degree and the coefficient of that highest degree term the leading coefficient is negative a half so the leading coefficient is negative because of those two pieces of information I know we have this scenario where it goes goes from Quadrant 3 to Quadrant 4 and before we move on one last tip for you if you look at the right side of any function if the right side is pointing up you know you have a positive leading coefficient and if the right side is pointing down you know you have a negative leading coefficient and if both sides are pointing in the same direction either both up or both down it's even degree but if both sides are pointing in opposite directions it's odd degree must know number three x intercepts and turning points a degree n polinomial function can have at most n x intercepts now there's something called the fundamental theorem of algebra which tells us that a polom function of degree n has exactly n roots or n zeros so why doesn't it have exactly NX intercepts well there might be fewer than NX intercepts if there are repeated or non-real Roots right only the real roots or real zeros of a polinomial correspond to an X intercept and let me give you a quick demonstration of the possibilities for the number of X intercepts let's say a degree 3 polom function could have a degree 3 polinomial function could look like this where there are 1 2 3x intercepts that's the most it could have but if it has a repeated root you'll see the shape at an X intercept change so it could look like this where there are only one two x intercepts or there could be a pair of imaginary Roots which would make it only have one x intercept like this but having zero X intercepts wouldn't be an option and actually it's not an option for any odd deegree polinomial because an odd deegree polinomial starts and finishes on opposite sides of the x axis it has to cross at least once so let's add that information to our table as well and now let me erase this graph and Shrink this writing here so we have room to talk about turning points a degree n polinomial function can have at most n minus one turning points so for example a degree 5 polinomial function could have at most 1 2 3 four turning points that's the most it could have could a degree five function have three turning points let's think about that I know degree 5 is an odd degree so it would start and finish on opposite sides of the xais if I were to only turn three times one time two times three times notice that the function is heading in the wrong direction that would have the wrong end behavior so what I'm trying to demonstrate to you is an odd degree polinomial function has to have an even number of turning points so an odd deegree polinomial could have 0 2 4 or so on up to whatever nus1 is equal to those are the possibilities for the number of turning points it could have and then similarly an even degree polinomial has to have an odd number of turning points so an even degree polinomial could have 1 3 5 or so on up to n minus one turning points and now let me make some room and let's do an example where we look at some equations and graphs and look at how many X intercepts some turning points they have the first example we have the graph of a function and we're just looking for how many turning points does it have how many X intercepts and what is the degree of the polinomial a turning point is a place where the function switches between increasing or decreasing or vice versa and I see that happening in 1 2 3 four places on this polinomial so it has four turning points how many X intercepts does it have I see it touch the the X AIS 1 2 3 times there are 3 x intercepts so what degree could this polom be well it has 3x intercepts so could it be degree 3 no it couldn't be degree 3 because a degree 3 could have at most two turning points could it be degree four well a degree four could have at most three turning points so no and also I can see based on this end behavior that this is odd degree so the least possible degree this function could be is degree 5 now let's look at example two where we have the equation of a degree for polinomial function an even degree polinomial function has to have an odd number of turning points so the possible number of turning points this function could have well I I don't have the graph of it so I don't know exactly how many turning points it could have but I know the options for what it could have are either one or three turning points and the possible number of X intercepts well the most it could have is equal to the degree or any number less so it could have 0 1 2 3 or 4 x intercepts notice 0 is a possibility for the number of X intercepts of an even degree polinomial but it wouldn't be an option for an odd degree so in example three where we have a degree 7 polinomial function the possible number of turning points since it's odd degree would be any even number less than seven so 0 2 4 or 6 and the possible number of X intercepts at most to seven or any number less but we can't include zero for an odd degree polom so one all the way up to [Music] 7 must know number four even and odd symmetry in this section I'll explain to you the characteristics of even functions and odd functions let's start with even functions we've talked about even degree functions but not all even degree functions can be further classified as an even function an even degree polinomial function is an even function if its graph has line symmetry over the y- AIS and in its equation every single term is an even degree term now because of this line symmetry over the Y AIS there is an algebraic rule that would hold true for any even function the rule states that the value of FX would be equal to the value of f ATX and I'll be able to explain that more to you as I show you an example of an even function so here's the function we're going to work with this polinomial is even degree because the highest exponent is four but it can also be further classified as an even function because every single term is even degree we have a degree four term a degree 2 term and a degree zero term those are all even which means if we look at its graph we would see that there is line symmetry over the y- AIS which means algebraically this property must be true so for example if I were to evaluate F at 2 subbing this two in for the X's into my equation I would figure out that F at 2 is equal to two that means this point right here is the point 22 because this is an even function I know it's got line symmetry over the y- AIS so moving two units to the right my point is at a height of two and because this is an even function I know this property holds true which tells me that if I were to move two units to the left the height would also be at two and if you were to evaluate F at -2 you would get an answer of two as well which means this point is the point -2 two now before I move on make sure you understand that not all even degree fun functions are even functions for example if I took this function here and I threw another term into it I've added this x Cub term now this equation has a mix of even and odd degree terms the yellow terms are all even degree but this blue term is odd degree that means that this even degree function is not an even function so let me take this term back out and then we'll move on to odd functions for odd functions an odd degree polinomial function is an odd function if its graph has point or rotational symmetry at the origin and in its equation all the terms are odd degree and algebraically it would follow the rule that says the value of f ATX is not equal to F ATX but instead equal to the negative of FX and now let me clarify all of this with an example for for this equation I see the highest exponent is three so this is an odd degree polinomial function but what also makes it be further classified as an odd function is that every single term that makes up this function is odd degree we have a degree three term and a degree one term those are both odd values so I know that its graph would have point or what sometimes called rotational symmetry about the origin and let me try and show you what that looks like so pay attention as I rotate this function 180° around the origin notice that it lines back up with the original function exactly that's what point symmetry is and because it's an odd function I know this rule would hold true and let me demonstrate that to you if I were to evaluate F at -3 subbing that -3 into my function for X I would calculate that the Y value is -5 so let me put that approximately on my graph moving 3 to the left of the origin my point has a height of -5 if this was an even function I would expect moving three to the right my point would also have a height of15 but the function is not down here it's actually up here when I evaluate F at three I actually get positive 15 so the 3 postive 15 is on the graph now look at the relationship between the y values of these answers this one's -5 this one's POS 15 so multiplying this one by -1 would make it equal to this one and that's exactly why this negative is in our rule so to make these values equal I would just multiply this function by -1 so F of 3 would be equal to [Music] -15 must know number five the factor form equation of a polinomial function the general structure of a factored form polinomial equation looks like this and what makes this a factored form equation is the equation is just a product of a bunch of factors this factor of a is just a nonzero constant these R parameters are just the roots of the equation because subbing in any of the values that you would see here for R subbing those in for any of the the X's would make one of the factors become zero which makes the whole product become zero so that's why they're the roots or the zeros of the function and any of those R values that are real numbers they're going to show up as X intercepts on the graph the exponents that we see on the factors that contain an X are the multiplicities or the orders of the roots so basically these exponents are just a way of describing how many times the root of that factor is repeated and the degree of a factored form polinomial function is equal to the number of roots or number of Zer the equation has counting their multiplicities let me give you a couple quick examples of that to make sure you understand it now this Factor would be zero if x was 2 which means two is a root of this equation and since this factor is repeated three times we say the root of two has a multiplicity of three the root of one has a multiplicity of 2 and this Factor would be 0 if x was -4 so the root of -4 has a multiplicity of 1 to find the degree of this polinomial function we have to add up all the roots counting their multiplicities so we would have to do 3 + 2 + 1 which is 6 so this is a degree 6 polom let me make a little more challenging one now this polom is in factored form it's a product of these factors since the degree is equal to the number of roots counting their multiplicities well I know this Factor would be zero if x was a half so a half is a root and it's repeated two times but this Factor doesn't have any real Roots that's okay it does have two imaginary Roots so we do include those when adding up the number of roots so this Factor has one root but it's repeated twice we count that as two roots and this Factor has two imaginary Roots so we count those as well so 2 + 2 is 4 so this is a degree 4 function and let me actually erase those examples we just did so that I have room to tell you about how the orders of the roots that cause the X intercepts affect the shape of the graph first of all it's worth writing down that all the real Roots correspond to the X intercepts any imaginary Roots don't show up as X intercepts on our graph and the Order of those real Roots affect the shape at the corresponding x intercept and then I'll draw a quick sketch of the shape of a couple different orders of X intercepts if a root of the equation is order one we would say that its corresponding x intercept is order one which means at that x intercept it's it's going to look exactly like a degree one power function which is just a straight line so it would go straight through an X intercept of order 1 but if the x intercept is order 2 that means at the x intercept it's going to look like a degree 2 power function which is a parabola so we know it would bounce off just like this and at an X intercept of order three it'll look like a degree 3 power function which makes this s shape through the x intercept and I should also mention that any even order higher than order two is going to have a similar shape to this at the x intercept where it just bounces off the x- axis but the higher the order is the flatter that it's going to get around the x intercept and the same thing for odd orders any odd order higher than order 3 is still going to make this s shape through the x intercept but the higher the order is the flatter that it's going to get around that x intercept and now let's try an example where I give you a factored form equation and we State a bunch of properties and do a rough sketch of its graph so for this polinomial function that's in factored form we're going to State a bunch of properties let's actually start with the X intercepts the X intercepts are the real roots of the equation so the real values of X that make the equation zero well a factored form equation would be zero if any of the factors were zero X - 1 would be0 if x was 1 and because the exponent on the factor that creates that x intercept is one we would say that x intercept is order one this Factor could be zero as well and if you needed to you could make yourself a little equation solve the equation for how it could be zero and you would figure out it would be zero if x was -5/2 which is just -22 and the factor that creates that zero is repeated three times which means that x intercept is order three and this Factor would be zero if x was two and that factor is repeated two times so the x intercept of two is order 2 now let's find the degree the degree is equal to the number of roots including their orders this equation had three Roots if I add their orders together 1 + 3 + 2 I get 6 which means the degree of this equation is 6 now to find the leading coefficient that means if we were to put this into standard form by fully expanding it what would be the coefficient of the X to the 6 term we could find that coefficient without fully expanding just by multiplying first of all the constant factor with the coefficients of all of the x's in any of the factors remembering to include the exponent that is on those coefficients so I would do one qu * 1 * 2 cubed * 1^ 2 and evaluating this you would get an answer of two now end behavior we have an even degree with a positive leading coefficient I know that goes from Quadrant 2 to quadrant 1 and we've already found the X intercepts with their orders let me change the color of this to Yellow to keep it consistent and now let's calculate the Y intercept I know wherever crosses the Y AIS the x value is going to be zero so to find the Y intercept all I have to do is evaluate P at 0 and if you do that you get -125 so the Y intercept is at the point 0 -125 and with all this information we could sketch a fairly accurate graph of this function I'll start by roughly plotting the X intercepts and then use my n Behavior which tells me I need to start in quadrant 2 and finish in quadrant one and at each x intercept make the shape that the corresponding order tells me it has to make so for example at my first x intercept of -2 and A2 that's order 3 so I have to make this s shape and then I'll continue the function and as I approach my next X intercept this x intercept of one is order one so it has to go straight through and then my last x intercept of two is order 2 so it has to make this Parabola shape at that x intercept so there we have a fairly accurate graph of this polom [Music] [Applause] function must know number six how to find the equation of a polinomial function from its graph let's find the equation of this polinomial function in factored form to write the equation in factored form we'll have to give the function a name we'll call it p x and that's equal to there's going to be some constant Factor we'll call it K being multiplied by a bunch of factors that contain an X and I'm going to leave room for three factors and the reason why is because on the function I see 1 2 3 x intercepts the first x intercept is at -4 and I notice at that x intercept it makes this Parabola shape where it bounces off the xais so I know that x intercept must be order two and then the X intercepts at both 0 and three the function just passes straight through those X intercepts so they're both order one and now we have to use each of those X intercepts to figure out what factors go in the equation each of these X intercepts would be a zero of one of the factors that are in the equation so let's start by figuring out what Factor would be zero if x was -4 so to do that I'll start with my answer I know X is equal to -4 when some factor is set equal to zero to figure that out let's reset it back to zero by moving the -4 to the other side and I figure out if the factor was x + 4 equal to Z that would be true if x was -4 so x + 4 must be a factor in my equation and because it's order two I know I have to put an exponent of two on that factor and now I'll do that same process for both of these X intercepts I'll set them as the answers x = 0 and x = 3 and then figure out what factor has those as their zeros well for this one I'll move the three to the other side and I figure out if the equation was x - 3 equals 0 then the answer would be x = 3 so one of my factors must be X x - 3 and then this one is already set equal to zero so that factor must just be X and now hopefully you can see what we did is we created an equation in factored form where any of these values would make the whole equation be zero4 would make this Factor be zero zero would make that factor be zero and three would make that factor be zero and in factored form if you can make any of the factors become zero it makes the whole product BEC zero meaning they are X intercepts of the graph but there are an infinite number of polom equations that could have those factors containing those x's and create those same X intercepts we want the very specific version that passes through this point that we see here the 15 that point has an X and A Y value so what we can do in our equation we can set the X's equal to 1 and we can set the Y value equal to 5 and then solve for what value of K stretches our function to be able to go exactly through that point so I'll set y to 5 and then change all of the X's to one and now we just have to isolate for K so I have 5 = K * 5^ 2 which is 25 * 1 I don't have to write a factor of 1 * -2 on the right side of my equation I have K * 50 which I'll write as K and then to isolate K I would just have to divide both sides by -50 those cancel and I figure out that k equals 550 which could simplify to -1/ 10 and now I can write my final equation of the polinomial function p x equals my constant factor is -1 10 multiplied by all three of these factors that contain the X when you have a factor of X all by itself typically you write that first and then the other two factors and there's our final [Music] [Applause] answer must know number seven synthetic and long division these are two methods for dividing pols let's start with long division let's say we want to do 4xb + 9x - 12 / 2x + 1 to do this division we can set up our division table underneath the table we write the dividend and notice a couple things about that dividend that I wrote I added in a 0 x^2 as a placeholder because when we do this method of long division with pols we want to make sure that we have a place for every degree equal to and less than the degree of the polinomial so if the degree of the polinomial is three I'm going to need a column for degree 3 terms degree 2 terms degree 1 and degree 0 but we were missing degree 2 so I threw in a 0x2 there as a placeholder and the second thing I want you to notice is that we put the terms in descending order based on degree from the highest degree three down to the lowest degree term the degree zero term and then we write the divisor the 2x + 1 outside the table and our answer our quotient is going to go at the top of the table now the process for long division with pols goes as follows step one first term of the dividend divided by first term of the diviser so I have to do 4X Cub / 2x dividing the coefficients 4 and 2 gives me 2 and dividing the variable parts of those terms X Cub / x 1 the quotient rule tells me to keep the base of X and subtract their exponents 3 - 1 is 2 so I get 2x^2 as the quotient of those two terms and what I want to do now is multiply this term in the quotient by the entire divisor so the 2x^2 has to be multiplied by 2X and 1 and write the answers down here in the dividend lined up in columns with their like terms so 2x^2 * 2x is 4X cub and 2x^2 * 1 is 2x^2 notice how like ter terms terms with the same exponents on their variable are lined up with each other in columns and what we do is we subtract the like terms that are in our dividend the first pair of terms should cancel out if we've done things properly 4X Cub minus 4X Cub is 0 and 0 x^2 - 2x^2 is - 2x^2 we can then bring down the next term in our dividend we can actually bring down all of the terms that we have left in the dividend if we want but we're only going to need one more for the next step so I'll bring down the 9x for now and then we repeat the process we do the first term that's in the dividend so -2X ^2 and divide it by the first term of the divisor -2X ^2 / 2x ISX and then thatx gets multiplied by the whole divisor and then in the dividend we subtract our like terms we then bring down our last term of -12 and repeat the process one more time 10 x / 2x is 5 and then multiply this 5 by 2x + 1 to get 10 x + 5 and when subtracting like terms there all I'm left with is -2 - 5 which is -17 so that's what we call the remainder of this quotient once your dividend is a lower degree than your divisor you know that you're done the division process and what you have left in the dividend is the remainder so there are a couple different ways to express this answer I'm going to show you how to express this in what's called quotient form we could say that the original division question is equal to the quotient plus the remainder over the divisor and now let me make a bit of room and show you another method that is a little bit of a shortcut that only works in certain situations and this method is called synthetic division now synthetic division only works to divide a polom by a linear divisor of the format xus B and what that means is we can only divide by a polinomial that is degree one with a leading coefficient of one so for example we could do this division question using synthetic division our divisor is degree one with a leading coefficient of one so that's perfect for synthetic division and how a synthetic division works is we set up a table I take the coefficients of all of the terms in my dividend and I write them across the top and then I take the zero of my divisor and write it up here so what makes x + 3 be Z would be if x was -3 and then I'm going to write a multiplication operator here and an addition operator here so that I can remember what steps to follow when doing synthetic division step one you take the first number in your dividend and you rewrite it at the bottom here that's going to be the coefficient of the first term in our quotient and then to get the remaining coefficients we follow this process where we do -3 * 6 and write the answer in the next column -3 * 6 is -18 I then add the numbers in this column 13 + -18 is-5 and then just repeat the process -3 * -5 is 15 -34 + 15 is -19 -3 * 9 is 57 adding those numbers gives me 10 -3 * 10 is -30 and then in adding those numbers gives me -2 now the far right number is our remainder and then the rest of the numbers are the coefficients of the terms in our quotient starting from the right the first number is the coefficient of my degree zero term then my degree one term degree 2 and degree 3 term so this is my quotient and this is my remainder we could write that just like the last one in what's called quotient form or there's another way we could write it in its multiplication statement form and how that would look you would split up the original dividend into a product of the divisor and the quotient plus the [Music] [Applause] remainder must know number eight the remainder zero and Factor theorems in order to be able to use those tools of synthetic and long division to get a polinomial function into its factored form we're going to have to understand four theorems that I'm going to show you here the first one is called the remainder theorem the remainder theorem says if P of X is / xus B the remainder is equal to P at B so basically this is saying if we take the zero of the divisor right xus B would would be zero if x was B we take that and we sub it into our polinomial function 4X we get what the remainder is if we were to do the division so we could extend this a bit further if we divide it by axus B the remainder would be equal to Well we'd have to sub the zero of that divisor into p x and what makes axus B Be zero if we isolate for X we ax = B which means x = b a we would just take that sub it in for x and that would give us the remainder so it would be equal to P at B A and let's do an example to see how this works the example says what is the remainder when dividing this polinomial function by x + 1 to find the remainder I don't have to do the whole long or synthetic division I can just take the zero of the divisor well x + 1 would be0 if x was -1 and sub it into this function for X so I just have to evaluate what is p at1 replacing all the X's with1 I get an expression that if I were to evaluate it it tells me the remainder and evaluating this I figure out the remainder is -4 and I'll make some room and show you the factor theorem the factor theorem is actually just an application of the remainder theorem it tells us that xus b would be a factor of P of x if well for x - B to be a factor of P of X that means it divides evenly into it or another way of saying that if we were to do p ofx / x - B we should get a remainder of zero so how could we check if there's a remainder of zero we could take the zero of the divisor x - B would be Z if x was B and sub that into our polom function so x - B would be a factor of P of x if P at B was equal to zero and then we could extend that one step further and say axus B would be a factor of P of x if when dividing them we get a remainder of zero and we could check their remainder by taking the Z of ax minus B which is b a and suming it into our P of X function so that would be if P at B A was equal to zero and let's do an example to see how this works so this example says is x - 3 a factor of this polinomial it would be a factor if when I divided this polinomial by x - 3 I got a remainder of zero and I can check to see if it has a remainder of zero by just subbing in the zero of x - 3 well x - 3 would be zero if x was 3 so just take that value of three and evaluate the polinomial function at that value of three and when I evaluate this I get zero which means the remainder is zero which means x - 3 is a factor of the polinomial and now I'll make some more room and give you the last two theorems so the next theorems that I want to show you are the integral Zero Theorem and the Rational Zero Theorem the integral Zero Theorem states that if B is a zero of P of X then B is a factor of its constant term and the best way for me to explain that is with an example for this function p x this polom is degree 3 So based on the fundamental theorem of algebra I know there are three zeros to this equation but some of the zeros might be repeated or some of the zeros might not even be real but if any of the three Zer of this equation are integers I know that those integers would be factors of the constant term so the possible integer zeros of this equation are factors of -6 so what numbers divide evenly into six well it could be plus or minus 1 2 3 or six so the positive or the negative of any of those four numbers would divide evenly into our constant term of the polom which means if there are any integer values that make that polom be zero they will be found in this list of eight numbers what about the Rational Zero Theorem if B A is a zero of P of X then B is a factor of the constant term and a is a factor of the leading coefficient let me explain this in more detail with an example I'll make a little bit more room and then let's write a polinomial function if I was interested in what integers might make this function be zero I know the only integers that would make it be zero would be factors of the constant term but I could write a more comprehensive list of the possible zeros if I use the Rational Zero Theorem it would allow me to make a complete list of all rational numbers that might make it be equal to zero and I can do that by doing factors of the constant term so factors of ne6 are plus or minus 1 2 3 or 6 and dividing each of those by the factors of the leading coefficient the factors of two are plus or minus 1 and two so I have to take each of those four numbers and divide them by each of those two numbers and that will give me my whole list of possible rational Zer of that polom so I would have plus or minus I have to do this one divid by both one and two so I'll have 1 over 1 which is 1 and 1 over two then I need to do this two / both 1 and two 2 / 1 is 2 and 2id 2 is 1 but I already have that written here here and then this three gets divided by both 1 and two and then this six gets divided by both 1 and two to give me six and three but I already have three written so if there are any rational numbers that make this polinomial be zero they will be found in this list and now that you know all four of these theorems we can apply them to be able to solve some polinomial equations and inequalities in the last couple sections of this video [Music] [Applause] must know number nine solving polinomial equations let's solve these two equations starting with the equation on the left I want to know when is this degree 3 polom equal to zero based on the fundamental theorem of algebra I know this polinomial has three zeros some of those zeros might be integers some might be irrational numbers some might be complex numbers and some might even be repeated but in order to find them we want to try and get this polom into factored form but in order to do that I'm going to have to find one of the zeros of this equation so that I can decide what to divide the polinomial by to get it into factored form but I don't just have to guess and check any number that might make it be zero I know based on the integral Zero Theorem that if there is an integer that makes the polinomial be zero it would be a factor of the constant term so the possible integer zeros of this equation are factors of 10 which would be plus plus or minus 1 2 5 or 10 now up to three of these numbers might work I just have to find one of them that works so we just test until we find one that makes the polinomial be zero the first test I'll do I'll test one in the equation so I'll replace all the X's with one and if I evaluate this it actually does equal zero since one makes this polinomial be zero I know that one of its factors would have to be xus one and that's because this Factor would be zero if x was one so I can start the factoring process by rewriting this equation and then I can split that polom into two factors the first factor is xus one and then the second Factor would be whatever the quotient is when I divide that polinomial by xus one and I can do that division with either long division or synthetic division I'll quickly do a synthetic division off to the side by writing the coefficients of that polinomial up here and the0 of x -1 right here and then run the synthetic division algorithm which tells me to bring the first term down do 1 * 1 which is 1 and then add the numbers in this column to give me ne3 and then if I repeat that process until I'm done I get the following answer getting zero for the remainder verifies that x -1 is a factor of the polom and the other factor is the quotient X x^2 - 3x - 10 and now that I have the polinomial in factored form I have a product of two things equal to zero I know that product would be zero if either of these factors was equal to zero this Factor would be zero if x was one and to figure out when this Factor would be zero I could either use quadratic formula or I could continue factoring it so I think I'll continue factoring it I just have to find numbers that multiply to -10 and add to -3 the numbers that satisfy that product and sum are 5 and 2 so that means I could further factor that quadratic into x - 5 * x + 2 and now that the polinomial is fully factored the product would be zero if any of the three factors would be zero this Factor would be zero if x was 1 this one would be zero if x was 5 and this one would be zero if x was -2 so those are all answers to this equation I'll write them in ascending order and you could verify any of those answers just by checking them in the original equation to make sure it makes left side equal right side let's move on to the second equation the one on the right the first thing I noticed is that it's not set equal to Z it's set equal to -12 so we'll start by moving that -12 to the other side and now that the polinomial is set equal to zero we want to get the polom into factored form but for this one we actually don't have to do a test of the possible Zer and then do a synthetic division to get it into the factored form there's actually a shortcut for this one that's called factoring by grouping for factoring by grouping you look at the first two terms take out a common factor so I could take out an X2 from the first two terms and when I divide them both by x^2 I get x - 3 and then for the last two terms we take it a common factor as well from the last two terms I could common factor a-4 and when I divide them both by4 I get x - 3 again getting this binomial means that factoring by grouping is working in the first term there's an xus 3 and in the second term there's also an x - 3 so what we can do is we can common factor out that x- 3 and when I divide both of those terms by x - 3 it cancels them out and we're left with x^2 - 4 as the second factor and this x^2 - 4 is a difference of squares it's an x^ 2us a 2^ 2 so it would factor to x - 2 * x + 2 and now that this is fully factored the product would be zero if any of the three factors were zero this one would be zero if x was 3 this one if x was 2 and this one if x was -2 so I'll write all three of those answers in ascending [Music] [Applause] order must know number 10 solving polinomial inequalities when solving an inequality we want to start by moving all of the terms to one side so I'll move the terms from the right over to the left and now we're interested in when is this polom less than or equal to zero so graphically speaking that means when is it below the X AIS so to figure that out I'll sketch a graph of this that includes the proper X intercepts and end behavior and then I'll be able to answer the inequality I know the X intercepts correspond to the zeros of that polom and I can find those if I get it into factored form form and to get it into factored form I'm going to first have to test and find one number that makes it be zero so that I can decide what to divide it by to get it into factored form based on the Rational Zero Theorem I know that if there's a rational number that makes this be zero it's going to be a factor of the constant term divided by a factor of the leading coefficient the factors of six are plus or minus 1 2 3 or 6 and the factors of five are just Plus or- 1 or 5 so to make my whole list of possible rational Zer I do each of these four numbers divided by each of these two numbers so of these 16 numbers that are here because this is degree 3 I know that up to three of these could make it be zero I just have to find one of them that works so I'll do a test and I'll show you that1 actually makes it be zero which means that a factor of the polom must be X x + 1 because x + 1 is 0 when X is1 so I can now split this polom into two factors a factor of x + 1 and then some other factor that is the quotient of this polom and x + one so I'll do that Division I can do that using synthetic division I'll do that off to the side here I'll use the coefficients of that polinomial and then the zero of the divisor is1 and then running the synthetic division I see that the quotient is 5x^2 - 17x + 6 so I can write that here as my second factor of the polinomial and then I want to factor the quadratic further by finding numbers that have a product of 5 * 6 so a product of 30 and a sum of -7 the numbers that work are -15 and -2 so I can split this middle term into -5x - 2x and then Factor this quadratic by grouping I'll leave the x + 1 but then I take a common factor from the first two terms I'll take out a 5x and then from the last two terms I could common factor a -2 I'll leave the x + 1 again and then common factor this x - 3 from both terms leaving me with 5x - 2 now that it's fully factored I could easily see when this polinomial is equal to zero it would be zero if x was either 1 3 or 2 over5 which means the X intercepts of this polom would be at -1 2 over 5 is 0.4 or 3 so now that we have the X intercepts of this polom we should be able to sketch a graph using its proper end behavior to figure out when it's less than or equal to zero but what is the end behavior of an odd degree polinomial function that has a positive leading coefficient remember that would go from Quadrant 3 to quadrant one so I'm going to erase what I've got over here so that I have some room to sketch My Graph and answer my inequality I'll make a cartisian grid I will plot my X intercepts roughly and then using this n Behavior I know it has to start down here in quadrant 3 and finish up here in quadrant 1 which means it's going to look something like this and we want to know when is this polinomial less than or equal to zero so basically when is it below the xais it's below the xais in this interval and in this interval so let's describe those two intervals as our answer to the polinomial inequality this section of the function is when X is less than or equal to1 and this section of the function is when X is between 0.4 and 3 so we could describe those intervals with inequality symbols like this or we could write it in interval notation by saying X is an element of numbers between infinity and -1 or from 0.4 to 3 and notice I used square brackets at each of the X intercepts because at those X intercepts this polom is equal to zero and the original inequality States we want to include when it's equal to zero so hopefully you learned a lot about polom functions in this video let me know what you want to top 10 about next Jensen move