Okay, we definitely want to you know get the long division the really big one done I'm just gonna lay out the equation here But what we're gonna end up doing is moving on to the bigger area the bigger bowl and doing it there right now We've got somebody riding around with their BMX so we can't go there yet so we're just gonna lay out the equation in the long division not the long division form but the fraction form so you Can take a look at it and that's the long division We're gonna end up doing okay now if you notice what we have is we got all the x's in descending order But we're missing the x to the power of one so what we're gonna do when we lay out the division statement is Put in zero x to the power of one which is zero x right just as a place marker So when we're doing the you know doing the long division we don't accidentally add Unlike terms and we know you know there's there's there's a missing term in there Okay, so keep this in mind and we're gonna take this and transform it into the long division form So the question is what are we gonna multiply negative three Negative three x squared by to give us negative negative three x to the power of five Well, they're both have negative three in front So we don't need any coefficients up front all we've got to do is match the powers right the axis So all we're gonna do is multiply by x to the power of three now slight correction over here It was supposed to be plus five x okay Excuse kind times negative x to the power of two is going to be negative x to the power of five x cube times Five x is going to be positive five x to the power of four and x cube times two is going to be two x cube now What we're gonna do is change all the signs and add on negative three x to the power of five plus x to the power three x to the power of five to kill each other Five x to the power of four minus five x to the power four these two end up killing each other and 17 x cube Minus two x cube is going to be 15 x cube So the question we're going to be asking ourselves is what are we going to multiply? negative three x squared by to give us 15 x cubed right because we're only concerned about matching the top numbers What I ended up doing as well is bringing all these terms down because it's Look up because every time I look up to try to bring something down I might make mistakes so every time I just transfer everything down transfer everything down transfer everything down So what we're going to multiply negative three x squared by to goes 15 x cube is going to be negative five x Right and that goes up top and that multiplies everything out. I'm going to lay it all down here again So what we're going to do now is change all the signs and add So 15 x cube minus 15 x cube these guys are going to kill each other 31 Negative 31 x squared plus 25 x squared is going to be negative 6x squared and that's just going to be Positive 10x because a 0x up top right 0x plus 10x is going to be 10x and we're going to bring it 400 424 straight down right because we're going to fill in everything we're going to transfer everything down at the same time So again, we're going to go back to the beginning What are we going to multiply negative 3x squared by to give us negative 6x squared? That's just going to be 2 right 2 times negative 3x squared is going to be negative 6x squared And we're going to put the 2 up here and multiply out with this guy this guy this guy put everything down here Change all the signs add them all together. So what do we got we got? Negative 6x squared my plus 6x squared. That's just going to kill each other 10x minus 10x those guys are going to kill each other right and we've got 424 minus 4 that's just going to be 420 right So we're left with 420 now the power of this guy is Less than the power of this guy the power of this guy is just 420 times x to the power of 0 right now If this was 420 x you would stop as well because the power over here was less than the power over here Right, so right now. We finished our long division We divided this guy into this guy and the answer is that guy That's the quotient and our remainder is going to be 420. Okay now. What are we going to do? We're going to move over and do a little division statement and write it out We're also going to look at it graphically to see what it looks like graphically. Okay It's just going to be very simple graph and you know just Very very basic graph of it right now. Keep this in mind We just divided this this guy by this guy So these two guys multiply together plus that guy are going to be equal to that guy But if we wanted to factor these further we could we could factor this further We could factor that further right either using simple trinomial factoring complex trinomial factoring with a quadratic form So we could and the top guide the quotient here is x to the power of 3 So we'd actually use synthetic division to factor that guy which we haven't talked about yet But we you know that's where we're going to we're building up, right? So what we're going to do right now is go to the other wall and write the division statement for this long division And hopefully this made sense because all it is is keep this in mind All you're doing is matching this guy with this guy right the first guy here has to match this guy here Everything else works itself out you get down To a place where the power of this guy is less than the power of the first guy in the new in the denominator Then you're done right you've done your long division and whatever ends up here is your remainder Okay, so let's write the division statement for this guy. Okay, so our division statement is our Big polynomial right our p of x what we're calling p of x in Division statement is equal to the quotient times the divisor plus the remainder right p of x is equal to q of x times d of x Plus r of x right now. What does this mean? It means this polynomial is equal to this polynomial times this polynomial plus 420 this function is equal to this function times this function plus this function Graphically, this is what it would look like So graphically, this is what this expression this division statement is actually saying it's saying this function It's actually the one thing to keep in mind with these graphs is they're not to scale So this guy is actually going up all the way to 420 that one's actually 420 a line Horizontal line at x at y is equal to 420 and these guys are smaller scale So the scales of the Cartesian coordinate systems on all these is not the same It's just sort of a presentation of what it looks like, right? So the division statement up here what it's saying graphically is this function Is made up of is equal to dysfunction times dysfunction Plus Dysfunction and when you're adding something adding a function It's just a translation. It's just moving around the Cartesian coordinate system, right? So if you multiplied This graph with this graph and moved everything 420 up 420 higher You would get dysfunction Okay, and that's what the division statement's saying here But we're saying it with terms with letters with numbers instead of visually as graphically, right That's what we did when we did the long division The long division took our original function. We knew or we didn't know we just took this other function, whichever it was it was the negative 3x squared the negative 3x squared we took this function and Divided it into here to see what would happen, you know this function times. What is equal to this? Well, this function times this guy Plus the 420 is equal to this function Okay I hope this makes sense because this is this is really what we're getting at because polynomials and non polynomials basically functions in general What we're doing is learning the different techniques to break things down and look at their core elements And you know mess around with functions take a little function from here take another function from here Combine them and see what happens take one function and take another function that we may like We may want to you know that that we've used somewhere, right? And you know may it may represent something that we have, right? So taking one function and dividing it into another function just to see what the end result is Is the end result what we want is the remainder what we want, right and keep in mind This guy here is Your y-coordinate when your x's are these guys, okay? So you can break these down even further, right? You can factor these down even further get your x values and if those x values you plug in to original function This is what you get out. You get 420 out, right? So there's a lot here and it's uh You know, it's just it's just a graphical way of looking at it and you know trying to Everything that you're doing when you're doing the long division when you're doing synthetic division When you're doing factoring when you're looking functions when you're looking polynomials What you should be doing is thinking about this in a graphical form, right? Everything that you're touching everything that you're working with in general is going to be a function, right? As long as we're dealing with polynomials and functions, right? Numbers. No numbers are just numbers basic operations They're just basic operations But those are our building blocks, right the real number set our building blocks variables our building blocks For us to be able to model things in real life and see how things work out What happens when we combine functions together? What happens when we multiply functions together? What happens when we multiply functions together and translate them over, right? Moving around the Cartesian coordinate system. Do they give us what we want? This was the last long division That we're going to do hopefully anyway and what we're going to do from here is learn synthetic division and uh, you know Probably might even do this one using some synthetic division So once we finish synthetic division, we've done all the different factoring techniques that we're going to talk about For the polynomial sections for functions, right? So we're going to have a whole arsenal of Weapons, we're going to have a whole bunch of powers to be able to Start going into polynomials graphing polynomials looking at functions developing our own functions. Okay, and uh, you know Stay tuned. We're going to go into synthetic division and once we're done with that again You're going to look at synthetic division. You're going to think it's super sweet because it's a lot easier than polynomial long division because you don't have to carry the x's over and You can eliminate things much faster and you can look at Functions and actually look at the last term and look at the first term and guess as to the numbers That might actually go into your function that you're given That you can factor properly or they give you they give you factors where your remainder is actually equal to zero Which means across is the x-axis. Okay, so, uh, next step next thing we're going to get into is uh, synthetic division and hopefully We'll have left long division behind us a little bit might come back to this with other functions But with other questions anyway, but synthetic division is what we're going to focus on. Okay. Talk to you guys soon