 one that's you know the remainder doesn't equal zero. Now all I'm gonna do is use the same polynomial on the top, the same numerator and in the bottom all I'm gonna do is change it from x minus 2 to x plus 2 so it's not a factor of the the top right the top quadratic equation. So your question would be or the expression that would give you is the yellow guy so our top divided by that so they would say simplify this expression right or do long division on this expression. If they said simplify you would factor the top and see if the top factors factors of this guy if any of them cancel out this guy then you're done. Now we already know that the factors of the top guy you know there is none of the factors of the top guy is equal to x plus 2 is equal to x minus 2 so they can't cancel each other off. So we can't simplify this just by factoring the top because factors of the top does not equal the factors of the bottom that means the remainder is not equal to zero. When we do the long division we're gonna find out the remainder does not equal zero right which means the bottom guy is not a factor of the top guy. So if they say simplify this or do this long division we're gonna have to do the long division because we don't know synthetic division yet but we will soon okay. So let's do the long division. Long division what we're gonna ask ourselves is this what are we gonna multiply x by to give us negative 3x squared well we're gonna multiply x by negative 3x right because negative 3x times x gives us negative 3x squared and then we're gonna see what else what the rest of it looks like right because the only thing we care about during the long division process is matching the first number here with the first number here and one thing to keep in mind and again this should be a given by now is whenever you're giving polynomials you have to have an in descending order right your highest order goes first and you go down from there. So if they give you you know an expression like this that's all mixed up you know you got x to the third and x to the 5 plus x to the 2 or whatever it is they're all in you know not in the right order descending order you have to put in descending order. Any polynomials you get and that's a given rule any polynomials you get any functions you get you put them in descending order okay and that's a given so if they try to trick you by mixing your expressions up the orders up take the x's and put them in descending order okay and these are single variable functions that we're really concerned about right now multivariate variable functions will get into you know later. So what are you going to multiply x by to give you negative 3x squared you're going to multiply it by negative 3x so you put your negative 3x on top of this each term follows each other you should keep things in order. So negative 3x goes up here and multiplies this guy and multiplies that guy. Negative 3x times x is going to be negative 3x squared negative 3x times 2 is going to be negative 6. Now again this is subtracting the bottom from the top so be minus negative 3x squared minus that that guy minus negative 3x squared and 5x minus negative 6x right now I don't like all those minus minuses all over place so I'm going to change the sign of that guy change the sign of that guy and I'm going to add them. They were both negative so they both turn into positive and I'm going to add them so negative 3x squared plus 3x squared is going to be 0 5x plus 6x is going to be 11x. That guy cancels that guy so usually what I end up doing is I don't put 0 down here just kill him right and 5x plus 6x is going to be 11x and then we're going to grab this guy and bring it down. So we've got positive 2 coming down and always remember the sign in front of the number always goes with the number right so positive 2 comes down so again the question that we're going to be asking ourselves is what do we multiply x by to give us 11x we don't care about this guy right now this guy's going to work itself out right if it doesn't work itself out it's just going to have a remainder so what do you multiply x by to give you 11x well you're going to multiply by 11 11 times x is going to be 11x 11 times 2 is going to be 22. Change the signs and add them so that becomes negative 11x minus 22 11x minus 11x they kill each other 2 minus 22 is negative 20. Now you can stop doing the long division as soon as the power over here is less than the power up here on your first term for your divisor for whatever's in the denominator you're done factoring right or you're done doing the long division so if this was an x squared right if this was x squared plus 2 and you know you went through your thing and you had an x term in the bottom here you would have to stop because if this was an x squared and you had an x over here this power over here is bigger than this power over here you can't go any further right so as soon as the power on the first term in the denominator becomes bigger than the power of wherever you've gotten to in the long division you're done right what we're going to do right now is write this in the form of the division statement okay so I'm gonna forget about writing the x's it's just going to be p is equal to q times d plus r so p is the top our numerator q is our quotient this guy up here d is the divisor and r is our remainder you see all that we sort of have to go on a sled because we're hitting this guy should have written down further down here right anyway so basically what we have is our product this guy is equal to that guy times that guy plus that guy so this the numerator is equal to negative 3x plus 11 times x plus 2 minus 20 right and what that means is if you foil this guy out and subtract it to 20 you would get this guy back so we broke down this guy into things multiplied together and you know whatever the remainder is so we try to factor it we basically wrote this guy in a different form and we got this guy right so both of these guys this guy this side is equal to this side okay now there there are reasons why we're gonna do this or there are reasons why we do this one of the ones just as a teaser what this means is if your x is equal to negative 2 because this would be x plus 2 is equal to 0 and you bring the negative 2 over when your x is equal to negative 2 y is equal to negative 20 when your x is equal to you bring the 11 over you set this equal to 0 and then do your division if you set x is equal to negative 11 over negative 3 which is 11 over 3 if you sub that into your original polynomial your y is going to be negative 20 so these guys are actually coordinate systems that we have there are two points on and this guy's a parabola okay so let's just let lay this out and we'll do the graph over here and you'll see where the points lie so if you actually rewrote these this guy the q and the d as x equaling something right so you took your quotient and set it you took your quotient and set it equal to 0 and solve for x and you took your divisor x plus 2 set it equal to 0 and solve for x what you would have is you would say x plus 2 is equal to 0 and solve for x so you got x is equal to negative 2 and you would set negative 3 x plus 11 equal to 0 and solve for your x and you got 11 over 3 if you took those x is equal to 11 over 3 and sub that into the top the numerator what that would give you is your remainder your y for a graph right so these are actual coordinates that you just figured out so when x is equal to negative 2 y is negative 20 when x is 11 divided by 3 y is negative 20 so if we put on a graph here we'll just put on a graph here in negative 20 well our x axis is going to be up here and we're going to graph it down here for a quadratic for a parabola when you got a negative sign negative sign here in the front in the highest order in front the x squared that means the parabola opens down so what we have for coordinates is these are two coordinates on the parabola negative 2 and negative 20 x is negative 2 y is negative 20 when x is 3 11 over 3 y is negative 20 so our parabola goes up and comes down again and those are two coordinates that we have negative 2 and negative 20 which is negative one is which one this one so that's negative 2 negative 20 and 11 over 3 is that guy right now we know by factoring that guy that it has factors right so if we're looking at that quadratic equation right there from the you know the quadratic formula if we look at the quadratic formula the discriminant that we've talked about in the quadratic formula section the discriminant of that is positive is greater than zero that means when you take the square root words x is equal to negative b plus or minus a square root of b squared minus 4 ac over 2 a b squared minus 4 ac that's positive greater than zero so you can take the square root of that so you're gonna actually have two x intercepts and we already knew that because we factored that guy right so we factored that we know what's got two factors that means it's got two x x intercepts that means the parabola crosses the x-axis and comes down and from before we know exactly what the factors of that thing are the factors of that thing are x is equal to 2 and x is equal to negative 1 over 3 so in the positive side this guy goes up and crosses at x is equal to 2 it hits it so there is another point that we know for the parabola and goes up the other one is x is equal to negative 1 over 3 so goes up here this guy goes up here I should have drawn a little bit closer and goes up here right another coordinate that we can find later on we'll get into it is the y-intercept and the way you find the y-intercept all you do is just set x is equal to zero and the y-intercept is just going to be 2 okay so anyone who's done graphing parabolas knows pretty much what I'm talking about hopefully knows what I'm talking about and if you don't hopefully this is a pretty good you know sort of a lesson of how you know what all these coordinates mean right so let's throw on our two factors up here our x-intercepts and graph that too so our parabola is going to look something like that hopefully you could draw it better than me because that should be you know looking like a parabola instead of going to like a little bulge so that's our parabola right and again all of this information we got to hear just by doing our long division our long division we got to the remainder knowing that there all the all the remainder is if there is a remainder it means just the y-corner when x is equal to whatever this is and x is equal to whatever that is when you set them equal to zero when you solve for them right so there's a lot of things going on here and you know usually when this is taught all they teach people is you know do the division do the long division you get to you get down to here and what does that mean right well that means that's the y when x is equal to either well for this guy anyway you set the denominator equal to zero you solve for x so x is equal to negative 2 so when x is equal to negative 2 y is y is negative 20 whatever the remainder is right and the quotient gives us another point now for parabolas there's an axis of symmetry going down here and we'll talk about a lot more about this there's a lot of things with with parabolas there's there's special things that you need to know about it right there's symmetry with it you know if you cut yourself down a middle hopefully you're this ear is the same distance from your center as the other ear right and there's symmetry in a lot of things in life and parabola parabolas are one of them there's symmetry to parabolas if you cut the parabola down straight down a middle it's symmetrical this side is symmetrical to the other side if you want to find the axis of symmetry for a parabola all you do is add that negative 2 plus 11 over 3 and divided by 2 and that will be your x your axis of symmetry going straight up so if you want to find out what the y coordinates what the vertex of a parabola is all you got to do is find the average of those two guys right once you find the average of the x's you plug it into your numerator here your product here right your dividend here right and once you plug that x value here the average of those two guys whatever it spits out is the y-coordinate for the vertex we'll do another another long division a little bit more complicated and again maybe we'll go ahead and graph it the graph it as well but I doubt it because it's going to be a higher order and we don't want to go that far yet right we will later but we don't want to go that far yet so let's do another one and and then when we get into synthetic division we'll come back and we'll use synthetic division for these guys as well and you'll see that we get the same answer