05 - Polynomial Long Division - Part 1 (Division of Polynomials Explained) - Free Educational videos for Students in K-12 | Lumos Learning

05 - Polynomial Long Division - Part 1 (Division of Polynomials Explained) - Free Educational videos for Students in k-12


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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