 So if the question if the question was do this long division, right? you would lay it out in this form and You're doing You're taking this guy and dividing this guy into it So when it comes to long division the way you end up doing it is the only thing you care about is matching the first guy in the Dividend to the first guy in the divisor the first guy in the numerator to the first guy in the denominator So what we what we should be asking ourselves is this what do we multiply x by to give us negative 3 x squared? It's the same thing that we had when we had 27 here and 2 here What we're gonna? What we're gonna multiply the 2 by to give us the first number 2 in 27 right so what we multiply x by to give us negative 3 negative 3 x squared is negative 3 x Right and what we do with that is we put it up top the same format that we had for integers right when we're dividing integers so What we're gonna do is put negative 3 x up here and multiply that out with these two terms and write them down here So what we do we put negative 3 x up here and negative 3 x multiplies the x and then it multiplies the 5 and again We definitely want to match the sign of these numbers to write that's a positive x and that was a negative 3 x squared So we need a negative number there to turn it into negative 3 x squared, right? So negative 3 x times x is going to be negative 3 x squared Which is exactly what we want to want to match it right and negative 3 x times negative 5 Yeah, I'm sorry negative 2. I'm looking at her backwards. It looks like a 5 so negative 3 x times negative 2 is going to be positive 6 x Now what you're gonna do is subtract This from the above term, okay? So usually the way it's the the way it's taught in most places that I've seen, you know, they go minus this And then minus that from those two terms and they go negative and a negative equals a positive So you add those guys and they kill each other I don't like that because I lose track of my signs. So the way I like doing long division is What I do as soon as you do more your multiplication and trust me This makes a life a lot easier because that way you don't have to worry about a negative and negative, right? Because again long division. This is a fairly simple one. It's a short one But for long ones, you're gonna have a lot of you know, it's gonna be It's gonna be there's gonna be a lot of things you need to keep track of right and the last thing you want to keep track of is You know when you have a negative and a negative so it turns into positive and then you do your you know Edition or whatever it is that you're doing right the simplest way to do it. I like adding Rather than subtracting. So what I end up doing is if you're subtracting the bottom from the top All it means is you're changing the signs of the bottom and adding it to the top So as soon as I do my multiplication here and lay down my terms here what I do I change the sign of all the terms in the bottom and I add things together So what I end up doing is Changing negative 3x squared to positive 3x squared and I'm gonna add it to the top the top is negative So negative 3x squared plus 3x squared is going to be zero in the bottom That's going to be that's 5x plus negative 6x instead of going 5x minus 6x right is going to be negative x Now what you do is just like you're you know dividing the numbers dividing your integers You're gonna grab whatever is over here and bring it down So you grab the tube bring it down now what you're looking for is the question you should be asking yourselves And this should become automatic it should be something that you know as soon as you get to the next line This is the question that should be you know clicking in your mind is What do you multiply x by to give you negative x and the answer to that is a negative one So what you end up doing is putting negative one up here and then multiplying it out through this guy and then multiplying it To the other term and writing it down here So you go negative one times x is negative x you're gonna write it down here and negative one times Negative two is going to be positive two So you have negative one times x is going to be negative x negative one times two is going to be positive two So what I end up doing is changing the signs in the bottom over here and adding these things together So that was a negative x it changes to a positive x and that was a net a positive two changes to a negative two So negative x plus x is going to be zero Positive two minus two is going to be zero So we just got down to zero which is Basically telling us that this guy divides evenly into that guy Another way you can think about it is as soon as you get a remainder of zero it means that this guy is a factor of the top guy, okay, and Going back to our terminology The symbols that we're going to call each one of these things is the following So what we have from our division statement is the top guy is q of x is the quotient, right? X minus two this guy is going to be our divisor d of x which is that guy right there This guy here is our Numerator or our dividend, but we're not going to call it big D We're going to call it p of x because it's the product of things multiplied together Okay, which is really what we're looking for right and you know it splits up the terminology So we don't have two different days in our division statement Down here is our remainder r of x right now it equals zero and whenever the remainder is Equal to zero it means what you were dividing is A factor of the top and this is super important because it's what we're looking for right So we're going to write our division statement and then write these out and you're going to see how everything fits together So our division statement is the following right p of x is equal to q of x times d of x plus r of x So what that means is? negative 3x squared plus 5x plus 5 p of x is equal to q of x Right, which is the quotient that whatever we found out out here Times d of x which is x minus 2 plus r of x which is zero, right? So right now we have our division statement and the above you can write in the following form We're gonna do a way more complicated version of this or more complicated version of this a longer one where the remainder does not equal zero and Where we're gonna have an r of x and again that r of x is just going to be our y when our x You know you set this thing equal to zero where x when you bring the number over x equals the opposite sign of this then If you sub that x into our original polynomial, that's going to be your remainder now We've talked about just you know factor in the polynomials breaking the polynomial down to those prime prime factors I guess prime prime polynomials, right? And whenever you get a quadratic you should right away look at it to see if you can factor it by using all the different Factoring techniques that we've learned so far simple trinomial complex trinomial Quadratic formula now difference of squares or even gcf to see if there's a gcf that you can take out and maybe that gcf Kills your denominator, right now the top guide We've already factored using two different techniques and its factors are the following so what you would end up having if you Factor that guy straight out using either complex trinomial factor in quadratic formula You would get the top guy is equal to x minus two times negative x minus one, right? Divided by x minus two well x minus two kills x minus two so if that guy kills that guy what you're left with is negative Three x minus one which is what we had over here, right? And I'll answer would be this guy so again right now learning those two techniques complex trinomial factor in quadratic formula It's given us two other choices for us to do this problem, right? And the way that's that's sort of the way it works in mathematics the more tools you have There you know the more the more ways you can tackle a problem and the more power you have, right? So there are easy ways to do things and There are harder ways to do things. It's up to you which way to solve Solve a question to answer a question, right? And the more of these things you do the more that the easier it's going to become for you to recognize Which you know which technique to use on which type of problem, okay? And this is exactly what we're doing right now We're taking our toolbox putting a whole bunch of different tools in their techniques in there to be able to attack problems From multiple angles because sometimes a lot easier coming at it from this side than it is coming at it from the other