AASL 1.9.2 Binomial expansion (finding a term) | Free Mathematics Analysis and Approaches (AA) Video | RevisionDojo

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AASL 1.9.2 Binomial expansion (finding a term)

Name: AASL 1.9.2 Binomial expansion (finding a term)
Uploaded: 2025-01-04T10:03:14.160Z
Duration: 11 min 44 s
Description: Learn AASL 1.9.2 Binomial expansion (finding a term) in this free IB Mathematics Analysis and Approaches (AA) video lesson for SL 1.9—Binomial theorem where n is an integer.

Learn AASL 1.9.2 Binomial expansion (finding a term) in this free IB Mathematics Analysis and Approaches (AA) video lesson for SL 1.9—Binomial theorem where n is an integer.

About this video

Learn AASL 1.9.2 Binomial expansion (finding a term) in this free IB Mathematics Analysis and Approaches (AA) video lesson for SL 1.9—Binomial theorem where n is an integer.

In this lesson, the focus is on finding a specific term in the binomial expansion of the expression $(-2x)^{23}$ . The process involves using the binomial theorem, specifically the formula for the $n$ th term, which is given by:

$T_{r + 1} = (\binom{n}{r}) (a$

)^{n - r}

(

b

)^{r}

T_{r+1} = \binom{n}{r} (a)^{n-r} (b)^{r}

Video transcript

00:00Hi everyone. So in the

00:02last lesson we learned about

00:04the binomial expansion. We learned

00:06what it was all about

00:08and how to fully expand

00:09binomial. In this lesson we're

00:11going to look specifically at

00:12how to find a specific

00:14term. This is probably the

00:17more likely question that you

00:21would get in an exam,

00:23but I think knowing how

00:24to fully expand it, well

00:26you could be asked that

00:27but it's certainly

00:28me is essential to you

00:29understanding the whole thing. I've

00:33deliberately here chosen quite a

00:35difficult example. Again, reason being

00:38if you can do this,

00:39you can pretty much do

00:40any of them. Now, I

00:44told you in the last

00:45lesson that we're going to

00:46use this part of the

00:48formula when finding a specific

00:51term. So we're going to

00:52be focusing on this part

00:53of the formula in this

00:54lesson. I'm going to show

00:56you

00:56two methods. One method is

00:58my method. Let's call this

01:01method one. Mr. Flynn. Let's

01:09say the Mr. Flynn way.

01:11And then the second method

01:15I'll show you is the

01:16kind of more technical mathematical

01:18way. Now there's absolutely nothing

01:19wrong with my way. In

01:22fact, I recommend you doing

01:23it this way.

01:24Okay, it just requires a

01:27little bit of not guessing

01:29what the word I like

01:30to use is deducing, deducing

01:32what to do. And look,

01:33I'll explain it, not explain

01:34what I mean by that

01:35as I do it. So

01:36it says find the term

01:37in x to the 25

01:39in the expansion of x

01:40where minus two x all

01:42to the power of 23.

01:43Okay, I'm gonna go straight

01:45to this, this part of

01:47the formula. So it's n

01:48choose or so it's 23,

01:50it's 23,

01:52choose or now I don't

01:54know what or is so

01:56I'm gonna put a box

01:57the first term is remember

02:01we do bracket brackets so

02:04the first term is x

02:05squared so I put a

02:06bracket an x squared and

02:08I'm not gonna put a

02:09box there because it's the

02:11power of something I don't

02:12know what the power is

02:13and then the second term

02:15is negative 2x and I'm

02:17gonna put a box here

02:19because I don't know what

02:19that is to the power

02:20Okay, now it says find

02:24the term in x to

02:25the power of 25. Look,

02:27this is hard for a

02:28number of reasons. One, I've

02:29picked large numbers like 25

02:32and 23, two, I've put

02:34a negative in there, and

02:36three, I have an x

02:37above the line and below

02:39the line, and this is

02:40x squared. Now, maybe the

02:43first thing you'd think of

02:44is right, let's go with,

02:47you want the term

02:48I'm going to X to

02:48the 25, let's put 25

02:50here. Well, that's certainly wrong

02:52for many reasons. These two,

02:55these two boxes have to

02:58add up to 23. So

03:00they always add up to

03:0123. That's why you have

03:0323 and zero, or 22

03:06and one, or 21 and

03:07two, or 20 and three.

03:10They always add up to

03:12N, which in this case

03:13is 23. So I need

03:15these two to add up

03:1623. But let's, let's say

03:20I put 23 and zero

03:21in there. If I put

03:2223 and zero, what happens

03:23is this actually becomes x

03:26to the power of 46,

03:27which is not 25. So

03:29that's definitely wrong. What if

03:32I put 20, what if

03:35I put 22 and one?

03:37Well, this is now x

03:39to the 44 and this

03:41is x to the one,

03:42which is underneath the line.

03:43So what happens is you've

03:44If you're 44 and you

03:46end up, so it's x

03:48to the power of 44

03:49over x to the power

03:51of 1. Now forget about

03:52the negative and the 2

03:53for a second. I just

03:54want to find that I

03:56need to find that x

03:57to the 25. Is that

03:58going to give me x

03:59to the 25? No, it's

04:00going to give me x

04:01to the 43. So it

04:03can't be that. So it's

04:06not 22 and 1. What

04:07if it was 20,

04:12If it was 20 and

04:133, well that would give

04:14me 40 minus 3, which

04:17is 37. No. And now

04:19let's jump down a bit.

04:20And this is what I

04:21mean by deducing and kind

04:23of guessing, but it's not

04:25this isn't really guessing. It's

04:28deducing. It's figuring out rationally

04:32what's going on. So I'm

04:33going to jump down now

04:35to let's go to

04:40Let's go to 15 and

04:428 imagine if I put

04:44if I put 15 there

04:45and 8 there because 15

04:47plus 8 is 23 so

04:51if I put 15 and

04:518 I'm gonna get 2

04:53times 15 is 30 minus

04:568 which is 22 All

04:59right, I'm close so I

05:01went down too far. Let's

05:02let's try 16 and 7

05:07now if I put 6

05:0816 and 7 there. I'm

05:10gonna get x to the

05:11power of 32 minus 7

05:16is 25. So that is,

05:20I have chosen the correct

05:22powers and that is the

05:24hard part of the whole

05:26process done. Once I have

05:2816 and 7, I've got

05:30it. This I'm gonna make

05:327. I could actually make

05:34it either. It can be

05:3416 or 7, but look,

05:36They have B to the

05:39bar of O so that's

05:40O. So, but 23 choose

05:4216 is the same as

05:4323 choose 7. Remember Pascal's

05:46triangle is symmetrical. And now

05:49I just have to multiply

05:52all this out. So I'm

05:54going to get 23 choose

05:577. So I need to

05:58do menu, probability, combinations, 23

06:03choose 7s is 22.

06:0423, comma 7, presenter. And

06:09it is, as I said,

06:11I chose a difficult question.

06:13245 ,157. So 2, 4,

06:185, 1, 5, 7. That's

06:21just this. How many ways

06:23can I choose 7 out

06:24of 23? This is now

06:27x to the power of

06:2832. So this is x

06:30to the power of 32.

06:32Let's put that

06:33bracket and this is negative

06:372 to the power of

06:387 is 128 and it's

06:43over x to the power

06:44of 7. So I've done

06:462 to the power of

06:467, I've done x to

06:48the power of 7 and

06:49negative to the power of

06:507 is going to give

06:50me the negative. This then

06:53equals negative. I need to

06:55multiply that by 128. My

06:57word is getting big so

06:59I'm going to do multiply

07:00by 1.

07:01to 8. So I have

07:063 -1 -3 -8 -0

07:08-0 -9 -6. 3 -1

07:10-3 -8 -0 -0 -9

07:13-6. That is the coefficient

07:17and then it's x to

07:19the 32 over x to

07:20the 7, which I knew

07:22and I know is going

07:23to give me x to

07:25the power of 25. It

07:27says find the term in

07:28x to the

07:29Power of 25, that is

07:30it. That is the term.

07:33That is the term in

07:35x to the power of

07:3725. So if you're lucky,

07:39I didn't ask you to

07:40expand this whole thing out,

07:41because if you did, that

07:43would just have been one

07:44of the terms. Okay, so

07:46that's method one, Mr. Flynn's

07:49way. Method two is the

07:51slightly more technical way. And

07:55you can choose, I have

07:56no problem.

07:57if you prefer to do

07:58it this way. So what

08:01we do is, we say

08:03it's 23 choose R. We

08:07say x squared to the

08:09power of, now it's here,

08:14so it's n minus R,

08:15which is 23 minus R.

08:17So this is gonna be

08:1923 n minus R. And

08:21this, the second term is,

08:25negative 2 over x. So

08:27here it is, this is

08:28negative 2 over x and

08:30this is to the power

08:32of r. And the key

08:35to the whole thing is

08:35to find r. What is

08:37r? I found it here

08:38was 7 by doing this

08:39little thing. The way we're

08:41going to find r in

08:41this situation is we're going

08:44to say, right, let's look

08:46at the x's. So this

08:48is x, just looking at

08:49the x part of it.

08:51x to the power of

08:532 times this, so 2

08:58times 23 is 46 minus

09:012 times or is minus

09:032 or so that times

09:08x to the negative or

09:12because look this is x

09:16to the part because it's

09:16underneath the line that's x

09:18to the power of negative

09:191 so this would be

09:20times x to the power

09:21of

09:21negative r. And this has

09:24to equal x to the

09:25power of 25 because we're

09:27trying to find the term

09:28in x to the 25.

09:30So it's like saying what?

09:31Let's find the r that

09:33gives us this. So this

09:35is 46. I'm going to

09:39add the powers because this

09:41is a law of exponent.

09:43First law is the first

09:44law of exponents. 46 minus

09:473r equals

09:4925 then 3 or equals

09:5746 minus 21 or minus

10:0025 which is 21 and

10:02then or equals 7 and

10:05once you have or equals

10:077 then you just go

10:09into this step here or

10:127 or 7 23 1

10:157 is 16 and then

10:16you're here and I'm not

10:17gonna repeat that again. So

10:20look, but my advice would

10:22be to try it this

10:23way and if you really,

10:24really get stuck or you

10:25can't do it, then go

10:26down to this way. But

10:27I find this is, this

10:32ends up being more, a

10:34more challenging way to do

10:35it because they never give,

10:37they never give you, they

10:40never gonna give you something

10:41worse than this. And I

10:43figured this one out without

10:45too much trouble.

10:45not fair enough. I'm the

10:47teacher, but I'm sure you

10:50can do it as well.

10:51If you have to, if

10:51you have to do them

10:52all, start with do 23

10:54and zero 22 and one

10:5621 and two 23 and

10:59just keep going until you

11:01kind of get the hang

11:01of it. You can, and

11:03you can start to jump

11:03down or whatever. Okay, that's

11:06the by no one expansion.

11:08Pretty much every single year,

11:10almost always in paper two,

11:11it comes up. It's normally

11:13question like this, find the

11:15term. In this or sometimes

11:17it's like they give you,

11:19they say, find two terms

11:21and then they say make

11:22them one is twice as

11:24big as the other. So

11:25you have to form an

11:26equation and solve it that

11:27way. Sometimes they say find

11:29the term and sometimes they

11:30say find the coefficient. If

11:32they say find the coefficient

11:33of x to the power

11:34of 25, then it's just

11:36this that they want, the

11:37negative 3, 1, 3, 8,

11:390, 0, 9, 6. Okay.

11:41See you in the next

11:43lesson.