05 - Polynomial Long Division - Part 1 (Division of Polynomials Explained) - By Math and Science
Transcript
| 00:00 | Hello . Welcome back to algebra and Jason with math | |
| 00:02 | and science dot com . The title of this lesson | |
| 00:05 | is called dividing polynomial is part one . Also I | |
| 00:08 | could have titled it polynomial long division . Part one | |
| 00:11 | . So we're actually gonna have three parts of this | |
| 00:13 | lesson where the problems get more and more complex and | |
| 00:16 | then we'll move on into some other skills that are | |
| 00:17 | related to this called synthetic division . We'll talk about | |
| 00:20 | that a little bit later . Here we have the | |
| 00:22 | topic of dividing polynomial . That's when you have a | |
| 00:24 | polynomial on the top like X squared plus two , | |
| 00:27 | X plus four . And you're gonna divide by another | |
| 00:31 | polynomial like for instance X plus three . Right ? | |
| 00:34 | So you can actually divide one polynomial by the other | |
| 00:37 | . Now let's face it . Nobody I have ever | |
| 00:39 | met actually likes this topic when you first learn it | |
| 00:43 | . The reason nobody likes it is because there's a | |
| 00:45 | process to it . You have to know what polynomial | |
| 00:48 | long division is . You have to follow the procedure | |
| 00:51 | . There's a lot of little opportunities to have some | |
| 00:54 | errors along the way . And so what I want | |
| 00:56 | you to do , what I'm going to do in | |
| 00:57 | this lesson is we're going to solve some problems . | |
| 01:00 | I'm gonna walk you through exactly how to do it | |
| 01:01 | to avoid all of those mistakes . But more than | |
| 01:04 | that in the beginning of the lesson , I'm going | |
| 01:06 | to walk you through where polynomial long vision really comes | |
| 01:09 | from . In other words in the beginning of this | |
| 01:11 | lesson , I'm going to spend a few minutes reviewing | |
| 01:13 | regular division division that you did in fifth or sixth | |
| 01:16 | grade . The reason is because once we refresh our | |
| 01:19 | memory with what we're doing with numbers , regular numbers | |
| 01:22 | , then the polynomial long division doesn't seem as scary | |
| 01:25 | . See if I would just throw you into a | |
| 01:26 | polynomial division like this and just not review anything . | |
| 01:30 | Then you would probably get lost really , really quickly | |
| 01:32 | . But if I review a little bit of some | |
| 01:33 | things that you already know in the back of your | |
| 01:35 | mind and then you're gonna immediately see how the polynomial | |
| 01:38 | long division looks similar to that , Then it's all | |
| 01:40 | gonna fall into place . So keep that in mind | |
| 01:42 | . Some of you are going to get a little | |
| 01:43 | impatient and want me to skip ahead , but please | |
| 01:45 | watch me . I know what I'm doing and doing | |
| 01:47 | it this way is gonna make it much , much | |
| 01:50 | much easier to understand it . So with that , | |
| 01:52 | out of the way , everybody at the end of | |
| 01:54 | this lesson is going to master long division of polynomial | |
| 01:56 | . So let's jump right in . Let's recall regular | |
| 02:00 | division , recall uh regular . I'll call it regular | |
| 02:06 | because it just involves numbers division . Remember the concept | |
| 02:12 | . What you're doing is you're taking a number and | |
| 02:14 | you're dividing it by another number . So , for | |
| 02:16 | instance , pick some very simple numbers . Let's take | |
| 02:18 | the number 19 . We're gonna divide it by the | |
| 02:20 | number five . Another way of writing that Is we | |
| 02:24 | can write it as a fraction , we can say | |
| 02:26 | 19 divided by five like this . However you want | |
| 02:28 | to write it . We can also use the little | |
| 02:30 | house thing that we're gonna we're gonna talk about here | |
| 02:32 | in a minute . But I'm using very small numbers | |
| 02:35 | because I want to get the concept down before we | |
| 02:37 | go into something more more detailed . Nobody has to | |
| 02:39 | do this uh with a long division to understand how | |
| 02:43 | many times will five actually go into 19 ? Well | |
| 02:46 | , the way you do it as you say , | |
| 02:47 | well five times one is five . That's not big | |
| 02:50 | enough . Five times two with 10 . Okay , | |
| 02:52 | that's closer . But that's not big enough . Five | |
| 02:54 | times three is 15 , but five times four is | |
| 02:56 | 20 and that's too big . Five times four is | |
| 02:59 | 20 . That's bigger than the 19 . So you've | |
| 03:01 | got to back up a step and say five times | |
| 03:03 | three is 15 . So what we say Is that | |
| 03:06 | this division can go three whole times , right ? | |
| 03:13 | Three whole times . But when when it goes three | |
| 03:16 | whole times and why ? By the way , does | |
| 03:17 | it do that ? Because three times five is 15 | |
| 03:20 | and that's as close as we can get to the | |
| 03:21 | 19 without going over . Right ? It's three whole | |
| 03:24 | times but there's another part of it and the other | |
| 03:27 | part of it is that there is a remainder And | |
| 03:32 | the remainder in this case is that once you get | |
| 03:35 | as close as you can three times five being 15 | |
| 03:38 | , the remainder that you have is 15 , -2 | |
| 03:41 | , 15 is remainder of four . And the way | |
| 03:45 | you get that , as you say the 19-15 . | |
| 03:47 | So this is all from basic arithmetic right ? You | |
| 03:50 | know that five can only go three whole times but | |
| 03:52 | you're gonna have some left over . And how much | |
| 03:55 | you have left over is the difference between the 15 | |
| 03:57 | that you ended up at in the 19 that you're | |
| 03:59 | trying to divide into . So there's two pieces of | |
| 04:02 | the puzzle here . How many whole times can the | |
| 04:04 | division happen ? And then what is left over ? | |
| 04:06 | Now ? Another way that you would write that is | |
| 04:10 | if I were going to write a little more compactly | |
| 04:13 | than that is I would say 19 5th . So | |
| 04:15 | you might see it written like this three with a | |
| 04:18 | little R . And then a four , right ? | |
| 04:21 | So whenever you're in fourth grade or fifth grade or | |
| 04:23 | whenever you learn division , this is the first way | |
| 04:25 | you learn to write the answer down . You write | |
| 04:27 | it down as three with a remainder of four . | |
| 04:30 | And you're taught to do that for a larger and | |
| 04:32 | larger and larger problems . Whatever remainder you have at | |
| 04:35 | the end of the thing Is you just write it | |
| 04:38 | as a remainder of four . Now notice that if | |
| 04:41 | the remainder were not four , if the remainder were | |
| 04:43 | five if there were five left over well then you're | |
| 04:46 | dividing by five . So you can never have a | |
| 04:48 | remainder bigger than this number . Because if I did | |
| 04:50 | have a remainder of five Then the that would just | |
| 04:53 | go one more time into it and the answer would | |
| 04:55 | have been four . So in other words if if | |
| 04:57 | it was 20 divided by five then you don't have | |
| 05:00 | any remainder . All you just increment this one more | |
| 05:02 | up because the remainder has to always be less than | |
| 05:05 | when you're what you're dividing by . Okay , um | |
| 05:08 | when you're in 3rd 4th , 5th grade , you're | |
| 05:09 | doing division , you learn to write it as three | |
| 05:11 | remainder of four like this , but when you punch | |
| 05:14 | numbers in your calculator , you never see that , | |
| 05:16 | you never see something remained or something ever . You | |
| 05:19 | always see it written as a decimal Right ? And | |
| 05:22 | the reason it's written as a decimal is because this | |
| 05:24 | remainder can be expressed as a decimal . What this | |
| 05:26 | remainder really means is that you have a leftover piece | |
| 05:31 | , you cannot go four times , you can go | |
| 05:33 | three whole times and this five can go only a | |
| 05:36 | fraction of the way into the 19 . Again , | |
| 05:39 | I'm going to say that one more time because it's | |
| 05:40 | important When you're dividing 19 divided by three , you | |
| 05:43 | can only go three whole times . This remainder really | |
| 05:46 | kind of expresses how many more times five can go | |
| 05:50 | in , but it can't go a whole time in | |
| 05:51 | , it can only go some fraction of the way | |
| 05:55 | in there and what this really means . This remainder | |
| 05:59 | of four , it means All this is going to | |
| 06:02 | be so important for polynomial long division , but the | |
| 06:04 | remainder of four means That five can go in 2 | |
| 06:10 | 19 , some fraction some fraction more , How much | |
| 06:14 | more ? Four divided by five more times four divided | |
| 06:22 | by five more times . Let me write a few | |
| 06:24 | things down and I promise it'll be clear . In | |
| 06:26 | other words , when you punch this thing into a | |
| 06:28 | calculator , you never see three remainder of four . | |
| 06:30 | What you see is the following 19/5 , it means | |
| 06:35 | that it can go three whole times with some remainder | |
| 06:38 | of fraction 4/5 the remainder you get you just write | |
| 06:42 | it divided by this because that is the fraction of | |
| 06:47 | the number of times . The thing can go in | |
| 06:48 | again , how many fractions of five can it go | |
| 06:51 | in again ? It can go 4/5 of the time | |
| 06:54 | , So it's three plus this remainder . And as | |
| 06:57 | you all know , this can be written as a | |
| 06:58 | mixed fraction , three and 4/5 right ? And if | |
| 07:01 | you punch this in the calculator , four divided by | |
| 07:03 | five and you add three to it , you're gonna | |
| 07:05 | get 3.8 . So when you put this in a | |
| 07:07 | calculator and you say 19 divided by five and you're | |
| 07:10 | gonna get 3.8 , you're never gonna get three remainder | |
| 07:13 | of four or anything like that . The reason it's | |
| 07:15 | important is because when we do polynomial division we're also | |
| 07:18 | gonna get remainders , but we're never gonna write it | |
| 07:20 | as something , something remained or something . You're gonna | |
| 07:23 | have to write it as this kind of fractional concept | |
| 07:26 | . Right ? So the bottom line is when you | |
| 07:28 | get a remainder you never just leave it as a | |
| 07:30 | remainder of something you say , it's going to be | |
| 07:32 | the whole number of times . The thing can happen | |
| 07:34 | . Plus some fraction of a some The five can | |
| 07:37 | only go in 4/5 of another time into 19 . | |
| 07:40 | And so if the remainder were five it would be | |
| 07:43 | five fits and it would be one more hole time | |
| 07:45 | , it can go in . So you basically add | |
| 07:47 | the remainder to how many times you can go in | |
| 07:50 | and you add it as a fraction . So bringing | |
| 07:52 | it all a little bit more into focus . Um | |
| 07:55 | You can you can write it like this . Mhm | |
| 07:59 | . As the following , What you say is 19 | |
| 08:03 | . If you're gonna do this long division way divided | |
| 08:05 | by five , what do you do ? Yeah . | |
| 08:08 | Okay . The algorithm goes something like this . Five | |
| 08:11 | can't really go into one so you don't really do | |
| 08:13 | anything with that . But you say how many times | |
| 08:15 | can five go into 19 5 ? Can we can | |
| 08:18 | go into 19 3 times once you have the number | |
| 08:20 | on the top , you multiply three times five and | |
| 08:22 | you get 15 Then you make sure and put a | |
| 08:25 | subtraction sign there and you subtract 19-15 and you get | |
| 08:29 | a four . Now you stop the division process when | |
| 08:31 | the number that you get at the bottom is smaller | |
| 08:34 | than the thing you're dividing by because if it if | |
| 08:36 | it will come out to a value of five then | |
| 08:38 | you could have divided one more time anyway . So | |
| 08:41 | this number is always going to be less and when | |
| 08:42 | it's less than you stop the division process and then | |
| 08:45 | you say um as I already said several times , | |
| 08:49 | you have three , it can go three whole times | |
| 08:52 | plus some fraction of the time . 4/5 4/5 of | |
| 08:56 | a time which we write as a fraction as three | |
| 08:58 | and 4/5 Right ? three and 4/5 now . Um | |
| 09:05 | Oftentimes you're taught when you're doing basic math to check | |
| 09:07 | your work , right ? And we're going to do | |
| 09:09 | that with the polynomial division two . So I'm gonna | |
| 09:11 | review very briefly on how we check our work with | |
| 09:13 | this stuff too , because it's extremely important . So | |
| 09:16 | let me go over here and you know what I'm | |
| 09:19 | gonna do actually on the bottom , I'm gonna do | |
| 09:21 | it on the bottom . There's two ways they want | |
| 09:24 | to do on the bottom . There's two ways to | |
| 09:28 | think about this , right ? When you're doing division | |
| 09:30 | problems to check yourself , you're saying that the answer | |
| 09:32 | is three remainder four . Also , you can write | |
| 09:34 | us three and a fraction . 4/5 . Okay . | |
| 09:38 | The way to check it is you multiply what you | |
| 09:39 | get this times this And you say well I'm gonna | |
| 09:43 | check it by saying three times 5 And that I'm | |
| 09:46 | gonna get 15 , that's But then I have to | |
| 09:49 | add to that whatever the remainder that I had left | |
| 09:51 | over which I had four which is 19 , which | |
| 09:54 | is a check because you should get back what you | |
| 09:56 | have underneath there . Right ? Alright . That's one | |
| 10:00 | way to think of it , multiply this , what's | |
| 10:02 | on the top times what's out in front ? You | |
| 10:04 | get 15 in this case and then whatever you get | |
| 10:07 | , you add the remainder back to it and you | |
| 10:08 | should always arrive with what you got in the beginning | |
| 10:11 | there . Okay . But there's another way to think | |
| 10:14 | of it which is the way I actually want you | |
| 10:17 | to think about it from now now on think like | |
| 10:21 | this and the reason you're gonna do it like this | |
| 10:24 | is because this is gonna be how we're going to | |
| 10:26 | end up checking our work when we do the polynomial | |
| 10:28 | division if we had 19 . And we divide it | |
| 10:32 | by five . We're saying essentially that the answer is | |
| 10:35 | three and 4/5 . So I'm gonna write it as | |
| 10:38 | three and the fraction of 4/5 . This is kind | |
| 10:41 | of like the answer is sitting up top . Okay | |
| 10:43 | ? So instead of multiplying the three times to five | |
| 10:47 | , getting the 15 and then adding the four when | |
| 10:49 | we express it as a fraction , then the answer | |
| 10:52 | is three and 4/5 . This is what you would | |
| 10:53 | get in a calculator will be 38 Right ? So | |
| 10:56 | whatever you get on the top is a whole thing | |
| 10:59 | . You can check it by saying yeah , the | |
| 11:03 | following thing you can say um mm five what's happening | |
| 11:07 | front times three and 4/5 and whatever you get is | |
| 11:11 | a total answer . There should equal 19 but three | |
| 11:14 | and 4/5 . Don't forget is three plus 4/5 . | |
| 11:19 | That's what we mean when we write them together like | |
| 11:21 | this it's three plus 4/5 . So we can distribute | |
| 11:24 | the five in and we're gonna get 15 and then | |
| 11:27 | the five goes into here five times 4/5 . But | |
| 11:30 | notice what happens the five cancels with the five . | |
| 11:33 | And so what you end up with is 15 plus | |
| 11:37 | 4 15 plus four because these go away and so | |
| 11:40 | you get 19 . Check so what you've done is | |
| 11:43 | I've shown you that hey , when you take a | |
| 11:45 | simple number and divide it it goes three whole times | |
| 11:47 | with a remainder of four . That's how you remember | |
| 11:50 | it kind of from third grade right ? And you | |
| 11:52 | say , well I'm gonna multiply these , I'm gonna | |
| 11:53 | get the 15 , I'm gonna add the remainder back | |
| 11:55 | . And that's what I get is my final answer | |
| 11:57 | in 19 . But I'm telling you when you grow | |
| 11:59 | up a little bit , you stop , stop thinking | |
| 12:01 | of . That . Remainder is just a number . | |
| 12:03 | You think of , that remainder is representing the number | |
| 12:06 | of fractional times five can go into 19 again . | |
| 12:10 | In this case you can only go 4/5 of the | |
| 12:12 | time . And so we say , the answer is | |
| 12:14 | really three and 4/5 3 times in plus 4/5 of | |
| 12:18 | another time . So the real way to think about | |
| 12:20 | checking it is to take this kind of answer three | |
| 12:23 | and 4/5 and as an entire unit and multiply by | |
| 12:26 | five . When you blow it out , you see | |
| 12:28 | that the fives cancel and what you're doing here , | |
| 12:30 | 15 plus four is exactly what you do when you | |
| 12:34 | do it . Kind of the other way of thinking | |
| 12:35 | about it . So there's two ways of thinking about | |
| 12:38 | checking division one is to multiply and then add your | |
| 12:41 | remainder brag back in . But I want you not | |
| 12:43 | to think so much about that . I want you | |
| 12:45 | to think of representing your answer as a as a | |
| 12:48 | fraction . Take that number multiplied by what's out in | |
| 12:51 | front here and you should always arrive with what you | |
| 12:54 | got underneath . That is going to be critically important | |
| 12:57 | when we check our work in a few minutes now | |
| 13:00 | before we get to the polynomial long division , I | |
| 13:02 | want to do one more quick little review problem like | |
| 13:05 | this And it's gonna go like this we're gonna have | |
| 13:08 | because I need to review the process here . If | |
| 13:11 | you have 349 And you divide it by two , | |
| 13:15 | how do you do this ? So we're gonna have | |
| 13:16 | to work on how we actually divide with the algorithm | |
| 13:19 | here . So you say well how many times can | |
| 13:21 | to go into three ? That can only go one | |
| 13:23 | time Once that number is there , you say one | |
| 13:26 | times two is two . You write it directly underneath | |
| 13:28 | , you have to put a subtraction sign there , | |
| 13:31 | Draw the line , and you do the subtraction 3 | |
| 13:33 | -2 is one . Once you subtracted you grab the | |
| 13:36 | next number from the next column over and you bring | |
| 13:38 | it down like this . Now you say how many | |
| 13:40 | times can to go into 14 and that can go | |
| 13:43 | exactly seven times . Then you say okay seven times | |
| 13:46 | two . Is itself 14 ? You have to subtract | |
| 13:50 | again 14 minus 14 , 0 . So I can | |
| 13:52 | leave it there but then I have to grab the | |
| 13:54 | next number which is nine . So that can kind | |
| 13:57 | of ignore the zero . If I'd like how many | |
| 14:00 | times can to go into nine ? Two times four | |
| 14:03 | is eight . So I have a four multiply two | |
| 14:05 | times four I get eight . Subtract nine minus eight | |
| 14:08 | is one . Uh I'm gonna say this one so | |
| 14:12 | notice that I have a number now less than what | |
| 14:14 | I'm dividing by . Remember that's exactly what happened here | |
| 14:17 | . The remainder I got is always got to be | |
| 14:19 | less than what I'm dividing by . If it ever | |
| 14:21 | gets bigger than I could have gone and divided the | |
| 14:23 | thing in one more time to begin with . So | |
| 14:25 | the answer you can think about it being written as | |
| 14:28 | two ways you can think of . The answer being | |
| 14:30 | 174 with a remainder Of one , which is the | |
| 14:36 | way you kind of write it in fifth grade or | |
| 14:37 | something like this . But the way I want you | |
| 14:39 | to think about it is the following way . This | |
| 14:43 | number can be divided in their 174 times , but | |
| 14:47 | it can also go another fraction of the time . | |
| 14:49 | How what's the fraction ? It's the remainder divided by | |
| 14:53 | what you're dividing by . So one half , so | |
| 14:56 | it can go 174 and one half times . That's | |
| 14:59 | 174 plus one half . So if you take 3 | |
| 15:02 | 49 and you put it in your calculator divided by | |
| 15:04 | two , you're never ever gonna see it written as | |
| 15:07 | a remainder one , You're gonna see it written like | |
| 15:08 | this . This is where the fraction part comes from | |
| 15:11 | , it's the remainder divided by By the two in | |
| 15:15 | this case . So let me just double check myself | |
| 15:17 | and make sure I'm on the same page . 1 | |
| 15:19 | 74 and one half . Now , after all of | |
| 15:22 | that is out of the way , we're finally ready | |
| 15:24 | to do an actual polynomial long division problem . Believe | |
| 15:27 | me spending a few minutes talking about this is going | |
| 15:30 | to make it much , much easier to divide polynomial | |
| 15:34 | . So we're gonna take a simpler problem first . | |
| 15:36 | But we're gonna gradually over the lessons here , increase | |
| 15:39 | the complexity . So what we want to do is | |
| 15:41 | we want to divide the following . We want to | |
| 15:43 | say x squared plus three , X -4 . And | |
| 15:49 | we're going to divide this by X-plus two . All | |
| 15:52 | right . Now it's a fraction , right ? But | |
| 15:54 | all fractions represent division . All right . So what | |
| 15:57 | you have to do is translate just like the fraction | |
| 15:59 | over there . We're gonna translate that to the actual | |
| 16:03 | format that we're gonna use . It will be X | |
| 16:05 | plus two sitting on the outside , and then we're | |
| 16:07 | gonna go under a house kind of thing . Right | |
| 16:10 | on the inside , you're gonna have X squared plus | |
| 16:12 | three x minus four . Now there's a couple of | |
| 16:15 | things I need to talk to you about before you | |
| 16:17 | actually do the division . The most important thing is | |
| 16:20 | that everything on the outside and everything on the inside | |
| 16:24 | , it has to be written in decreasing powers of | |
| 16:27 | X . So see I have an X square term | |
| 16:30 | and then I have the three X term and then | |
| 16:32 | I have the term with no exit all on the | |
| 16:34 | outside . The highest power was the X term . | |
| 16:37 | And then the no X term . It has to | |
| 16:38 | be written like that in other words on the outside | |
| 16:40 | , I can't do it is to plus X . | |
| 16:42 | The algorithm won't work on the inside . I can't | |
| 16:44 | flip it around so it's like three X plus X | |
| 16:47 | squared minus four . The algorithm won't work , you | |
| 16:49 | have to ride it in decreasing powers of X on | |
| 16:51 | the inside and on the outside . That's important . | |
| 16:54 | And the second thing is if you're missing a term | |
| 16:59 | , for instance , if I was doing X squared | |
| 17:01 | minus four and there was no three X at all | |
| 17:04 | , then when I write it down I have to | |
| 17:05 | fill in the blank . If there's no X term | |
| 17:07 | , I have to say X squared plus zero x | |
| 17:10 | minus four . So you have to write them in | |
| 17:12 | decreasing powers of X . And if you're missing a | |
| 17:14 | term you have to put a zero term in place | |
| 17:17 | , you have to have the placeholders . It's kind | |
| 17:19 | of like when you think about um you know , | |
| 17:22 | you're dividing you know 504 divided by two . You | |
| 17:26 | have to have the zero here , you can't just | |
| 17:29 | write it as 54 divided by two . That's a | |
| 17:31 | different problem if you have it . Uh in this | |
| 17:34 | case this is the ones place and this is the | |
| 17:36 | tens place . The tens place doesn't have any value | |
| 17:39 | , but it exists as a placeholder . To show | |
| 17:42 | me that the five is worth 500 That's what it's | |
| 17:44 | for . I still have to have a zero there | |
| 17:46 | so that the five retains its meaning . If the | |
| 17:49 | three X term were not there at all , I | |
| 17:51 | would have to put a zero X here for the | |
| 17:53 | same reason , it has to exist and have its | |
| 17:55 | place so that the X square term retains its meaning | |
| 17:58 | . Okay , we're gonna get to a problem where | |
| 18:00 | we have to kind of put the zeros in as | |
| 18:02 | we go . Now , let's get on to business | |
| 18:04 | and start actually dividing . What you do is you | |
| 18:08 | ignore the two , You ignore everything after it . | |
| 18:11 | And you only look at the first , the leading | |
| 18:14 | term here and the leading term here . That's the | |
| 18:16 | biggest problem I see is students will start trying to | |
| 18:19 | figure out how to divide X plus two into this | |
| 18:21 | thing and you you don't do it that way . | |
| 18:24 | You only look at the X and you only look | |
| 18:26 | at the X squared so much like when we're doing | |
| 18:28 | this here , how do we figure this out ? | |
| 18:30 | You know that two goes into 31 time ? Why | |
| 18:33 | did you know that ? Because , you know , | |
| 18:34 | your multiplication terms of tables , you know that two | |
| 18:37 | times one is two and that's the closest I can | |
| 18:40 | get to going to three without going over . So | |
| 18:43 | you're saying two times some number is as close as | |
| 18:46 | I can get to this . So X times something | |
| 18:50 | up here can give me X squared . What do | |
| 18:51 | I have to put up here ? X times X | |
| 18:54 | is X squared ? I multiply them and get X | |
| 18:56 | squared . I'm trying to equal what is under here | |
| 18:58 | . In exactly the same way that I'm trying to | |
| 19:00 | get close to the number three here . Right ? | |
| 19:03 | So I only look at this , I'm ignoring the | |
| 19:05 | two . I'm ignoring everything . I'm saying X times | |
| 19:07 | X gives me what X squared . Ok ? Now | |
| 19:10 | , whenever I have the number written at the top | |
| 19:12 | , what's the next step ? I multiply by two | |
| 19:14 | . I write it down and I subtract . Now | |
| 19:17 | , here's the thing when you multiply down , you | |
| 19:19 | have to multiply by each term because you're distributing it | |
| 19:22 | in , right ? So X times X is X | |
| 19:25 | squared and then you have to multiply X times 22 | |
| 19:28 | X because it's positive times positive . So when you | |
| 19:31 | multiply , you have to write the whole thing down | |
| 19:33 | now Here , the next step is probably the absolute | |
| 19:36 | biggest way to make a mistake with polynomial division . | |
| 19:39 | It's easy with numbers because when you do the subtraction | |
| 19:43 | it's just numbers . You just subtract numbers . No | |
| 19:45 | problem . But here it gets tricky . I have | |
| 19:48 | to subtract here so I'm gonna put a minus sign | |
| 19:51 | but I'm gonna put a circle around it . This | |
| 19:52 | is my own notation . You're not gonna see it | |
| 19:54 | in a book anywhere . I put it in there | |
| 19:56 | as a reminder . Let me actually , first of | |
| 19:59 | all take it away and show you what you can | |
| 20:01 | do . What , how you can screw yourself up | |
| 20:03 | . If you just put a minus sign there , | |
| 20:05 | then it looks like X squared minus X square , | |
| 20:07 | which is zero . But then when you do the | |
| 20:10 | three and the to the negative , it's nowhere close | |
| 20:13 | to the two anymore . So you forget to subtract | |
| 20:15 | . what you need to do is I put a | |
| 20:17 | little circle around it to tell me X squared minus | |
| 20:20 | X squared is zero and three X minus the two | |
| 20:25 | X . Is gonna give me a one X . | |
| 20:27 | So I'm gonna write it as one X . Or | |
| 20:29 | I could just put it as X if I want | |
| 20:30 | to if you leave that alone . Oftentimes the negative | |
| 20:34 | sign is so far removed from this that you'll forget | |
| 20:38 | that you're subtracting and you'll add them together because it | |
| 20:40 | looks like you're supposed to add these but you're not | |
| 20:42 | you're supposed to subtract three minutes to is one . | |
| 20:45 | Okay Now you have to just like we did hear | |
| 20:48 | once we do the subtraction we get the three months | |
| 20:50 | to is one . The next step is we grab | |
| 20:52 | the next term , the next number and pull it | |
| 20:54 | down . Here . We have to grab the next | |
| 20:57 | term and pull it down but it's a negative for | |
| 20:59 | so we have to bring that down so now we | |
| 21:01 | repeat the process . But again we only look at | |
| 21:03 | the first term in the first term . X times | |
| 21:06 | something will give me X . And that's something has | |
| 21:09 | just got to be a one because X times comes | |
| 21:12 | one will give me X . So then I go | |
| 21:13 | back and multiply X times one is X . One | |
| 21:17 | X . And then one times two . Don't forget | |
| 21:19 | to multiply that one . You're gonna get a positive | |
| 21:21 | two and then I'm gonna have to subtract . So | |
| 21:23 | you see I have mixed signs here . If I | |
| 21:25 | don't put a minus sign here then I'm going to | |
| 21:28 | probably just add these together . But here's what you | |
| 21:31 | gotta do . You gotta remember you're subtracting one x | |
| 21:33 | minus one X zero . You have a negative four | |
| 21:36 | minus a two negative four minus or two -4 - | |
| 21:41 | or two is actually -6 . If you don't put | |
| 21:45 | the negative sign there at all , most people will | |
| 21:48 | just add these together because they'll forget what they're doing | |
| 21:50 | . They won't remember that it's minus a positive which | |
| 21:53 | means it's like a negative to there . And so | |
| 21:56 | you won't get a six , you'll probably get a | |
| 21:57 | negative too . If you just add these together negative | |
| 22:00 | four plus two , you're not adding them . You're | |
| 22:02 | subtracting them . You have to take the first thing | |
| 22:04 | , subtract off the second thing so it becomes a | |
| 22:07 | minus six . All right now what we do when | |
| 22:12 | it's in numbers is we continue down this process until | |
| 22:15 | we get a number down here that is smaller and | |
| 22:17 | value to the numbers that it's on the outside . | |
| 22:20 | But in polynomial division it's not so clear because you | |
| 22:23 | have X . Is everywhere . So here you stop | |
| 22:27 | when one of two things happens okay ? You get | |
| 22:31 | a remainder . I guess I should say you stop | |
| 22:36 | when the remainder is either one or two things . | |
| 22:45 | If the remainder is zero then you stop because the | |
| 22:47 | whole thing is done and there's no remainder at all | |
| 22:49 | anymore . Or um the you have a remainder . | |
| 22:53 | That is a lower degree then divisor . This number | |
| 23:03 | out here is called the divisor . So you see | |
| 23:06 | in the regular numbers we stop when the remainder is | |
| 23:08 | a lower value than the divisor . Here we stop | |
| 23:11 | the process when the remainder we give is either zero | |
| 23:13 | or it's a lower degree than the divisor . So | |
| 23:16 | some students will think , well I can't stop because | |
| 23:18 | I have a two and a six . But no | |
| 23:20 | no no . You're looking at the degree of this | |
| 23:22 | . The degree of this X . Term is a | |
| 23:24 | one X to the first power degree of one . | |
| 23:27 | The degree of this is actually zero because it's like | |
| 23:30 | negative six X to the zero power . So there | |
| 23:33 | there is no X term there it's a lower degree | |
| 23:36 | than this . This is a degree of one and | |
| 23:37 | a degree of zero because this extra like an invisible | |
| 23:39 | X to the zero power here . So we're clear | |
| 23:42 | to stop . So the way that you would write | |
| 23:44 | the answer to this thing is that X squared plus | |
| 23:48 | three X minus four divided by X plus two is | |
| 23:51 | the following answer . The answer is X-us one on | |
| 23:57 | the top with the remainder of negative six . That's | |
| 24:04 | the way you would write it down when your very | |
| 24:06 | first learning this maybe in a lower level class or | |
| 24:08 | if you were going to try to write it the | |
| 24:10 | way that we write it in fourth grade or fifth | |
| 24:11 | grade with the remainders . But I told you I | |
| 24:14 | don't want you to think about writing him as this | |
| 24:16 | whole number of times with a remainder . I want | |
| 24:18 | you to think about it as this fraction you say | |
| 24:21 | , well This can go in 174 whole times plus | |
| 24:25 | another half . Because why you had this much left | |
| 24:28 | over and the fraction left over is one half . | |
| 24:31 | Or in terms of this one , you can go | |
| 24:33 | three whole times with a remainder of four . So | |
| 24:36 | as a fraction it's 4/5 of the time they can | |
| 24:38 | go back in again . So the way that I | |
| 24:41 | really want you to write this answer is the following | |
| 24:45 | . I want you to say it like this . | |
| 24:47 | I want you to say it can go X plus | |
| 24:48 | one times plus the value of the remainder , which | |
| 24:53 | is negative six divided by the divisor out here , | |
| 24:56 | X plus two . This is the fractional part that's | |
| 24:59 | out here and this is the answer . This whole | |
| 25:02 | thing is the answer . If you just give me | |
| 25:05 | this part it's wrong . If you just give me | |
| 25:07 | this part it's wrong . If you give me part | |
| 25:09 | of one part of the other , it's wrong . | |
| 25:11 | If you if you forget the one it's wrong . | |
| 25:13 | If you forget the X . It's wrong . If | |
| 25:15 | you forget the negative six it's wrong . The whole | |
| 25:17 | thing has to be there . It's kind of like | |
| 25:19 | saying 100 . If I tell you to divide 3 | |
| 25:23 | 49 divided by two and you say oh the answer | |
| 25:25 | is 174 we'll know what happened to the other half | |
| 25:28 | . I mean that's that's part of the answer . | |
| 25:30 | Same thing . If you miss any of the stuff | |
| 25:32 | then it's wrong , you have to have the whole | |
| 25:34 | thing there . So just like we express the fractional | |
| 25:38 | part and the whole number part this part here , | |
| 25:40 | the part that's kind of written above . That's how | |
| 25:43 | many whole times it can go in . This is | |
| 25:45 | the whole part , the whole number of times it | |
| 25:49 | can go in . This is how many fractions of | |
| 25:52 | the time it can go in . This is the | |
| 25:53 | fractional part . Yeah , the remainder . The fractional | |
| 25:59 | part , the remainder . So this is like from | |
| 26:03 | our example before this is like three from before and | |
| 26:07 | then 4/5 was the remainder part , right ? So | |
| 26:10 | what we got before was three plus 4/5 Which we | |
| 26:14 | all right . As three and 4/5 right Like this | |
| 26:18 | . So when you are asked to do polynomial long | |
| 26:21 | division , you have to right underneath the house what | |
| 26:24 | you're dividing into outside what the divisor that goes out | |
| 26:28 | there , you have to make sure it's in descending | |
| 26:30 | order of powers of X . If there are any | |
| 26:33 | missing terms , you have to put zeros in there | |
| 26:35 | and I'll do that in a second and show you | |
| 26:36 | how then you go through the algorithm that we've done | |
| 26:39 | . You can watch as many times as you need | |
| 26:40 | to get the feeling for that . When you get | |
| 26:42 | a remainder , that's a lower degree than what you | |
| 26:44 | had outside you stop . But to write the answer | |
| 26:47 | , it's the whole part plus a fraction of which | |
| 26:50 | is the remainder divided by what's on the outside . | |
| 26:53 | That is why I spent so much time doing this | |
| 26:55 | because it's exactly mirroring what we're doing with numbers . | |
| 26:59 | All right . Now , what we want to do | |
| 27:02 | before we go onto the next problem is I want | |
| 27:06 | to check this answer right ? Because remember with division | |
| 27:10 | , you learn how to divide and then you learn | |
| 27:12 | how to check the division to make sure it's the | |
| 27:14 | case , right ? How would you check this division | |
| 27:17 | ? Well we know that two going into this or | |
| 27:20 | 3 49 divided by two is 174 and a half | |
| 27:24 | as a whole thing . If we were going to | |
| 27:26 | check it , if we're gonna check it , we | |
| 27:28 | would say I'm gonna check this guy , I would | |
| 27:30 | say 174 and one half . I would multiply that | |
| 27:35 | by what I have out in front And I should | |
| 27:37 | equal what is underneath here . 349 . If you | |
| 27:40 | grab a calculator or if you do it by hand | |
| 27:42 | , you multiply this out , you're going to get | |
| 27:44 | what is under there , but you have to have | |
| 27:46 | the remainder part , otherwise you're never going to get | |
| 27:48 | the correct answer . So what I wanna do is | |
| 27:50 | I want to check this . So this whole thing | |
| 27:53 | that we got here , this is the answer , | |
| 27:54 | the whole part and the fractional part this is the | |
| 27:56 | answer . If we take the answer and we multiply | |
| 28:00 | it by what's out in front Then we should come | |
| 28:03 | back and get what's underneath here . Just like we | |
| 28:05 | take the answer multiply what's out in front over there | |
| 28:07 | . So we have to take this whole thing and | |
| 28:09 | multiply by X-plus two . So I'm going to check | |
| 28:12 | it over here on this other board , I'm gonna | |
| 28:15 | say um X plus two right ? And on the | |
| 28:21 | inside uh So it's X plus two . I'm gonna | |
| 28:24 | have X plus one plus the negative six over X | |
| 28:26 | plus two . And it's a monster . It's a | |
| 28:28 | lot of stuff but you have to keep it together | |
| 28:29 | so it's gonna be X plus one plus negative six | |
| 28:33 | over X plus two . Now it looks completely crazy | |
| 28:38 | to check this . It looks like oh my gosh | |
| 28:39 | how am I going to do this so many terms | |
| 28:41 | ? But it's not that bad . So first you | |
| 28:43 | have this whole thing , the X plus two is | |
| 28:45 | a unit distributed into the X . So let me | |
| 28:48 | go underneath here . It's gonna be X times X | |
| 28:51 | plus to save this multiplication for later . Right now | |
| 28:55 | you're just saying X times this is the first term | |
| 28:57 | . Then you're going to distribute multiply by the one | |
| 29:00 | that's going to give you the X plus two . | |
| 29:02 | So here I did this times the X . Gives | |
| 29:04 | me this this time is the one gives me this | |
| 29:07 | . Then I'm gonna have to go times and third | |
| 29:09 | term . So it's going to have X plus two | |
| 29:12 | . Then multiply by this giant thing negative six over | |
| 29:15 | X plus two . What we're saying is that we | |
| 29:19 | multiply all this together . We should get back what | |
| 29:21 | we started with . But look what's happened . You | |
| 29:23 | have an X plus two in the top and then | |
| 29:25 | X plus two in the bottom . So this completely | |
| 29:27 | cancels with this term right here . In fact you're | |
| 29:31 | always going to see that cancellation . So what you | |
| 29:33 | have is multiplying this out X squared plus this will | |
| 29:37 | be X times two is two , X plus X | |
| 29:40 | plus two . Plus this all divides away . You | |
| 29:43 | have a minus six plus a minus six means I | |
| 29:45 | have a minus six . And so what I'm gonna | |
| 29:48 | have is X squared and I have two X plus | |
| 29:51 | one , X . Is three X . And then | |
| 29:54 | I have to minus six is four . So negative | |
| 29:56 | four . So I get X squared plus three X | |
| 29:58 | minus four X squared plus three . X minus four | |
| 30:02 | bingo . You know you have the right answer because | |
| 30:04 | you've checked it and you've checked it using a totally | |
| 30:06 | different technique than the way you divided it . Use | |
| 30:09 | this ugly looking algorithm to divide it but you do | |
| 30:12 | multiplication to check it . So you know you've got | |
| 30:14 | the right answer . Alright ? Notice if you don't | |
| 30:17 | have the remainder here then you're never gonna have this | |
| 30:19 | term here and you're never gonna get the right answer | |
| 30:22 | . But also noticed that whenever you multiply with that | |
| 30:25 | remainder , you're always going to get this cancellation business | |
| 30:27 | going on . So this third term is gonna look | |
| 30:29 | simple and you'll always get back what you should . | |
| 30:32 | So that problem should be enough . Mhm . Two | |
| 30:38 | . You should sort of kind of understand polynomial long | |
| 30:41 | division . I mean you're not going to be an | |
| 30:42 | expert now but you should understand the concept now let's | |
| 30:44 | do one more to solidify this before wrapping it up | |
| 30:48 | . So here we're gonna divide these two things , | |
| 30:50 | we're gonna say 6 -2 x minus x squared . | |
| 30:55 | And we're gonna divide that by two minus x . | |
| 30:58 | So the first thing you notice , we want to | |
| 31:00 | write everything in the proper way . What goes on | |
| 31:02 | the outside is over here , but you always have | |
| 31:05 | to write it in decreasing powers of X . So | |
| 31:07 | you cannot right it is two minus X . You | |
| 31:09 | have to write it as negative x minus two on | |
| 31:11 | the outside . I'm sorry not negative x minus two | |
| 31:15 | . And negative X . Plus to the negative X | |
| 31:16 | . Term goes first the two term goes later and | |
| 31:20 | then you're gonna build your little house . Same thing | |
| 31:23 | here you have to write it as negative X squared | |
| 31:25 | . Then this term negative two X . Then this | |
| 31:28 | term positive six . If you have it all in | |
| 31:31 | the reverse order , you won't get the right answer | |
| 31:33 | with the algorithm . Okay , the next thing you | |
| 31:35 | look for is you say do I have any missing | |
| 31:38 | terms ? In other words ? Do I have any | |
| 31:39 | missing ? Like Yeah , I just said missing terms | |
| 31:42 | . So you have an X squared term in X | |
| 31:43 | term and a constant term . So there's nothing missing | |
| 31:45 | there . You have an X term and a constant | |
| 31:47 | term . Nothing missing there . In future problems will | |
| 31:50 | have missing terms . Like if we had if there | |
| 31:52 | was no two X term there at all , then | |
| 31:54 | I would have to put X negative X squared plus | |
| 31:56 | zero X minus six . You have to put the | |
| 31:59 | zero in there as a placeholder for its value . | |
| 32:02 | So here I'm ready to do the actual multiplication . | |
| 32:04 | I'm sorry . Division And again , we look only | |
| 32:07 | at the first terms . How many times will negative | |
| 32:10 | X . Go into this ? A better way of | |
| 32:11 | saying it is negative X times something will give me | |
| 32:14 | negative X squared . Well what if I just put | |
| 32:16 | an except there ? Then I multiply these . I | |
| 32:19 | have the negative times the positive giving me this X | |
| 32:22 | times X . Is X squared . So that gives | |
| 32:23 | me negative X squared . So then I do the | |
| 32:25 | multiplication , I get negative X squared . But then | |
| 32:28 | I have to multiply this , I get to X | |
| 32:32 | . Now you have to put a subtraction sign and | |
| 32:34 | circle it to remind yourself that you're subtracting both things | |
| 32:37 | . If you don't do that , most students will | |
| 32:39 | accidentally add these and get a zero . What you're | |
| 32:42 | trying to say is negative X squared minus a negative | |
| 32:45 | X squared . They go away to give you zero | |
| 32:47 | negative two X minus two X is negative two minus | |
| 32:51 | two . Gives you negative four X . It's very | |
| 32:53 | helpful to write that with a little circle there . | |
| 32:56 | I like to put a circle around it so it | |
| 32:58 | doesn't get too close to the other thing . It's | |
| 33:00 | just the way I do it do it whatever however | |
| 33:01 | you want . But you have to remember to subtract | |
| 33:03 | the value so that you have a flipping of the | |
| 33:06 | sign here going on because of that . Once you | |
| 33:09 | do the subtraction then you drag the next term down | |
| 33:13 | and you do the process again negative X . Times | |
| 33:16 | something will give me negative four X . It has | |
| 33:18 | to be plus four because four times negative X is | |
| 33:22 | negative four X . And then you go and say | |
| 33:25 | four times two is eight . So I need to | |
| 33:28 | subtract them . But again you're gonna accidentally add these | |
| 33:31 | if you don't put a little subtraction symbol there so | |
| 33:33 | this minus this is zero . What is six minus | |
| 33:36 | eight ? Six minus eight is negative two . And | |
| 33:39 | you're gonna get a negative two . Now at this | |
| 33:41 | point we have a degree of zero which is lower | |
| 33:44 | than the degree of one . So we stop . | |
| 33:47 | Uh there's no where else to go because the remainder | |
| 33:50 | we got is less as a smaller degree than what | |
| 33:52 | we had over here . So we say we're done | |
| 33:54 | with the division process . So then to write the | |
| 33:57 | answer down we're gonna write it down the proper way | |
| 34:02 | , we're gonna write it down in terms of its | |
| 34:03 | remainder . So the answer that we got was X | |
| 34:06 | plus four whole times X plus four whole times plus | |
| 34:10 | a fractional remainder which was negative two over what's outfront | |
| 34:14 | , negative X plus two . So this is the | |
| 34:17 | fractional part of the remainder . This is the whole | |
| 34:19 | number part of the remainder , this is the answer | |
| 34:21 | . So you have X plus four plus a negative | |
| 34:23 | two over negative X plus two . It looks like | |
| 34:25 | a nightmare . I agree with you . It does | |
| 34:28 | . But this is the answer . If you don't | |
| 34:30 | give me the remainder , it's wrong . If you | |
| 34:31 | don't give me any part of this , it's wrong | |
| 34:33 | . You have to have the whole thing there . | |
| 34:35 | All right . So now what we wanna do and | |
| 34:36 | we're not gonna do this for every single problem is | |
| 34:38 | we're going to check our work . We're claiming this | |
| 34:42 | is the answer . Which would be like if I | |
| 34:44 | had written the whole thing up here . So I | |
| 34:46 | take everything on top multiplied by this . So I'm | |
| 34:49 | gonna check it by saying that negative X plus two | |
| 34:53 | times the answer I get should recover what I have | |
| 34:57 | . So what do I have here ? I have | |
| 34:58 | X plus four plus negative two over negative X plus | |
| 35:03 | two close parentheses . So this times this whole thing | |
| 35:07 | and then I have to write all the terms out | |
| 35:08 | . It's gonna be this times the X term . | |
| 35:11 | So I'm gonna say X negative X plus two . | |
| 35:15 | And have this times the four term . Save this | |
| 35:18 | multiplication for later . Don't Too many do too many | |
| 35:20 | things at one time . All right . So then | |
| 35:23 | I have to take this and multiply by the large | |
| 35:25 | term at the end . So , I'm gonna have | |
| 35:27 | a negative X plus two , multiply by negative two | |
| 35:31 | on the top and then negative X plus two on | |
| 35:35 | the bottom . Notice what you have . This entire | |
| 35:39 | term cancels with this entire term . And so now | |
| 35:41 | I'm ready to to finally finish the whole thing . | |
| 35:43 | X times the negative X is negative X squared X | |
| 35:47 | times the two is two X . Four times a | |
| 35:50 | negative X is negative four X four times the two | |
| 35:53 | is eight . And this cancels leaving only a negative | |
| 35:56 | two . So what I'm gonna have is negative X | |
| 36:01 | squared what is two X minus four X is negative | |
| 36:04 | two X eight minus two is going to give me | |
| 36:07 | six . And so the answer I get is negative | |
| 36:11 | X squared minus two X plus six , negative X | |
| 36:15 | squared minus two X plus six . That checks out | |
| 36:18 | . So in order to check it , you have | |
| 36:20 | to multiply by the whole thing including the fractional remainder | |
| 36:23 | and you just have to be careful the way you | |
| 36:24 | do it . We're not gonna check every one of | |
| 36:26 | these problems but I do want you to know how | |
| 36:28 | to check it because you're inevitably going to be asked | |
| 36:30 | at on an exam . So here you have been | |
| 36:33 | introduced to the to the glorious concept of polynomial long | |
| 36:36 | division . I say it that way because honestly it's | |
| 36:38 | not a lot of fun , especially the first time | |
| 36:40 | you learn it . But we spent a lot of | |
| 36:42 | time in the beginning essentially telling you and showing you | |
| 36:45 | how it relates to regular division . When you do | |
| 36:48 | regular vision , you get a remainder . But the | |
| 36:50 | right way to express it is a fraction of the | |
| 36:53 | time that this can go in one more time . | |
| 36:55 | Once you get that concept down , then when you | |
| 36:57 | do your first polynomial division problem , when you get | |
| 37:01 | the answer and the remainder , it doesn't seem so | |
| 37:03 | weird to take this guy and add to it , | |
| 37:05 | the fraction that's left over . That's why we did | |
| 37:08 | that to show you that . And then once you | |
| 37:09 | have it in this form , it becomes a simple | |
| 37:11 | matter to check things because you just multiply by the | |
| 37:14 | divisor on the outside . You always get a cancellation | |
| 37:16 | , you simplify and you should always arrive with what | |
| 37:19 | you have kind of uh started with underneath the division | |
| 37:23 | symbol there . So we have several more problems . | |
| 37:26 | We're going to get longer polynomial is we're going to | |
| 37:28 | have some missing terms . We're going to do different | |
| 37:31 | things to make it a little harder . Maybe some | |
| 37:32 | fractions in there , but ultimately it's all gonna be | |
| 37:34 | the same process . So make sure you can solve | |
| 37:37 | all of these yourself . Follow me on to the | |
| 37:39 | next lesson . We'll get more practice with polynomial long | |
| 37:41 | division . |
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05 - Polynomial Long Division - Part 1 (Division of Polynomials Explained) is a free educational video by Math and Science.
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