A Level Maths Made Easy - Binomial Expansion Positive Powers

hey guys and welcome back to yet another one of these maths Made Easy videos today we're going to be looking at binomial expansion and in Brackets it says positive integer Powers because in A2 we get to learn about fractional powers and negative powers and all that fun stuff that we love so first things first what is binomial expansion let's quickly discuss all the different ways that I'm going to help you in your GCC or A- level maths careers first of all I wanted to talk to you about the YouTube members this is a really cost effective way of getting tutoring so I'm going to do we live streams for GCSE and AEV maths and that all comes for £18 a month and you get at minimum an hour of tutoring per week in these live streams at the moment they're quite small so you basically get tuition for about £450 an hour which is really really cheap the second thing is the patreon for those of you doing a level maths I have already completed as stats on my patreon is a full class with worksheets mark schemes videos I answer all the questions we're going to do exam papers and everything that needs doing we're halfway through mechanics as of yet and lastly there's tuition I still have places in my group classes for a level maths and GCSE Maths for GCS it's £3 an hour and for a level is £40 per hour so if you want to let me know if you're interested just email me or join the Discord where I'm very very active all the time anyways back to the video why do I care well let me take you through a little story so to speak so if I asked you to expand this bracket right here you would say that's nice and easy it's x + 1 then if I asked you to expand something like x + 1 2 I'm hoping that you'd be able to get to x^2 + 2x + 1 because what you'd have is you'd have two separate brackets that you can then expand and nice and quickly get to the final solution and then if I ask you to do x + 1 cubed well then you probably have to expand it like this x +1 x + 1 x + 1 and then you pick two brackets and you do this thing where you expand the first two brackets which be X2 + 2x + 1 and then you'd still have the x + 1 so now you have to expand the bracket again so then you'd have X2 X cubed sorry plus X2 + 2 x^2 uh + 2 x + x + 1 and then you would close that all up nice and neatly to get X Cub + 3x 2 + 3x + 1 right well okay and then what if I asked you to expand x + 1 to the power of four well then you probably tell me to uh you know well I mean I'm not going to I'm not going to say what I was about to say you probably tell me to go and do something right because it just gets unreasonably difficult and in fact it gets exponentially difficult because you have to expand more and more and more and more terms now funny enough there's actually a pattern between expanding with one the power of one power of two and power of three now what would make this really easy if you have a look if you notice the power of X just goes down right so it goes from X to the 1 x to the 0o we have x to the 2 x the 1 x 0 x 3 x the 2 x the 1 x 0 so if we could somehow work out what the coefficients were we could expand even x^ 4 really really easily because all it would be is x 4 plus something X cubed plus something X2 plus something X plus a constant 1 now if I just work out what the numbers in front are I would have expanded the bracket in a much quicker way now there is a way to do this actually the first way is called Pascal's triangle and it is the same Pascal who has a unit of pressure named after him so Pascal's triangle and this is how it works so we start off with just one like this this is for nals 1 in fact I'm going to move this slightly further away so this is going to work for x + 1 to the^ of n it's going to give me the coefficient so if you notice that Nal 1 we get one and one and if you look the coefficients of both of these is one then you keep kind of spreading out this and the middle is going to be these two added together which gives me 1 2 1 and if you have a look at the coefficients for any equals 2 you get 1 2 1 cool right so one that's going to be three now cuz we add one and two uh that's going to be three and that's going to be one and if we have a look at the coefficients of X cubed we have 1 3 3 1 okay so now to do xal sorry Nal 4 all we have to do is spread out the one again and we get 4 6 4 1 so then if I go over here my answer will be well there's a one in front of the X to the four so don't write anything uh then we have 4 6 4 1 how cool is that expanded a bracket really really quickly okay but then what if I had something like x + 1 to the^ of 10 could you see writing out all these numbers all the way to Nal 10 is going to make me want to die it's too tricky or at least it's too long right it's not tricky adding numbers is nice and easy for us but it's going to take us a really long time so is there yet another way the answer is yes if you pull out your calculator Tada I want to expand this a little bit just to make it a bit easier for you to see you have a very special button called the NCR button hopefully you can see that so it literally has the letters N C and R there and we get there by pressing the shift key and the Divide symbol now if you have a different calculator then of course it's going to be different I know shocking so I'm going to just quickly look at write down NCR n is simply the power of the bracket and R is I like to call it it's the position that you're in now if any of you do programming or know anything about programming Position will make a bit more sense it's like an array a string and we start counting from zero I know weird but you will get used to it extremely quickly which is good so for example if I want to generate the first coefficient in an Nal 4 expansion I would simply put four here and i' put zero on the end and it would give me one if I change that to a one now I get four if I change that to a two for the Third third value I get six so can you see I'm generating them all on a calculator nice and easy perfect this is what we're going to use to expand brackets of any size really bloody useful now let me show you how to do this with a question now of course we don't always just have x + one we may have 2 plus X over 2^ 2 or 2 x over2 sorry something along those lines so let me show you how this works first thing it says is it find the first three terms in ascending powers of X okay so what I want you to do now first thing first is we're just going to write the coefficients down so what is our power our power is six so I'm going to change the four to a six and we're looking at the first coefficient zero and if you notice from Pascal's triangle it's always going to be one and the last one's also going to be one but you know in for a penny in for a pound might as well do it one then I'm going to leave a bunch of space and I'm going to put the next coefficient one gives me six cool plus six big space you don't need to leave that much space but I like to and if you notice it says the first three terms so that's the good thing about this is we don't have to generate all of them it will normally be the first three terms sometimes four but usually it's three terms in exam questions nice and easy and then two we can just do that and then boom we get 15 that's the first step now this is the really important part ascending powers of X so you know how I said usually it's the first three terms in I mean almost always it's the first three terms it's also always ascending powers of X which is the which is really really really nice so what that means is you're starting actually from you're going backwards now that might sound more tricky but it's not because I'm going to show you what happens do you remember how I said the power of the X term goes down right so it goes from x^2 x and then x 0 here it goes x 4 x Cub X2 X1 x0 okay so if you're going in ascending Powers we're going backwards it be x to the 0 first okay so this x term which is X over2 goes like that now something that we didn't spot because we had x + one is that this plus one term actually increases in power as we go along so in this case we start off with the highest value of x and we start off with the lowest value of the constant term and then as we reduce the power by one we increase the power of the constant by one so since we're going backwards as I said we're going to be over time increasing the constant um sorry increasing the power of the X term decrease the power of the constant so what's the biggest power of the constant it in this case it's 2^ 6 now that might seem a bit confusing but Watch What Happens we reduce the power by one and increase this power by one reduce this power by one increase this power by one because that's how it works when you expand brackets however when we expand brackets we normally decrease the power of X by one increase the power of this by one however with binomial expansion because we're going into ascending powers of X we're going to increase the power we're going to it backwards increase the power of X over time decrease the power of the number the constant over time that's it four marks all we have to do now is simplify it so so easy X over 2 to^ 0 is going to be one so literally all we're doing is doing 1 * 2^ 6 so we have 64 as the first term now what we do for the next one well all we have to do is change the coefficient change this power to five now of of course we actually still have 1/ two right don't write X on the calculator because it won't know what you're talking about so it's 96x and again what we're doing is we're changing the constant the front reducing the Power of Two by one increasing the power of the half by one and then that's going to be 60 x^2 that is four marks four marks two lines of working out they did it all on a calculator so what are the problems why is it worth four marks well a people tend to forget uh to reduce or increase the power in fact a lot of students actually skip this first bit here they ignore it so then they don't put the one which means that they can screw up the later Parts other reasons is because people don't take it slowly they've given you four marks it's a roughly 4 minutes use that time take it slowly work through it you have a lot of time so it then says use your expansion to find an estimate for the value of 2.05 to^ 6 so how do we do that well it's quite simple this binomial expansion that we've got here looks like this and it's an approximation so the wiggly equal signs means that it's a oximation it's it's shaky you know it's roughly like this and we want to find an approximation for 2.05 power 6 if you notice the power is the same right so in other words what's inside this bracket should equal 2.05 so the first thing I do is I minus two from both sides we get X over 2 = 0.05 times both sides by two we get x equal to 0.1 so in other words all they want me to do is substitute in 0.1 in for X but you need to show that you've worked out X okay so that will give me 2.05 to^ 6 being approximately 64 + 96 0.1 + 60 and by the way this is actually what binomial expansion is used for in real life is approximations because certain calculations are too long for computers to do so we approximate instead because if you look or if you notice if you're a bit math Savvy the actual value of each term is decreasing right because here it's 0.1 so we're dividing that by 10 then we're dividing this by 100 so the numbers are going to get smaller and smaller and smaller and we get 74.2 so what I like to do sometimes to have fun cuz I'm a loser is I like to just do 2.05 power six and see what happens not bad 7 4.22 not bad at all hopefully that makes a lick of sense and we'll move on to the next one now of course there are always different types of questions that we're going to do so I'm going to give you an overview of every question that we can do part a very similar so what I would like you to do right now is pause the video and have a go yourself at doing this again uh with maths is a very active subject you need to practice yourself so I'm going to assume that you've done that you better have but all we're going to be doing is once again using the NCR button on our calculator to work out the coefficients first so whip out the calculator we want shift NCR button what is the power the power is seven start off with zero and we know it's going to be one so one leave a gap plus and what does it say the first three terms again so we're only going to go up to two seven and plus 21 like so now what do we do we start off with the highest power of the constant and the lowest power of the X so we have 2^ 7 and then we have Min - x over 8 so it is a negative so we need to make sure we keep that negative and then we keep going so reduce the power of the constant by one increase the power of X by one very formulaic very formulaic but you may have made a couple mistakes in you're working that is completely fine it happens as you uh practice you will make fewer and fewer mistakes I guarantee you so let's work it out well of course the first one is just going to be two 7even technically it's Times by one but you know it's not that is it so 128 then um I'm not going to put as plus cuz I can see it's probably going to be a minus so we have seven Open Bracket 2 p 6 and then we have minus 1 over8 again we don't bother writing the X we have - 56x and then the next one should be a positive again if you're not sure you can always up first and so let's change that to a squared let's change the six to a five and let's change the seven to a 21 and we get 10.5 x^2 like so again four bloody marks it's all on a calculator but part B looks a bit scarier and honestly it sounds a bit scary so we have FX is equal to this this where A and B are constants given that the first two terms in ascending power of X are this and this find A and B so what they're trying to tell me is if we were to expand This truly gargantuan bracket here we will get two terms the first two terms in ascending power of X is 384 and4 x well we've already kind of got the first two terms all we need to do is times it by a X plus b so this is not an approximation in this case because we're only looking at the first terms in ascending powers of X anyway so now I'm going to use my now we have to expand this now if you use your nogin a little bit you realize you don't have to actually expand that much so why is that well we're looking at the first two terms in ascending power of X in other words we're looking for the X to the zero term and the X terms so when I multiply out this bracket anything that gives me x^2 or X cubed I can ignore so let's have a look at this ax * 128 is completely legit so 128 ax but I can't do ax * 56x cuz that give me x s and I can't do ax * 10.5 X2 because going to be X cubed it's too big I'm only looking at the first two terms so let's go to B well 28 * B that's completely legit too and B * 5656 BX is also correct so now let's put this in like order ascending powers of X because again it tells me in ascending powers of X so we start with the 128 B then we have 128 ax minus 56 BX so these are my first two terms we have 28b is my first term and all of this is my second term and it tells me what they're equal to so 128b the first term should be equal to 384 divide both sides by 128 and I'm going to pray that this isn't an ugly number 128 actually I think it goes in yeah three times perfect so now I know that b is three and it tells me that the X term the second term should equal -4x so in other words we have 128 ax - 56 BX but B is three don't forget equals -4x so what's my first move here well the first thing I'm going to do is I'm going to divide everything by X so in other words I'm going to delete all of the X is that will give me 128 a - 56 * 3 let's work that out so 56 * 3 gives me 168 equal 104 add 168 to both sides gives me 64 so then a is equal to 64 / 128 which is a half so a is a half and B is equal to three not that bad really it looks like it would be bad but it's not that bad by the way this is practically guaranteed to come up in your year 12 marks so just keep that in mind eight marks huge huge about what 10% of Pap of of the paper you're do in year 12 so overall 5% of your grade just on this not bad increase your grade by 5% that's not bad at all that's quite close to a grade boundary here we go now here's where things get interesting so with um a level maths they really really like to do something called symbiotic maths so that sounds really fancy and it's not about like Venom or anything what it is is it's using different uh parts of maths in other parts of maths so this right here is actually a Statistics question I know right so they've taken statistics put it in pure maths with regards to this so um this bit here is fully expand I know it kind of sucks right it kind of sucks but since we're fully expanding it doesn't matter to the order but the first thing we need to do is once again get our good good old fashion NCR button and work out all of the coefficients this will not take too long I promise the power is five we're going to start with zero and we're going to end with five right so we have one and then plus uh five yeah so this it doesn't actually take that long because there's only going to be five terms well six terms in total [Music] 10 uh and again since we don't actually have any numbers it's literally just going to be like P to the 5 P4 q1 P3 Q2 like it's not actually that bad um when you get used to it uh five and then I know that the next one's going to be one and if you notice it is symmetrical when you have an odd number it can't be symmetrical when you have an even number of course but when you have an odd number it's always um symmetrical which is nice and then again like I just said it doesn't matter if you start with P or Q to be honest because it's fully expanding but it's going to be P to 5 q 0 P to 4 q1 p 3 q^2 p to 2 Q cubed Q cubed interesting P Q4 and then Q to the 4 so if we simplify this we literally just have P 5 + 5 p 4 q+ 10 P cubed q^2 God I hate saying that 10 p^2 Q cubed + 5 P Q4 Plus Q to the 5 I forgot to actually increase the power thank God I spotted that there we are oh there we are not too bad um slightly more annoying but we should keep in mind that it is technically easier to an extent but now here's where things get spicy so we've got four Mark we've already got most of the marks in the question and we haven't even done the hard bit what in well in my opinion the hard bit it then says uh let the probability of Dave being late on any school day 0.1 so0 so p is equal to 0.1 right so then Q would have to equal 0.9 and it said it I mean it tells you use the last two terms of the expansion so these two yeah find the probability that Dave is late no more than one time in a school week well if you notice the probability of him being late once is going to be he's late once and then he's on time on time on time on time so if you notice that actually gives us the answer and the other probability the other outcome is that he's never late so he's either never late or he's late once we're not including he's late twice three times or four times because it says no more than one time so it's just these two terms here and all we need to do is literally sub in the probabilities that they've given us and that's three marks by the way three marks for doing a simple substitution on your calculator but again it's the conceptual knowledge knowing to do that which is a I mean it's obviously very tricky and it's brand new as well right like you're not really expected to do stuff like this in GCSE so the fact that we have to do it now is scary but it's not that bad and I'm going to get you through it okay just practice learn and if you are ever unsure about what I'm doing or why I'm doing something then let me know leave me a comment and I can help you out I can explain it again so that right there was the binomial expansion in that shell so I've gone through a few different exam questions uh that can turn up in your year 12 papers so all you need to do now is go and practice it it's very similar every single time it's not that bad and I know that you're going to get it but anyways I'm going to leave you to have a wonderful day and I'll see you in the next one