okay welcome to this video where we're going to have a look at everything to do with binomial expansion now all the topics you can see on the screen are the ones that we're going to look at throughout this video and just so you know how to use it let's just have a look at that now so if you've used this video before you will know that you can click into the description and have a look at all of the other chapter videos for each of the topics and if you scroll down you will also find the video there where i go over everything in the asp curriculum if you also click on the video you can see that all of the chapters are bookmarked so you can click on them and you can scroll through the video and have a look at the individual topics in the chapter so with that being said let's get started [Music] okay so when we're looking at binomial expansion essentially we're thinking about from gcse level where we've looked at expanding and simplifying a triple bracket we're going to look at sort of how this all sort of interlinks with then higher powers when we have sort of a bracket to the power of five or to the power of 8 and how we go about finding those powers using this process now when it comes to binomial expansion it's all sort of interlinked with um sort of the the separate powers when we've looked at brackets now if we think about sort of a nice easy one something like x plus 5 in bracket squared we get a certain pattern that sort of comes out of these and when we expand one of these we get the x squared we get that first piece there's always to that power of 2. then we get the 2 in the middle we get plus five x plus five x and we always get these two there and then you get that five at the end which is squared as well to make 25 and then obviously those two in the middle are added together but we get this sort of little pattern and it's all linked to pascal's triangle and pascal's triangle is nice and easy to draw we get a one at the top then you add together uh the numbers either side of it to get the ones below and obviously one at the top has got a zero either side so we just have one and one below that but then when we add these together we get one on the left those two ones add to make two in the middle and then we get one on the end and we can just keep following this process of making this pattern so always one on the outside then the one and two above this next one makes three two and one makes three and one and we can keep on going one three and one makes four three and three make six three and one makes four again and then one at the end we can keep on going basically with this now when it comes to this double bracket that i've drawn here that is this row here so one two one so we get one of the pieces at the start the x squared we get the two pieces in the middle okay which we've obviously just shown there we got the two in the middle and then we get the one at the end and following that little pattern of one two one now if i come away from that and we have a look at this triple bracket which is what we're gonna have a look at to start with that is obviously linked to the next row down that next row down is one three three one now it's not as easy to spot the pattern with this one um but obviously if we were to expand this all and have a look at all the pieces we would see we get the one at the start we always get an x cubed at the start we get the one number at the end which is going to be that three cubed which we'll have a look in a sec and we get these two lots of the three pieces in the middle i'm gonna have a look at those now so if we have a little think when we were to expand this at gcse obviously what we would do is we'd expand two brackets to start with so we'd do x plus three and we're gonna fully expand this i'm to do it nice and quick though and x plus 3. we'll expand that which gives us x squared plus 6x plus 9 and then we expand that by another x plus 3. there we go and we do that and we expand this all out we get the x squared times x gives us x cubed the x squared times the 3 gives us 3x squared and then we go on to the 6x and just keep following this process so 6x times x is 6x squared the 6x times the 3 is 18x 9 times x is 9x and 9 times 3 is 27. okay we simplify this all down so eventually we get x cubed plus and we've got those two lots of the x squareds there which gives us 9x squared the 18x and 9x gives us 27x there we go plus the 27. and we end up with these four pieces here and as you can see in that line we've got those four pieces now it's not quite as clear with the triple bracket here or with the bracket to the power of three as to how we got those two lots of three in the middle but essentially we kind of got and let's have a look we've got the x squared here one two three pieces of x squared and we also got three pieces of x we've got the 18x the 9x the 6x and then obviously we've got the numbers on either end as well we've got the x cubed and the 27. so you can kind of see where this pattern comes from you've got the one lot of the x cubed the one lot of the number at the end and then the three of each piece coming on uh coming in the middle there obviously eventually we got two x cubed plus nine x squared plus 27 x plus 27 but we're gonna have a look at how we can actually get to that using binomial expansion and using this process uh within pascal's triangle now obviously we don't want to have to always draw this triangle out every time and thankfully there's a nice little button that we can use to get that on the calculator and if you have your calculator this is a calculator topic it's a button that looks like this and it tends to be that you've got to press shift to get there first but it says n c r and it's normally behind the divide button so you still normally have to press shift divide and it gives you this ncr now in order to get these numbers here which was what we're going to use for the first one we have to press obviously it's a power of three so we press three shift that ncr button and you'll get a c and then we press zero for the first one so three c zero and that always gives us a value of one we always get that one for the first one which is just there now for the next one we press three again three c one and we get three c one and that gives us three then we do it again so swap the one but go for two this time so three c two and that gives us let's just have a look we get three again and then for the last one we're going to do three c3 which gives us one so make sure you know where that that button is in your calculator it's normally just behind the divide so three shift ncr and then whatever number we're looking for we start with zero and then one two three for those next ones so we can actually find out what the pattern's going to be just using this button on our calculator i'm going to be using that throughout the video so this is going to be a very important little button i'm going to be using that every question and i'm mentioning that as we go but as i said you've got to press shift first to get there so let's have a look at how we can actually use that on this question um making sure that we can obviously expand this triple bracket using binomial expansion before we start looking at higher powers so if we get rid of this obviously we can get rid of all of this we know what ants we're trying to get to and that's this one here x cubed plus nine x squared plus 27 x plus 27 okay so we obviously know that from gcse level that's going to be the answer let's look at using binomial expansion to get there so once we've got the pattern then and we've got nc zero is one we've got one three three and one and that's going to be our pattern now what we do is we look at the two pieces within it within our bracket so we have an x as the first one and a three is the second one okay and it's just actually just highlighting different colors we've got an x and we have a three so that first piece and that one there the x is obviously the first piece in our pattern and we have one of those so we have one lot of and that's going to be x and everything here is going to have to be up to a power of 3 so that's going to be x to the power of 3 and we have one of those so all of these little pieces we're going to write now we need to make sure there's always a power of 3 in all of them now the next one we're gonna have three lots of this next piece so we're gonna add to that three lots of and the next piece is ever so slightly different now x the first one we're gonna drop down that power now because we only have one of those x cubes in there so that x piece is now gonna go down to a power of two so one x to the power of two but we need to make sure we balance out these powers and the next piece that we're gonna throw in there is that three so we have then a three and that's going to be to the power of one not that we need to write the power of one there okay so all we're going to do is reduce that first piece which in this case is x so that's gone down to a power two and introduce that next piece in the bracket which is a three and we're just going to swap these round now so the x is going to keep reducing and that number the three is going to keep going up in powers until we get to a power of three for that one as well so the next one and again we've got three lots of this next piece so we're going to add to that three lots of and we have x to the power of one this time that's going down to the power one and that three is going to go up to a power of two there we go that's our third piece in that pattern done and then onto the last piece we have one of these so we're going to add to this one of now x is going to go down to the power of zero so that's gone and we just have this three at the end which again has to balance out to that power of three so one lot of three cubed and if we expand these all out and see what we get one lot of x cubed is x cubed there we go then we're going to have 3 lots of x squared times 3. well let's just do the 3 times the 3 to start with which is 9 so that's 9 lots of and then we've got x squared with that so 9 lots of x squared then for the next piece let's have a look we've got three lots of three squared three squared is nine three times three squared so three times nine is 27 again with that x to the power of one so plus 27 x and then at the end there we have one lot of three cubed three cubed is 27 so plus 27 at the end and there we go that matches obviously what we've got from expanding our triple bracket at the top we've got x cubed plus 9x squared plus 27x plus 27. so that's all we're going to do for binomial expansion obviously it gets a little bit more complicated as we start to go further down pascal's triangle and start to get larger numbers involved but basically that's all we're going to do we're just going to write down what what the numbers are in obviously in that line of pascal's triangle that we're going to be looking at and obviously just balancing that out with our powers sort of moving through but let's have a look at another one and see how we can apply this to a slightly harder question okay so this question says find the first four terms in ascending power of x of the binomial expansion of three minus a third x to the power of five giving each term its simplest form so let's have a look at this one now we've got a power of five so before we start dealing with the fractions and having a look at how we're going to deal with that we'll find the first four terms in the pattern here so we'll go for five c zero five c one five c two and five c three and just work out what that pattern actually is now obviously we know the first one's one so we go for five c one which is five the next one five c two is ten and the next one 5 c3 is also 10. there we go so that's how many lots of this pattern we're going to have now this time the first piece in our pattern is a three that's okay and the second piece in our pattern is negative a third x now obviously that's okay as well but we're just going to be careful we're typing it into the calculator another way that you could write this here is you could write negative x over 3 so you could relieve it as that as well i might actually write it as negative x over 3 instead just to sort of um make the terms a little bit easier to write but let's go for this then so we have one lot of the first piece which is three to the power of five so we've only got three more to do now and we're going to add to that we've got five lots off for the next piece let's just highlight these as we go so now we're on this five lots so we've got five lots of and we have three to the power of four and then we've got this negative x over three as well so negative x over three there we go on to the next piece again so we have 10 lots of this one so we've got 10 lots of so we've got 10 lots of and we've got 3 getting down to the power of 3 and then we've got this negative x over three which is now being squared and there we go and then for our last piece here again we've got 10 lots of this so we have 10 lots of three now i've gone down to a power two and we've got the negative x over three which is now going to be cubed right so let's deal with all these pieces the first one's okay we've got three to the power of five work that out nice and easy that's 243 so 243 is our first one you can always tick these off as you go so that's done the next one five times three to the power of four well let's work that out to start with that's 405 so 405 and that's going to be multiplied by negative x over 3. we'll come back to that and deal with that in a sec onto the next one so we have 10 lots of so 10 lots of 3 cubed so we have 3 cubed which is obviously 27 times that by 10 is 270. so we have plus 270 lots of let's have a look negative x over three well if we square that's going to become positive so we're gonna get x squared on the top and on the bottom three squared is nine so x squared over nine there we go and that's that p start with and onto the next one obviously just being careful with those when you're squaring that remembering that's going to make it positive so we get the x squared on the top positive x squared nine on the bottom on to the next one ten times three squared is 90. so we've got 90 and that's going to be and we've got negative x over 3 cubed so when we cubed it's going to stay negative so we're going to have negative x cubed on the top and 3 cubed which is 27 on the bottom there we go and that's that's the last piece that with right now simplifying all of this then so we've got 243 at the start that's all good and then we've got to be careful on this next bit now negative x over three just obviously we've got we could have had the negative a third there we basically just want to do a third of 405 so if we do a third of 405 we have 405 divided by 3 is 135 and obviously it's negative there so it's going to be negative 135 x there we go on to the next piece we have 9 on the bottom there so we want to divide that 270 by 9. obviously it's positive this time so we'll divide that by 9 270 divided by 9 is 30 so we get 30x squared onto the next piece 90 divided by 27 is not as nice it gives us a fraction here and gotta be careful because it's negative so that's not going to be plus there it's going to be minus so we get minus 10 over 3 which is what 90 divided by 27 comes out as with an x cubed and there we go and that is those four first four terms finished so obviously just be careful on this because obviously sometimes you can get these fractional pieces or these fractional coefficients here and that is quite common on these questions so you just need to be nice and careful obviously just watching out for any pieces here that are negative like the two there just making sure that you show those with your negatives here as well when you do have these fractions involved just remember obviously you're just finding a fraction of that coefficient there again it's just like 405 multiplied by negative a third in terms of actually getting these coefficients so here we've got the 135 a third of that and one ninth of 270 gives us the 30. obviously our fractional piece at the end as well there so just being careful there's the first four terms and ascending powers of x and we've simplified each term okay so this question says find the first three terms and descending power of x of the binomial expansion of one plus p x to the power of ten it says where p is a non-zero constant and give each term in its simplest form so obviously p is an actual number that we're going to find uh and obviously it hasn't asked us to find that yet it's just asking us to find those first three terms in the next part the process though or the next part of the question it says given that in the expansion the coefficient of x squared is nine times the coefficient of x find the value of p so we're going to find that value of p uh you know later on in the question and then it says hence write down the coefficient of x squared and we're just going to follow this step by step and have a look at how to approach this then you've got one so if i go up so finding the first three terms then we've got a power of 10 and just bearing in mind our first piece is a one which is always nice and our second piece is this plus px so as we've got a power of 10 we need to do 10 c 0 to start with so we've got 10 c 0 which is 1. we're only finding the first three terms we've got 1 10 c 1 is 10 and then we have 10 c 2 which is 45 there we go so that's how many lots of each we're going to have on this particular one here so for this one then we've got one lot of and then that's 1 to the power of 10 the first piece we're going to add to that we've got 10 lots of and it's now 1 to the power of 9 and then px as our second piece and then for the final one we have 45 lots of 1 to the power of 8 and then p x to the power of 2. there we go right so if we simplify this down then 1 to the power of 10 gives us 1 for our first piece 10 times 1 and then times the px gives us 10 px so we have 10px for the second piece and then for our final piece there 45 times 1 is 45 and px in bracket squared squares both those letters so we get p squared x squared there we go square in both of those letters so 1 plus 10 px plus 45 p squared x squared and there's our first three terms in ascending powers of x now it says given that in this expansion the coefficient of x squared is 9 times the coefficient of x well the coefficient of x squared let's have a look and let's highlight that it's 45 p squared that's what's in front of the x squared and the coefficient of x is 10 p so what it's saying is okay if one is nine times the other it's saying if we multiply this coefficient here by nine it would be equal to this coefficient here or in other words nine times ten p is going to be equal to 45 p squared or you could do it the other way around we could do 10 p is equal to 45 p squared divided by 9. completely up to which one you do obviously 45 nicely divides by nine so that would be quite nice way to do it i'm just going to follow this process as i've written it down this way now it doesn't really matter which way it's probably easier actually to divide it by 9 because i think we're going to divide it by 9 anyway in a sec but let's have a look so if we times that by 9 anyway we get 90 p equals 45 p squared and then we just need to solve that as an equation so dividing both sides by p would obviously remove the p from the one side so we'd get 90 equals 45 p just dividing both sides by p there then we could divide both sides by nine or we could actually divide both sides by 15. completely up to you or just divide both sides by 45 and that will tell us what p is so if i divide both sides by 45 just write that down divide by 45 as we've got 45 p there we get two equals p or p equals two and there we go there's our answer for that part so p equals two equals p then it says hence write down the coefficient of x squared well the coefficient of x squared is 45p squared there we go and we now know that p is 2 so that's 45 multiplied by 2 squared which is 45 times 4 and 45 times 4 is 180 and there we go there's our coefficient of x squared 180 right there we go so there's our final answer so in terms of actually what we did there obviously we had a different piece uh within our bracket we had a px i'm just being careful with that i think the point where you need to be careful with it is obviously this bit here where we get the p squared x squared i think other than that the rest of it was quite nice and simple and then obviously just reading the wording carefully it said one coefficient was nine times the other so obviously multiplying that by nine or dividing the other one by nine there gave us a little equation to solve we got our value of p and then sub that in to find the coefficient of x squared okay so when having a look at binomial estimations we can have questions like this so it says find the first four terms and ascending powers of x of the binomial expansion of and then we have 1 minus x over 4 to the power of 10 and use your expansion to estimate the value of 0.975 again to the power of 10 giving your answer three decimal places so the first thing we want to do is do our binomial expansion so for a power of 10 we'll want to use 10 c1 which is equal to 1. we're going to want to use 10c2 which is equal to 10 10c3 which is equal to 45 and then 10 c4 for our fourth which is equal to 120 so putting this into our binomial expansion our first piece is one so we have one to the power of 10. our next piece we're going to have plus 10 lots of and then we're going to have 1 to the power of 9 plus oh no sorry multiplied by negative x over 4 and that is to the power of 1. we're now going to move on to our next piece so plus 45 lots of one drops down to a power of eight and our negative x over four jumps up to a power of two and then for our final piece and it's quite a large question so let's just try and move some of this out of the way so that we've got enough space there and for our final piece then we're going to have 120 lots of one to the power of down to seven and negative x over four is going to go to the power of three so we just need to simplify all of this one to the power of ten at the start is going to stay as one then we have ten times one times the negative 1 on the top there so that would be 10 over 4 and that is going to come out as 2.5 so it's going to be negative as it is and let's just highlight that the negative x over 4 so rather than being a plus we're going to have a minus so that'd be minus 2.5 and again you can just type that into your calculator but minus 2.5 x for the next piece we have got 45 times 1 times the negative and obviously we need to expand that bracket as well so negative x over 4 squared would come out as x squared over we can just write this to the side x squared over 16. so we're going to divide by 16 so 45 divided by 16 and if we type that into our calculator gives us plus 2.8125 so 2.8125 and that's going to be x squared and then on to our final piece x uh minus x over 4 cubed is going to be negative and we're going to have x cubed over 4 cubed which is 64. so i'm going to be dividing by 64 on this piece and it's a negative x cubed at the end there so it's going to be a minus 120 times 1 and then divided by the 64 gives us negative 1.875 [Music] and that is x cubed and we're only finding the fourth first four terms here but it does continue so we'll put that it continues there so that is our binomial expansion and now we need to go about solving this estimation so it says in the question here use your expansion expansion to estimate the value of 0.975 to the power of 10 which matches what's in here so what we want is that piece in the bracket the 1 minus x over 4 to be equal to 0.975 so if we set them equal to each other to start with to find out our value of x and then we can actually sub that into our binomial expansion so if we set them equal to each other we get 1 minus x over 4 which has to be equal to as stated in the question 0.975 and we just need to go about solving that and you can do that in two ways we could add the x over four to the other side and then subtract the 0.975 from one and that leaves us with x over four which would be equal to naught point 0.025 again multiplying that by 4 then we would get a value of x which comes out as 0.1 so x is 0.1 we can now use that to substitute it back into our binomial expansion so you can write this down you can write down substitute x equals 0.1 into the expansion but if we just go about substituting that in we've got one at the start and again you could write sub x equals 0.1 but 1 minus 0.1 times 2.5 comes out as 0.25 and sub it into the next part so 0.1 sub into x squared gives you naught point naught 1 and then we need to obviously multiply that by 2.8125 and that comes out as 0.028125 and we've got our last piece there and again you're just typing these all into a calculator take away one point eight seven five multiplied by naught point one cubed and that comes out as naught point naught naught let's read off the calculator one eight seven five there we go and if we simplify all of that uh again just type it all into the calculator it comes out as naught point seven seven six quite a lot of decimals here not point seven seven six two five that comes out as i've just typed in the 0.975 to the power of 10 which is a lot the more decimals so when we sub that into our binomial expansion we get 0.77625 now the question here says giving your answer to three decimal places so if we type in 0.975 to the power of 10 into our calculator that gives us the answer and that's where i've got all my decimals from it comes out as 0.776329 there we go there are more decimals there but i'm going to stop there because it only wants it to three decimal places so if we look at these two decimals you can see 0.776 on the top one and 0.776 on the bottom one and we've got a matching amount of decimal places there so they round to the same number so we would give our answer as 0.776 to 3 decimal places and we've not got much space to write that down but if we just put it over here to the side we would just write naught 0.776 and just put that that is two three decimal places and there we go that would be our final answer for using estimation with binomial expansions okay so that's the end of looking at binary expansion hopefully that was useful and helpful if it was don't forget to like don't forget to comment and don't forget to subscribe and i will see you for the next one [Music] you