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3-18 AIM3 7.4 Polynomials in Powers of x - a

Added Mar 18, 2024 •

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hi then everybody so y'all to visit this concept that we referred to as polynomials and powers of x minus a where a has some particular value so please notice the here we have a nice dinner looking cubic in standard form but it turns out through some magic that we could ride it this way and noticed what we have no sir this is a cubic X Cube and we have something * x - 3 3 because we're doing it around the point three don't worry about why right now we have something * x - 3 cubed and then something comes x -3 squared something * x - 3 the first and then hearing I think of this is something * x - 3/20 knows are the powers of x - A and chosen art extra person is polynomials and powers of x minus whatever the value isn't since its 3 here we have x - 3 is this -3 then this would have been x - -3 / x + 3 another name for these very very common is a Taylor expansion and you might want to pause the video and we did some of these examples were Taylor expansions or Taylor series can be you for our point we'll just be learning how to create them you'll be really using them in a discovery kind of sense more in other classes and perhaps some of your budding physicists will use it to work more sooner than later anyway so how do we create these polynomials in powers of x minus a I got nothing to do all this all right you can just bring your own and Page 532 and I will just leave that up to you to a technique there's a technique for me in the future actually next year called partial fraction decomposition involves things called partial fractions and partial fractions work something like this suppose you had the fraction 56 so what are the factors of 6 not counting one 46 now they're two and three so it should be possible to rewrite this is something over 2 + something over 3 2 and 3 being the factors of 6 and this day over to deal with these known as partial fractions they also can be used in a more non child erase and so for example I have 3x over where's -4 the factors of x squared - 4r x + 2 + x - 2 so I can create a partial fraction of something over what's 2 + something over x minus 2 on 5 picking the correct values using some particular technique I could create these partial fraction and the reason for that is as we've we've done various time sometimes the form something then say this form isn't this convenient isn't terribly convenient for what we're trying to do but if we write it in a different form example these partial fractions then that form would be more convenient never this was trying to achieve so the question be how do we take partial fractions now it said we will deal with that in the future but part of that is inspired by what we're going to chat about here all right so first how do we find the coefficient different powers of x - 3 take a look on page 529 but let me warn you first you better be seated when you read this cuz it's it's something that you know something you want me to do before I'm going to ask you to pause the video and take a look at page 529 then come on back right well I hope you got filled you with Dread wasn't that just an impressive bit of arithmetic you have to do there so I actually have it I will trick don't make it a whole lot easier I know I was a huge amount of a ride involved was actually nice little move we can do that and make it easier and I've got it hidden right here let me show it to you here you go we're not going to do that without that is not a fun wave doing things I've got a different style that I think you will like much more such take a look at that so hopefully you remember the technique of synthetic division if you don't I suggest you stop this crap to go to Khan or some other YouTube video remind yourself how to do synthetic division I will remind you when we take a polynomial and / here's the important thing a linear Factor well of course we could do long division but it turns out there's this other techniques synthetic division that we're going to use that will typically be more convenient but only works when I have linear factors so let's do it on this example I have here notice by the way that I'm missing the quadratic term and we can't have any missing term do I have to consider there is a quadratic but I can't change this this expression is such so I'll just added 0 x squared how's the Padre, doesn't really change but the expression is so how does synthetic division work it's always of the form x - something this is x minus positive 3 so we're going to quit 3 out here and that's what we can you divide 2 divided we stand take the coefficient of all these terms are so we have a 1 - 5 Mendes 0 x squared 3 is the linear coefficient and then minus one and if you remember how to do it just remind yourself let's go through it so the way it works is you actually do this you carry down the first term and then you will multiply so 3 * 1 is 3 and then you simply adding together - 5 + 3 is -2 and then you're going to multiply just repeat the process 3 * -2 is -6 we get -6 and then we're going to multiply -18 add them to get - $50 - 15 and one last time we're going to multiply we get - 45 and then we add we would get - 46 this was just the end of it then we would say that the X to the fourth-year / x - 3 this would be our solution and this last term is the remainder so x - 3 is not a factor which we could have predicted free can't go into one and so but we could say that X cubed - 2x squared - 6x - 15 with the remainder of -46 is there thought you did he divided this quartet by x - 3 we're looking for polynomials in all the powers we need something comes x- Q that's my sweet squared and so on so how do we get to that well it turns out we just keep repeating I'm not going to go with my color changes anymore I'm going to repeat the process that's all remain we can ignore what we just worried about this this part right here till bring down to 13 * 1 is 3 add them up 3 - 3 - 9 - 24 I'm at the end I'm done I just separate them now I repeat bring down the one to 34129 dumb are we on the one re7 and as my roommate has done done how does that help me well what this tells me set my function X the 4th - 5 x cubed + 3x one can be written in powers of x - 3 as and we start down here so we have one I'll go out there and have Friday to cbx - 3 to the fourth is after all the time coretec then I just keep taking these outside value so + 7 * x - 3 + 9 + happening and this would be our polynomial and powers of x - 3 lettuce no I would like you to try you might want to go back and look at the process on the previous slide but I would like you to try turn Powers at 3 for the same X my strength at this time just ask you and I would like you to try and do what I did hear this process of synthetic division been using these values out here to create the vehicle I would like you to try that on the function X cubed and so you should pause the video a try it and it come back and see what I have for you okay I certainly hope you pause the video and came back and let me share with you July 1st yo yo self since it was X Cube One X Cube 0 x squared + 0x + 0 remainder and then I went through the process so I have these values 1927 27 and so that's what we get for the expansion of X cubed around the -3 until that's the technique you don't want to practice with this lesson when you look at problems 1 and 2 after you done Parts A and B see if you can use them to make life easier for the other parts of the problems cuz you don't have to go through this whole thing completely you can enough it would work fine but if you can take advantage of what you've done Parts A and B and see what's happening in the remainder I think you can see why to make your life a bit easier and I hope you find it good luck to take care

Uploaded by Brian Macky.

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3-18 AIM3 7.4 Polynomials in Powers of x - a

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