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 . |
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