21 - Pascals Triangle & Binomial Expansion - Part 1 - Free Educational videos for Students in K-12

21 - Pascals Triangle & Binomial Expansion - Part 1 - By Math and Science

Transcript


00:00	Hello . Welcome back to this lesson . The title
00:02	of this lesson is called pascal's Triangle and binomial expansion
00:06	. This is part one of two . I'm really
00:08	excited to teach this because when you look at it
00:11	in a textbook , it looks very , very confusing
00:14	. There's tons of exponents flying around in what we're
00:16	trying to do in this lesson , but I'm gonna
00:18	break it down so that literally like a second grader
00:20	or maybe 1/4 grader can certainly do this , although
00:23	it will look very intimidating . It'll be very ,
00:25	very simple . First one I want to do is
00:27	show you what we call a pascal's triangle . It's
00:29	a very simple , kind of a neat little concept
00:31	. I'll show you what that is , but ultimately
00:34	keep in the back of your mind , what we
00:35	want to do is want to learn how to expand
00:38	binomial . What I mean by that is just visualize
00:40	the binomial A plus B all raised as a parentheses
00:45	to the second power . We already know how to
00:47	do that with foil . But the problem is what
00:49	if you get to a higher binomial power ? Like
00:51	what about A plus B in parentheses raised to the
00:55	sixth power . To the sixth power . We don't
00:57	know how to Well , we can certainly multiply it
00:59	out by hand , but it's a giant pain .
01:01	What if you have a plus B raised to the
01:04	17th ? Power ? Tons of multiplication . You'll have
01:07	to do by hand . Uh I'll take pages of
01:09	work to get the answer . This lesson is going
01:11	to show you how to get those answers . Really
01:13	, really , really fast . And in order to
01:15	do that , we need to understand something first called
01:17	pascal's triangle . So let me introduce that first and
01:20	then I'll connect the dots and show you how we're
01:22	going to ultimately use it . Here . We have
01:25	uh pascal triangle . Obviously the triangle means triangle .
01:31	Alright , here's what you do . First to start
01:34	the triangle off , you put a number one right
01:36	here and then to the left and to the right
01:38	of that one , you put another number one and
01:40	another number one . So you kind of make a
01:42	little triangle of ones . This is how it begins
01:45	, it always begins the same way one on the
01:47	top flanked right underneath by another one and another one
01:50	. Okay , next in the next row , you
01:52	go down to the left . See all the diagonals
01:54	here of this triangle are all going to be a
01:56	one . You'll see why in a second but put
01:57	it one right here now to get the number that
02:01	goes over here , because this is 11 line ,
02:04	one element to element . So this is gonna be
02:06	three elements to get the one in the middle here
02:08	. What you do is you look at what's above
02:10	it and you add them up . So one plus
02:12	one is two . So we put a two right
02:14	here And then off off to the end of the
02:17	edges of the Triangle Always has a one . So
02:19	again we now have 11 element to elements three elements
02:23	to go to the next line of the triangle .
02:24	We always started with the number one . And then
02:26	to get what goes between these guys , we add
02:28	them up . One plus two is three . And
02:31	then we skip over this to to this spot here
02:34	. They're only really putting numbers in the kind of
02:37	in the spaces between the numbers above . So we
02:39	skip over here , one plus two again is three
02:41	. And then on the edge of the triangle we
02:43	always put a one . So you see to form
02:45	uh formulae pascal's triangle , all you're doing to write
02:48	the elements of the next line is you're just adding
02:50	up the two numbers that are directly kind of above
02:53	it into the left and to the right . So
02:55	to go to the next line again , you always
02:57	start it with the one . So put it one
02:59	there and then to plug a spot into this location
03:02	, it's one plus three which is four . To
03:05	put a spot here , it's three plus three which
03:07	is six . To put a number here , it's
03:08	three plus one which is four . And then on
03:11	the end you always have , one's always on the
03:13	edge of the triangle . Okay , we'll do two
03:16	more lines just to show you how it works .
03:19	But before we get too much farther I'm gonna draw
03:21	a little a little line here showing that to get
03:23	the number three here . What you're doing is your
03:25	adding a one plus two to get the number four
03:28	here . What you're doing is your adding the three
03:30	plus one . So I'm not gonna draw these little
03:31	things everywhere , but these are how you're getting the
03:33	numbers . This would form a little little triangle like
03:36	this and so on , showing you how the addition
03:38	happens , we'll do two more lines and then we'll
03:41	kind of talk about how this is useful for doing
03:43	algebra here . So in the next spot it's 1-plus
03:46	4 is five . In the next spot , four
03:49	plus six is 10 here , six plus four is
03:52	10 here , four plus one is five . And
03:55	then here we have a one always on the edge
03:58	. The last line that we will do , we
03:59	always have a one in the diagonal . One plus
04:02	five is 65 plus 10 is 15 , 10 plus
04:06	10 is 20 10 plus 5 , 15 , 5
04:10	plus +16 And then always a one out here and
04:14	again , just to drive at home to get ,
04:16	for instance , this one you're adding those two numbers
04:18	together , for instance , to get something like this
04:20	, you're adding these two numbers together . That's how
04:22	you get . So let me spend just a second
04:24	double checking my triangle . +111121 13311464115 10 , 10
04:31	5116 15 2015 6 and one . That's correct .
04:35	So a couple things I want to point out before
04:36	we go any farther . First of all noticed that
04:39	you can generate any number of rows of this triangle
04:42	that you want to right now we stopped right here
04:45	. But if I need you to go one more
04:46	level deeper , you could easily do it by just
04:48	adding the appropriate spots in the triangle . Uh to
04:52	get the next line and then the line after that
04:55	, we would just add the numbers right above it
04:57	and so on . So if I wanted to get
04:59	the 100th row of this table , I would have
05:02	to generate all the rows in between . But eventually
05:05	I could get down to road number 100 . Just
05:06	by edition , a computer could do that pretty easily
05:09	. Right . More importantly than that , it is
05:12	noticed that this thing has symmetry . If you cut
05:14	a line right here in the middle , then the
05:16	numbers to the left and the numbers to the right
05:19	or mirror images here , you have 15 61 for
05:22	that role . Here you have and the 20 was
05:25	in the middle , so it doesn't really have a
05:26	mirror image . But here there is no middle number
05:28	. So 10 is a mirror . Five is a
05:30	marijuana is a mirror again , six is in the
05:32	middle so it's not a mirror . But on the
05:34	other side of that four and one you can see
05:36	the three , the two and the one and so
05:38	on . So everything is a mirror image . And
05:39	that's always going to be the case . The right
05:41	side of the triangle should be exactly the same numbers
05:44	as the left side of the triangle split right down
05:46	the middle . All right now , the last thing
05:49	I want to do is I want to label the
05:52	rows of this thing . Now I know you won't
05:54	quite understand why just yet , but I promise you
05:57	in a minute there will be a really good reason
05:59	that we're doing this this first row here . Even
06:01	though most people would call it road number one ,
06:03	we're actually going to call it row zero . Okay
06:08	, we're gonna call this one row one . I'm
06:12	going to call this one row to , we'll call
06:15	this one row three , we'll call this one row
06:20	four , we'll call this one row five and then
06:25	this one right here we'll call it ro six .
06:29	Now again , I'm gonna show you and tell you
06:31	exactly why we're labeling the first row zero instead of
06:34	one , but it's just gonna get confusing . Just
06:37	let me get into the meat of it a little
06:39	bit more and you'll instantly understand why we call that
06:41	rose zero . Now this is a neat little game
06:44	, mathematical game . You can construct this triangle ,
06:46	pascal's triangle has a lot of uses . The use
06:49	that we are going to use it for is to
06:50	help us expand by no meals . So what I
06:53	want to do is go down memory lane with you
06:56	and expand a few simple by no meals . And
06:59	we're going to use that to generate a general way
07:01	to expand any binomial with any power using pascal's triangle
07:06	. So for instance , if we want to expand
07:09	A plus B To the power of zero , what's
07:12	the answer ? Well , no matter what A and
07:15	B are , it's raised to the zero power ,
07:17	so it's equal to one . Okay . What about
07:21	A plus B raised to the first power ? Well
07:25	, it's it's if it were squared , it would
07:27	be multiplied by itself again , but it's just the
07:29	first power . So really what you get here is
07:31	just A plus B . Okay , so far this
07:34	stuff is really easy . You're saying why is he
07:35	going through this ? Well , let's continue on down
07:38	through here . What if you do uh a plus
07:41	B quantity squared ? Now , I know that a
07:43	lot of , you know the shortcut tricks and memorize
07:45	how to do that . But if you did it
07:47	manually , what you would say is A plus B
07:50	. Multiply by a plus B . That's what this
07:53	means , right ? And how would you multiply it
07:55	out ? You would say first terms A squared inside
07:58	terms is a . B . Outside terms is also
08:01	a B . And last terms is B squared .
08:05	So when you multiply these together , you get a
08:07	squared plus two here , times a B plus B
08:13	squared . Now you might say why do we care
08:16	about this ? Now ? I'm gonna do one more
08:18	. But before I do anything else , let me
08:21	just point out to you something really quickly notice pascal's
08:24	triangle has a one in the top , Notice the
08:27	coefficient here and the answer is a one . The
08:30	second row of pascal's triangle has a one and then
08:32	a one those are the coefficients of this , There's
08:35	a coefficient of one in front and a coefficient of
08:38	one here notice and row three of the triangle ,
08:40	that is one , then two , then one notice
08:43	those are the coefficients one than to than one .
08:46	Now I understand that we have an A squared and
08:49	an A B and b squared and you're like ,
08:50	oh it looks confusing , but all I want you
08:52	to know right now is that it appears that the
08:55	triangle predicts the coefficients that are going to be in
08:58	our answers as we keep making larger and larger powers
09:02	of A plus B . So when we say binomial
09:04	expansion , remember binomial is just anything with two things
09:08	added together ? Bicycle means two wheels . Um uh
09:13	so binomial means too little terms . So we're expanding
09:16	them to a power and so we can see the
09:18	terms match up . Now let's do one more .
09:20	I'm not going to go down this whole triangle and
09:22	prove it to you . But let's do one more
09:24	A plus B to the power of three . How
09:26	would you do this ? We don't have a ready
09:29	made formula . But you know that this is the
09:30	same as A plus B times A plus B squared
09:35	. You know that because you can add the exponents
09:37	and get the three , but we just calculated what
09:40	A plus B squared is . So really this is
09:42	A plus B times we just arrived at a squared
09:47	plus two A B plus B squared . So this
09:52	results reduces down to having to multiply this . Now
09:55	you can see where the pain comes from because to
09:57	do this multiplication , you'll distribute in , distribute in
10:00	distributing the A . Then you move to be distribute
10:03	B . B . And B in . So let's
10:04	do it real quick . I think it's worth doing
10:05	eight times A squared is a cube eight times this
10:10	is two times then a squared B . And then
10:15	the eight times the third term is a B squared
10:19	. Alright now we move our finger to be and
10:21	push it in so it's gonna be be a square
10:23	. We're gonna write it with the A . Term
10:24	first a squared B . Then be times this will
10:27	be to a B squared to a B squared .
10:32	Then the B . Times to be square will be
10:34	be cube . See how many terms I have .
10:35	It's really ugly but I have uh I can combine
10:39	some of them so I have a cube notice I
10:41	have to a squared B . And then I have
10:44	an A . Squared B . Here so I have
10:45	another one of those so I can add those together
10:47	and make it three times a squared beat . So
10:51	I've added that now I have a B squared and
10:53	I have a to a b squared , so I
10:54	can add that for a three A B squared .
10:57	So I've really taken care of all of these terms
11:00	. The only thing left I have is this one
11:01	be cubed , so this is equal to a plus
11:05	B , raised to the third power . So if
11:08	our theory is correct , the coefficient in front of
11:11	here , in front of here , in front of
11:12	here and in front of here should be 1331 The
11:17	next row of the triangle is 1331 So I'm not
11:20	gonna go any farther . But I hope that I've
11:21	proven to you that this pascal's triangle actually does predict
11:25	the coefficients of all of the answers of what we
11:27	call binomial expansion . So if you need to expand
11:30	the binomial , like in this case we've done A
11:33	plus B to the first , A plus B to
11:35	the second , A plus B to the third .
11:36	If you needed to do for instance , A plus
11:38	B to the fourth power , then I'm gonna show
11:41	you in a second how to get all of these
11:43	A . Bs and stuff together . I'm gonna show
11:45	you how to do that . But the coefficients themselves
11:47	have A plus B to the fourth power must be
11:50	one than four than sixth and four than one .
11:54	That's why I label this row four because A plus
11:57	B to the fourth power , you just look at
11:59	row four and there you go . A plus B
12:01	to the third power . You read this line off
12:04	A plus B to the second power . This one
12:06	to the first power this one . And now you
12:08	know why this is labeled Rose zero because when you
12:12	take A plus B to the zero power , that
12:15	gives you one , which is the top of the
12:16	triangle . So if we were to label this row
12:19	one , it would get confusing . But it's very
12:21	easy . Now if you're just expanding A plus B
12:23	to the , let's say the sixth power and your
12:26	label rose zero , then +123451 more road down here
12:30	is six , then A plus B to the sixth
12:33	power is going to have all of these coefficients .
12:35	And the answer . All right . So what I
12:38	want you to do is take a look at what
12:40	we have here and try to keep them in your
12:42	mind . First we had a one , then we
12:44	had an A plus B . Then we had a
12:46	square plus do A B plus B squared . Then
12:48	we had this large answer with the +1331 I've actually
12:52	filled out a similar table to this on the next
12:55	board already ahead of time . I just didn't want
12:57	to hit you over the head with it . This
12:59	is the first six actually the 1st 012345 Yeah the
13:04	first six uh levels of pascal's triangle here actually the
13:09	1st 123456 1st 6 of them . Uh lines here
13:17	. And so what I've done is I've kind of
13:19	tabulated here , so A . Plus B to the
13:21	first power is one , and I've colored the one
13:23	in because this is gonna now match pascal's triangle what
13:26	they want . Then you have a plus beat of
13:28	the first , it's 18 to the first plus B
13:31	. To the first . Or you can think of
13:32	it as just A plus B . The one in
13:34	the one comes straight out of the table and you
13:36	can just read it down , 1 to 1 come
13:39	from there . Then we already did . 1331 comes
13:43	straight out of there . If you were to continue
13:45	multiplying A plus B to the fourth power . You
13:48	have a ton of things to add multiply and add
13:50	together . But you would arrive at this and it
13:52	would be 18 of the fourth for a third B
13:56	and so on . But the coefficients 14641 exactly matched
14:00	the table . And then again a plus B to
14:03	the 5th . Power . 15 10 10 51 match
14:08	this 15 10 10 51 Now I took this table
14:11	one more row and I didn't bother to write one
14:13	more row here because I'd be riding forever . So
14:16	I just wanted to show you that when you expand
14:18	these binomial it does form a triangle that can be
14:21	predicted exactly from pascal's triangle . So it allows you
14:24	to do something like A plus B to the 17th
14:27	power . As long as you know the elements of
14:29	that triangle . Alright , now I have to give
14:32	you a couple notes here before we do any problems
14:34	because ultimately the problems I want to give you as
14:36	I want you to expand by no meals , right
14:38	? So I need to show you some really important
14:40	things . Okay , the first thing is you may
14:43	not have noticed before , but now when they're all
14:45	in the same board , look at A plus B
14:47	to the power of two . Notice that this is
14:50	a squared and the last term is B squared and
14:53	the middle term is just eight times B . But
14:55	A has a power of one and B has the
14:57	power of one . If you in your mind add
15:00	one plus one , you get to . So really
15:02	when you're taking a plus B in your squaring it
15:04	every term , the some of the exponents must be
15:08	equal to . In this case to notice the exponents
15:11	add to to the exponents here , add to to
15:13	the exponents here add to two . Okay try it
15:16	for the next line . This is a plus B
15:17	cubed . So this is a cube . So the
15:20	exponents add 23 matching this . These exponents add 23
15:25	These exponents add 23 These exponents add 23 It's the
15:28	same pattern for all of it . If you look
15:30	at A plus B to the fifth , every exponents
15:33	uh in every one of these terms they must sum
15:36	up to the power of the thing . You're expanding
15:38	too . So the some of these exponents must be
15:41	five , must be five , must be 53 plus
15:44	two must be 52 plus three must be 51 plus
15:47	four and then must be five B to the fifth
15:50	. So the trick is if you if I just
15:53	asked you tell me what a plus B to the
15:55	five is . The first thing you need to know
15:56	is what these coefficients are . You can get those
15:59	just by having your pascal's triangle written down on paper
16:03	. The second thing is you need to know uh
16:05	I'm talking about doing this without multiplying it all out
16:08	. You saw how much of a pain it was
16:09	just for the cube . Imagine if you did it
16:12	for 1/6 power . It would have numbers and letters
16:14	everywhere . So you have to know the coefficients that
16:17	comes from past house . Tribal . How do you
16:19	predict all of these little terms here ? Let me
16:21	show you a little secret . Let's go look at
16:24	this one for instance , that's probably the easiest one
16:25	to look at . To the cube power . What
16:27	you do for the first term is you start with
16:29	a cubed just by itself . Then for the next
16:32	term , what you do is you're always reducing a
16:35	down by one . So notice this is a two
16:38	, but in order to make it all equal three
16:39	, B has to come up from a zero power
16:42	. It has to come up to a one power
16:44	. So be goes up a power in the next
16:46	term . A again goes down to power and be
16:48	then goes up a power And then the last term
16:51	be goes down the power 8-0 , which means it's
16:55	one , it disappears and be goes up a power
16:57	. This pattern is really powerful . It allows you
17:01	to predict any binomial expansion . You want to ,
17:04	All you do is you start let's go with the
17:06	next one . You start with a to the fourth
17:08	, that's the first term . Then to write the
17:10	terms down , All you do is you make a
17:12	go down by a power bi goes up . Then
17:14	you go down here , it goes down by another
17:16	power bi goes up . Then you say it goes
17:19	down by a power from that one , B goes
17:21	up , it goes down by a power to aid
17:23	to the zero , which means it disappears and then
17:25	be goes up . Same thing is happening here ,
17:27	you say eight to the fifth is your first one
17:29	. It goes down , he goes up , it
17:31	goes down , he goes up , it goes down
17:33	, he goes up and so on , it goes
17:35	down , he goes up all the way to the
17:36	end and then the coefficients in front of all of
17:39	those terms . Just come from the triangle . Okay
17:42	, let me make sure I have everything I have
17:43	in my notes here . Some of the exponents power
17:45	. The binomial coefficients are from pascal's triangle . And
17:48	then as one of the variables decreases , the other
17:51	one increases in such a way that they always add
17:53	up to the power that you are expanding to .
17:56	Okay , so by knowing this , you can write
17:59	down an expansion of any binomial you want . The
18:01	only limitation is that you have to know the numbers
18:04	in this triangle . So , if you're going up
18:06	to 1/6 power , I haven't written for you here
18:09	, but if you need 1/7 power or an eighth
18:11	power or 1/9 power or 10th power , you're gonna
18:13	have to fill this triangle out even deeper than I
18:15	have here . All right . So , what I
18:17	want to do now , I think is solve a
18:19	couple of problems . All right . I think I
18:22	can do Let's see here . Yeah . I think
18:25	I'm gonna do a couple of them right underneath this
18:29	pascal's triangle because they're very simple and they powerfully illustrate
18:32	how to how to do this . So , let's
18:35	do the first one . Let's just go ahead and
18:36	write down X plus Y to the power of three
18:41	. Now , the way we did this before is
18:44	we actually did the foil multiplied it all out at
18:46	it all the terms and we end up so we
18:48	know what the answer is . But that's no fun
18:51	. We want to use it . Do it using
18:53	this more powerful method . Here's what you do .
18:56	The first thing we need to do is say okay
18:58	, we're raising to the third power . So we
18:59	go to row three in the triangle . The numbers
19:02	from this triangle are exactly written here . 1331 So
19:06	, I recommend that you can write on your paper
19:09	pascal Triangle 1331 . This tells me the numbers I
19:14	have to use in the expansion of this thing right
19:17	here . Mhm . Next . All right . So
19:20	the first thing is just gonna be the first term
19:22	, the first thing in your binomial cubed with a
19:25	coefficient of one . So I don't need to write
19:27	a one down . I can just say it's going
19:28	to equal X cubed because I have a one here
19:31	. That's the first coefficient . Then I have a
19:32	plus sign . The next term is going to have
19:35	a coefficient of three . And then what do I
19:37	do ? I say X goes down a power So
19:39	it's gonna be X . To the power of to
19:41	And there's a Why here why has to go up
19:43	a power ? It was why to the zero here
19:45	. Which means why to the zero is one .
19:47	So it goes up a power just up to y
19:50	. To the first power then the next term grabs
19:53	the next three . And then again X goes down
19:55	a power X to the first power . And why
19:57	then goes up a power like this . And then
20:01	for the last term it's just one is the coefficient
20:03	. And then again X goes down a power .
20:05	So you can say X 20 making it just one
20:09	and then why to the third ? So this is
20:11	the answer . And notice , let me just double
20:13	check X . Q plus three X squared , Y
20:15	plus three X Y squared plus Y cube . Notice
20:18	that I was able to write this down more or
20:19	less without doing very much work and compare that to
20:23	something like this . I had to first foil this
20:26	, then cross multiply everything and then add the terms
20:28	. and then I really hope and pray that I
20:31	didn't make an error because it's really easy to make
20:32	errors . All right . All of us make errors
20:34	. I make errors all the time . All right
20:37	. Um And so it can just happen . So
20:41	that's how we do it . Now . What I
20:42	want to do is just do a few more just
20:45	to give you a little practice . What if I
20:47	give you something a little bit more challenging than this
20:50	is just X . Plus y to the third power
20:52	. Now let's do one a little bit different .
20:54	What if you do C minus D . Raise to
20:57	the fifth power ? A couple of things that makes
21:01	it difficult . You have 1/5 power . So I'm
21:03	gonna be using the road number five of the pascal's
21:07	triangle , right ? But also there's a minus sign
21:09	in here . And also we're using C . And
21:11	D . This is what I want you to do
21:13	. Okay anytime you have anything other than A .
21:16	And B . In there , this is what I
21:17	want you to do . I want you to work
21:19	on A Plus B to the 5th Power . Then
21:23	sub . What I mean by that is if you
21:26	know it's to the fifth power , just expand A
21:28	plus B . To the fifth power . And once
21:31	you have the answer then substitute A . Goes to
21:35	see . And then uh be here once I get
21:39	the answer . In terms of A . And B
21:40	. I'll just put negative D . In for that
21:42	. So I'll say B . Goes to negative D
21:44	. So instead of trying to do too many things
21:47	at once which you will almost certainly make an error
21:49	. Just say okay I'm raising to the fifth power
21:51	. I'm gonna expand this . I know how to
21:53	expand this very easily . Once I get the answer
21:55	I'll just substitute for A . And make it this
21:57	and substitute for B . And make it negative D
22:00	. And then that will be my final answer .
22:02	Okay um that's exactly what I'm gonna do and then
22:05	I'm gonna say pascal's triangle . What are the coefficients
22:10	? So I go over here to pascal's triangle ,
22:11	it's 1/5 . Uh power of five . So is
22:14	row number five . So the numbers I use are
22:16	15 10 10 5115 10 10 5115 10 10 51
22:24	All I need to do is write them down on
22:26	my paper so that I have some reference when I'm
22:29	solving the problem to actually use it . Okay now
22:32	what do we do ? We are no longer working
22:34	with this , we're working with this . What is
22:36	the first term ? It's a coefficient of one A
22:38	. to the fifth . So we c we just
22:40	say a to the fifth . The next coefficient five
22:44	. So we put a five . Now , what
22:46	happens A has to drop down by a power and
22:49	be goes up by a power ? Next coefficient is
22:52	10 A drops down by power . B goes up
22:57	by power . Next coefficient after this again is 10
23:01	A . Goes down by a power making it a
23:03	squared B goes up by a power making it to
23:06	the third . Next coefficient is five A . Drops
23:10	down by a power . Making it out of the
23:11	first . Be goes up by a power making it
23:13	be to the fourth . And then the final guy
23:16	is just a one . And then we drop a
23:18	down to 80 which means it disappears be goes up
23:21	to be to the fifth . And the coefficient is
23:23	one . It's a good idea at this step .
23:25	All of these terms should have exponents that add together
23:29	to give me five because I'm raising this thing to
23:31	the fifth power . So that adds to five .
23:34	That adds to five . That adds to five .
23:36	That adds to five . That adds to five .
23:37	That adds to five . So it looks like I'm
23:39	correct . I probably should double check myself . So
23:41	let's do that . 10 A . QB 10 a
23:44	square be cubed five A . B . To the
23:46	forest and so on the the fifth . All right
23:47	. So the next thing I need to do is
23:49	I need to everywhere . I see an A .
23:51	I'm just gonna put the letter C . In there
23:54	everywhere . I see A . B . I'm gonna
23:56	put a negative D . But I'm gonna tell you
23:57	right now when you're substituting for be with that negative
24:00	D . That you want to put their rapid in
24:02	parentheses or else you're going to make a sign error
24:04	somewhere . I virtually guarantee it . So a becomes
24:09	see . So what I'm gonna say here this is
24:11	c . to the 5th plus five C . To
24:15	the fourth . But I have A B . But
24:17	B means I'm putting a negative D . Open the
24:19	princes making negative D . To the first power .
24:21	So I don't have to really write a power there
24:23	. I'll take care of the signs in the next
24:25	step . Then I have a 10 A . Becomes
24:27	C . C cubed . B becomes negative D .
24:31	Negative D . Is squared because B squared . I
24:35	put a negative D . There . Then I have
24:37	10 . This is going to become C . Square
24:41	negative D cubed because B . Is now negative D
24:45	. Then I have five A . Is C .
24:49	And then I have negative d . to the power
24:51	of four . Last term here is just be here
24:55	. So it's just gonna be negative D . To
24:58	the power of five . And now I think I'm
25:01	ready to write the final answer . So here I
25:04	have C . to the power of five . This
25:08	negative sign is going to come outfront -5 c .
25:11	to the 4th d . Then notice here this is
25:15	going to be squared so the negative D . Is
25:17	going to be squared . It would be positive D
25:19	. Squared . So really that sign goes away .
25:20	So to be 10 C cubed D . Squared ,
25:24	the negative sign goes away . But this negative sign
25:27	doesn't because it's an odd power . So negative times
25:31	negative times negative D . You're still gonna have one
25:33	minus sign that comes out here 10 C . Squared
25:36	. But D . Is still cube . This negative
25:38	comes out because of the Cuban and D of course
25:41	is still cubed . That this is an even power
25:43	. So this sign goes away to have five C
25:46	. D . To the power of four and then
25:49	this is an odd power so it's negative sign will
25:51	survive and you'll have D to the power of five
25:54	. So I'm gonna check myself but this is the
25:56	final answer . C . To the fifth minus five
25:58	. C . To the four D . Plus 10
26:00	C cube . D squared minus 10 C squared eq
26:04	plus five C . D . To the fourth minus
26:06	D . To the five . And you also should
26:08	double check since you do all this . This work
26:10	here that the exponents should add to five . These
26:13	at 25 these at 25 these at 25 these at
26:16	25 these add to five and then that one of
26:18	course adds to five . Now the coefficients look a
26:21	little different than what's in pascal's triangle . You have
26:23	negative and positive alternating signs . But that's because we
26:27	really didn't solve this problem . The problem we wanted
26:29	was this one . So this negative sign is going
26:31	to ripple through and and cause an alternating sign .
26:33	And that's something that you're gonna see a lot .
26:35	When you have a minus B or x minus y
26:39	raised to the power of something . Often you'll see
26:41	these alternating signs in the final answer . If you
26:43	did this manually , it would be a nightmare .
26:45	You'd have to multiply it by itself five times you
26:47	have so many negative signs to keep track of .
26:49	But ultimately , if you went through all of that
26:51	process , you would arrive at the same answer .
26:54	Mhm . Okay , now , I have one more
26:56	problem . I want to do do I have room
26:59	? Where do I have room ? You have room
27:01	right here . One more problem I want to do
27:03	. Um I want to generate the following or expand
27:07	a plus one to the eighth . Power To the
27:11	8th power . All right . Um Now , the
27:15	problem is senses to the eighth power . You need
27:17	to have a road number eight of pascal's triangle ,
27:20	right ? Road number eight of pascal's triangle . Actually
27:23	, what I think I'm gonna do to make it
27:24	easy for us is I'm gonna write down row six
27:28	on on this page because what we need to do
27:29	is generate two more rows . Right ? So let's
27:32	go and do that right now , just to make
27:33	it , you know , 100% clear . So that
27:36	row right there above is one , then six ,
27:39	then 15 , then 20 then 15 than six ,
27:44	then one . This is road number six , row
27:48	six . And when you wrote seven in row eight
27:50	. So now we have to put a one out
27:51	here . Six plus one is seven . This becomes
27:54	21 . This becomes 35 . This right here again
27:59	becomes 35 . This right here becomes 21 this becomes
28:04	seven and then the one out at the end .
28:06	So I have 17 21 35 35 21 71 This
28:10	is row seven . This is what you would use
28:13	if it were to the seventh power , but it's
28:14	not . So we have to do one more ,
28:16	then we have an 87 plus 21 is 20 .
28:20	Um Let's see here . seven plus 21 , Like
28:24	this , so we have eight and 28 . Then
28:26	here we have 56 . This becomes when we adam
28:29	70 then this becomes 56 this becomes 28 this becomes
28:34	eight and this becomes one , this is row eight
28:37	. So let me double check . 18 28 56
28:40	70 56 28 8 and one . So now we
28:44	have what we need to generate an expansion which is
28:48	a monster to the eighth power like that . Yeah
28:50	. Okay . So , what do we do next
28:53	? All right . So , we say that A
28:57	plus uh This is A plus one to the eighth
29:00	. So what we want to do is just go
29:02	ahead and do A plus B to the eighth .
29:04	And then at the end of it we'll just substitute
29:06	and make be equal to one . That's the easiest
29:08	way to do it . So , what do we
29:10	do ? We first say we need this road ,
29:12	we need a one and then the first term is
29:14	going to be A to the eighth power . So
29:15	, A to the eighth power . Like this Next
29:18	term is going to be eight . Then what do
29:20	we do ? A comes down to seven and B
29:23	goes up to one . Then it's going to be
29:25	28 . Then what happens a comes down to a
29:29	. six and B goes up to a square .
29:33	After 28 comes 56 56 then what happens a comes
29:37	down to the 5th and be goes up to the
29:40	third . Now I've got to go to the next
29:42	line here . So after 56 . Let's see here
29:45	. That was this 56 . I have a 70
29:48	And then a 70 is going to be a coming
29:50	down to the 4th b . Going up to the
29:54	4th . That's 70 . Then I have a 56
29:59	. Uh in the 56 is gonna be a cubed
30:02	B to the fifth again coming down in a .
30:05	And up and be . And then I'm going to
30:08	have After 56 28 , I should have given myself
30:12	more room here . It looks like having a squared
30:15	B . To the sixth . Okay go down in
30:18	A . And up and be and then after 28
30:20	I'm gonna have an eight . I'm gonna go down
30:23	in A . And up and be and then I
30:25	have a one and then down in A . Is
30:28	80 So it goes away and then be to the
30:30	eighth power . So let me double check myself .
30:32	88 A . 78 A . Seven B . Then
30:35	28 8 of six B squared then plus 56 8
30:38	of the 50 cubed plus 74 4 56 . 3
30:43	and five 28 2 and six . Uh And then
30:46	eight A . One at seven and then be eight
30:49	. And you should just kind of scan and make
30:50	sure these exponents should always add together to be eight
30:53	for every one of these terms everywhere you look ,
30:56	it should always add to eight and that's always true
30:58	of any kind of polynomial expansion like this . Okay
31:02	, so if we were asked to find this ,
31:05	it would be simple , but we're not we want
31:06	to find this so we want to set be equal
31:09	to one . So what do we do there ?
31:12	It's just exactly like it sounds A to the 8th
31:16	is right here , then we have 88 to the
31:19	seven , but then we have be so just put
31:21	a one right there , then you'll have 28 8
31:25	to the six . Then you have one square putting
31:28	a one in for this be Then you'll have a
31:30	56 , 8 to the fifth , one cube ,
31:34	putting a one in for this , then you'll have
31:36	a 70 A to the fourth , then one to
31:40	the fourth , putting a one in here . Let's
31:42	go way back over here after 70 becomes this 156
31:47	A cubed one to the fifth . This one then
31:51	we have a 28 A squared . One to the
31:55	sixth right here . Then we have an eight A
31:59	. One to the seventh . Then we have B
32:01	becomes one . So it's one to the eighth power
32:03	. So finally we're almost done . I know it's
32:05	a ton of work but we're almost done . We
32:07	have A to the eighth power . This is just
32:10	A . One . So we have eight A .
32:12	to the 7th , Then we have 28 A .
32:16	to the 6th , then we have 56 A .
32:20	to the 5th , Then we have 70 a .
32:24	to the fourth , then we have 56 . I'll
32:27	start over here , 56 A cubed , then 28
32:32	a squared , then ate a then one just writing
32:37	all of these down here . This is the final
32:38	answer . Let me double check . Eight , I'm
32:41	sorry . Eight to the eighth . Power . 88
32:43	to the 7 28 . Eight of the 6 56
32:47	. 8 of the 5th 78 of the 4th 56
32:49	A cube 28 A squared eight A . And one
32:52	. This is the final answer . So you can
32:54	see that . It is still work to apply pascal's
32:58	triangle to expand by no means , but it's way
33:01	less work than actually multiplying all these terms out and
33:04	actually adding them all together , canceling terms if needed
33:07	. And collecting everything all the while , trying to
33:10	make sure you get all the squares correct and you
33:12	can go back of course and look uh that you
33:15	might say oh these don't add up to eight anymore
33:17	because of this . But that's really not true because
33:20	really you have an invisible one to the first power
33:23	. The exponents do add up to eight , you
33:24	have an invisible one to the second power . These
33:27	exponents do add up to eight . These add up
33:29	to eight . These add up to a it's just
33:30	it doesn't show up in the final answer because we've
33:33	evaluated and kind of calculated some of those exponents .
33:35	So of course they don't show up all the time
33:37	in the final answer . This is how a computer
33:39	might do it . Or one way in which a
33:40	computer might do it . So what I'd like you
33:42	to do is sit down with a piece of paper
33:43	and convince yourself by doing these problems that you can
33:46	get the answers that I get , do it yourself
33:48	. There's no substitution for you doing it yourself .
33:51	Then follow me on to the next lesson . We'll
33:52	get a little bit more practice with evaluating and expanding
33:55	. Binomial is using pascal's triangle .

Summarizer

DESCRIPTION:

Quality Math And Science Videos that feature step-by-step example problems!

OVERVIEW:

21 - Pascals Triangle & Binomial Expansion - Part 1 is a free educational video by Math and Science.

This page not only allows students and teachers view 21 - Pascals Triangle & Binomial Expansion - Part 1 videos but also find engaging Sample Questions, Apps, Pins, Worksheets, Books related to the following topics.