Intro to the Polynomial Remainder Theorem

so let's introduce ourselves to the polynomial remainder theorem and as we'll see it a little do you feel a little magical at first but in future videos we will prove it and we'll see you well like like many things in mathematics when you actually think it through maybe it's not so much magic so what is the polynomial remainder theorem well it tells us that if we start with some polynomial f of X so this right over here is a polynomial polynomial and we divide it we divide divided by X minus a then the remainder then the remainder from that essentially polynomial long division is going to be f of a it is going to be it is going to be F of a F of a I know this might seem a little bit abstract right now I'm talking about f of X is and X minus a s let's make it a little bit more concrete so let's say that f of X f of X is equal to I'm just gonna make up a I say a second-degree polynomial this would be true for any polynomial though so 3x squared minus 4x plus 7 and let's say that a is I don't know a is 1 so we're gonna divide by we're gonna divide that we're gonna divide by X minus X minus 1 so a in this case is equal to 1 so let's just do the polynomial long division and I encourage you to pause the video and if you're not unfamiliar with polynomial long division I encourage you to watch that before watching this video because I will assume you know how to do a polynomial long division so divide 3x squared minus 4x plus 7 divided by X minus 1 see what you get is your remainder and see if that remainder really is f of 1 some of you mean you've had a go at it so let's let's work through it together so let's divide X minus 1 X minus 1 into three x squared into three x squared minus four X plus seven all right a little bit of polynomial long division is never a bad way to start your morning night it's morning for me I don't know anyone it is for you all right so hi how many I'll look at I always look at the X term here the highest degree term and then I'll start with a highest degree term here so how many times does X go into three x squared well it goes three x times 3x times X is three x squared so I'll write three x over here I'll try to get in the I guess you could say the first degree place 3x times X is 3x squared 3x times negative 1 is negative 3x and now we want to subtract we want to subtract this thing and this is just the way that you do traditional long division and so what do we get well three x squared minus three x squared that's just going to be zero so these just add up to zero and this negative 4x this is going to be plus 3x right a negative of a negative negative 4x plus 3x is going to be is going to be negative X so let me do this in a new color so it's going to be negative negative X and then we can bring down we can bring down the 7 complete analogy to how you first learn long division and maybe I don't know third or fourth grade so all I did is I multiplied three x times this you get three x squared minus three X and then I subtracted that from three x squared minus four X to get this right over here or you could say I subtracted it from this whole from the whole polynomial and then I got negative x plus seven so now how many times does X minus one go to negative x plus seven well X goes into negative x negative one times negative one times negative one times X is negative x negative one times negative one is positive one but then we're gonna want to subtract this thing we're gonna want to subtract this thing and this is going to give us our remainder so negative x minus negative x that's the same thing as negative x plus x so that's these are just going to add up to 0 and then you have seven this isn't gonna be seven plus one remember you have this negative out so if you distribute the negative this is going to be a negative 1 7 minus 1 is 6 so your remainder here is 6 one way to think about it you could say you could say that you could say well actually I'll save that for a future video this right over here is the remainder remainder and you know when you got to the remainder and this is just all a review of polynomial long division is when you get something that has a lower degree this is a I guess you could call this a a zero degree polynomial this has a lower degree than what you are actually dividing into or then the X minus one then you're devised or so this is a lower degree so this is the remainder you can't take this into this anymore any more times now by the polynomial remainder theorem would if it's true and I just picked a random example here this is by no means of proof but just a kind of a way to make it tangible of what the polynomial remainder theorem is telling us if there a polynomial remainder theorem is true it's it's telling us that f of a in this case 1 f of 1 should be equal to 6 it should be equal to this remainder now let's verify that this is going to be equal to 3 times 1 squared which is going to be 3 minus 4 times 1 so that's just going to be minus 4 plus 7 3 minus 4 is negative 1 plus 7 is indeed is indeed we deserve a minor drumroll is indeed equal to 6 so this is just kind of at least for this particular case we're saying okay it seems like the polynomial remainder theorem worked but the utility of it is if someone said hey you know what's the remainder if I were to divide 3x squared minus 4x plus 7 by X minus 1 if all they care about is a remainder they don't care about the actual quotient all they care about is the remainder you can say hey look I can just take that you know in this case a is 1 I can throw that in I can evaluate F of 1 and I'm gonna get 6 I don't have to do all of this business all I had would have to do is this to figure out the remainder of 3x squared when you take 3x squared minus 4x plus 7 and divide by X minus 1