08 - Learn Synthetic Division of Polynomials - Part 1 - Free Educational videos for Students in K-12 | Lumos Learning

## 08 - Learn Synthetic Division of Polynomials - Part 1 - Free Educational videos for Students in k-12

#### 08 - Learn Synthetic Division of Polynomials - Part 1 - By Math and Science

Transcript
00:00 Hello . Welcome back to algebra . The title of
00:02 this lesson is called synthetic division . Part one .
00:05 We're gonna have several lessons here increasing the problem complexity
00:08 . But the most important point here in the beginning
00:10 is to really understand what synthetic division is , why
00:13 we use it and to understand where it comes from
00:16 . I could just give you the kind of the
00:18 procedure and you could do it , but I want
00:20 you to know where it's actually coming from . So
00:22 in the last few lessons we learned how to divide
00:24 polynomial . And if you're anything like me or like
00:27 most students , you pretty much find long division of
00:29 polynomial is to be a pain . Uh it's just
00:31 because there's a lot of things to write down ,
00:33 a lot of cumbersome subtractions and so on . So
00:36 there is a method to do that long division in
00:39 a simpler way for certain kinds of problems . I
00:42 need to say that one more time . There's a
00:43 way to do that long division in a much much
00:46 simpler way , but only for certain kinds of problems
00:49 . There's a certain type of division where you can
00:52 use this uh concept called synthetic division to make it
00:55 simpler . Specifically , if you're dividing by the very
00:58 specific form , if you're dividing something by like X
01:02 minus a number or X plus a number , then
01:05 you can use this synthetic division . So for instance
01:07 , if you're dividing some polynomial , if you're dividing
01:10 it by x minus one , you can use synthetic
01:12 division . If you're taking some polynomial divided by x
01:14 minus 10 , you can use synthetic division . If
01:17 you're taking some polynomial , dividing it by X plus
01:20 three or X plus seven or X plus or anything
01:23 like that , X plus a number or X minus
01:25 the number . If you're dividing by that , you
01:27 can use synthetic division and it's much much faster uh
01:30 than the long division . So you cannot use synthetic
01:33 division for problems where for instance you're dividing by X
01:37 squared plus to the X squared , kills it .
01:39 You can't use synthetic division . If you're dividing by
01:41 X cubed minus nine , you can't use synthetic division
01:44 . So when you're divided by X plus a number
01:47 or X minus a number , we can make our
01:48 lives much much much easier by doing this . So
01:51 what we're gonna do now is I'm gonna do a
01:53 long division problems to kind of keep it in our
01:55 mind and we're gonna do that long division problem in
01:58 parallel by using the synthetic division procedure . So you
02:00 can see where it comes from . And then we'll
02:02 work a few problems , additional problems in synthetic division
02:05 so that you can get some practice . So before
02:08 I can introduce the synthetic division , we need to
02:10 do one quick problem in long division . And I
02:12 know that we just did this uh in the last
02:15 lesson . But it's really really really really helpful to
02:18 have a problem in your mind when we do the
02:21 synthetic division . So this one is not going to
02:23 be hard . So we're gonna take the polynomial two
02:27 X cubed minus seven X squared plus zero X plus
02:33 five . Notice we have a zero X . So
02:35 remember when we do division , we have to pad
02:37 any missing powers . We have an X cubed X
02:40 squared X . To the first . And then no
02:41 X at all . We have to pad any missing
02:44 X's variables there with uh with a zero there and
02:48 we're gonna divide that by x minus three . Now
02:50 notice right away that this problem is set up .
02:52 I'm going to do it in the long division format
02:54 . But because we're dividing by X minus and number
02:57 or alternatively X plus a number . I'll show you
02:59 in a few minutes . Then we know we can
03:01 use synthetic division . So we will do that in
03:03 just a minute . Um So in order to crank
03:07 through the long division procedure , what do we do
03:09 ? We take a look at the first term here
03:11 in the first term here , X times something has
03:13 to give me this . So there's a one out
03:15 here . So one times two is two and X
03:18 times something is X cubed . This has to be
03:20 an X squared . All right now we go backwards
03:23 and we multiply the two times the one gives me
03:26 a two and the X squared times the X .
03:29 Gives me X to the third power . And then
03:33 the two times the negative three is gonna give me
03:35 negative six . And then the X squared times one
03:37 is X squared . Now I need to subtract both
03:40 of these from their partners above . So I'm gonna
03:41 put a minus sign . I'm gonna put a circle
03:43 around it to remind me that I'm subtracting directly from
03:46 above . And by the way if you have none
03:49 of this looks familiar to you then it's just because
03:51 you haven't watched my lessons on long division . So
03:54 if this looks crazy already then go back and watch
03:56 those lessons . I have entire lessons explaining every little
03:58 part of this process . So we have the two
04:01 X . Cubed minus two to execute . That's gonna
04:03 give me a zero . So I don't need to
04:05 really write it there . But this negative seven X
04:07 squared subtracting a negative six X squared really becomes negative
04:12 seven minus minus six . So negative seven minus a
04:16 negative six is basically negative seven plus six . So
04:19 it's gonna be negative one . I'll put it like
04:22 this -1 X . Squared . So you have to
04:25 be careful when you do this attraction negative seven minus
04:27 a minus becomes negative seven plus six . So once
04:30 I've done the subtraction I take my next digit and
04:33 I bring it down which is zero X . And
04:36 then I start again X . Times something will give
04:38 me the negative one X . Square . It has
04:40 to be a negative one X . So I take
04:43 and multiply this direction I'm gonna get negative one X
04:46 . Squared . And then the negative one X .
04:48 Times the negative three is positive three X . Again
04:52 I need to subtract I'll put a negative sign there
04:54 with a circle to remind me that I'm doing both
04:56 terms . This minus this gives me 00 minus three
05:00 gives me negative three X . Gives me negative three
05:04 X . And then after I've done this I dragged
05:07 my next term down which is a five . And
05:10 I look at the first terms X times something gives
05:12 me negative three X . I have to have a
05:14 negative three here and then I multiply . This is
05:16 gonna give me negative three X . And then the
05:18 negative three times the negative gives me positive nine .
05:21 Almost done here , we subtract both terms . These
05:24 give me 05 minus nine , gives you negative four
05:28 . Okay , Gives You -4 . So we have
05:32 done this numerous times before and now that we have
05:35 the whole number of times this can be divided in
05:37 and the remainder part , the answer to this problem
05:41 is what we have written above here two x squared
05:44 minus x minus three . And then we have a
05:47 fractional remainder part which is negative four over what we
05:51 have out here X minus three . So this is
05:53 kind of the remainder is a fraction of what you're
05:55 dividing by . So this is kind of the fractional
06:00 This is kind of the whole number part of the
06:04 the answer now , why am I doing this ?
06:05 Because in order to explain what long division is you
06:08 have to have in your mind a concrete example from
06:11 , I'm sorry , in order to explain what synthetic
06:13 division is you really need to have in your mind
06:15 . What a problem feels like from long division .
06:18 We can all agree that this is a real pain
06:20 . What are the real problems with this ? The
06:23 first problem is I have to write all these exes
06:25 down everywhere . There's excess everywhere . It's just completely
06:27 filled with X X squared x cubes . And so
06:30 uh I know we're doing it to keep track of
06:33 it , but that's obviously something that's not good .
06:35 Another thing that's not good is that uh this subtraction
06:39 process is really cumbersome because you have to be careful
06:42 when you're doing negative six minus a minus six .
06:45 It's really negative . I'm sorry negative seven minus of
06:47 minus . It's really negative seven plus six . And
06:50 so it's very very easy to make sign errors when
06:53 you're subtracting negative numbers and they're all spread out like
06:56 that . That's the number two things . So synthetic
06:58 division is a faster way to do this process .
07:01 It's actually lightning fast . You don't have to worry
07:03 about the crazy subtraction . You don't have to do
07:06 that . And also as a bonus you don't have
07:08 to write all those X . Is everywhere . So
07:10 the problems don't take up the whole page . So
07:13 what we're gonna do now is we're gonna talk about
07:15 the concept of synthetic division . Send fucking spell synthetic
07:24 division . Right ? So that take division . Right
07:29 ? Alright . So I've already told you a few
07:31 times , but basically you can only use it .
07:33 It's a faster process . But it lets us to
07:35 divide when the divisor that means what you're dividing by
07:41 must B . I'm gonna write it as X minus
07:46 a number . But keep in mind this number can
07:48 be positive or negative . So when I say x
07:50 minus a number , it can really be like x
07:53 minus two . I can divide by this . I
07:55 can divide by X minus 19 . That's fine .
07:58 I can also divide by X plus three . I
08:00 can also divide by X plus , you know whatever
08:02 I want 18 . Right ? So when you see
08:04 x minus see it doesn't mean it has to be
08:06 x minus something . Consider that C can be positive
08:09 or negative . So really it's better to say that
08:12 you can use synthetic division when you divide by X
08:14 plus a number or x minus a number . All
08:17 right . So what does it look like first one
08:18 ? I'm not gonna show you how to do it
08:20 yet . I'm gonna show you what it will look
08:21 like and then I'm gonna show you how to do
08:23 it . Here's what you do this three out here
08:27 . See how it's X minus of three . In
08:29 synthetic division , it's X minus three . You're dividing
08:31 by but you don't write the minus three . You
08:33 write it as a positive three and then everything on
08:36 the inside that you're dividing by . You write all
08:38 the numbers down , right ? But you don't write
08:40 these these variables now . So I'm gonna show you
08:43 how to do it and we're gonna talk about it
08:44 . Uh Just a second . So the to the
08:46 negative 70 and the five . And then instead of
08:50 doing a traditional division bar you kind of go upside
08:53 down and kind of right it upside down . You'll
08:55 see why in just a minute . So you can
08:57 see when we write the problem statement like this .
08:59 All of the important information is contained because all polynomial
09:03 are going to have these coefficients even if there is
09:05 a zero . We have to include it when we're
09:07 doing a long division . So really we're just not
09:09 writing all these exes everywhere . But we know that
09:11 this has to be a two X cubed because this
09:14 has to be constant . So this has to be
09:16 zero X . So this has to be negative seven
09:19 X squared . So this has to be two times
09:20 X cubed . So you kind of go backwards and
09:22 you can reconstruct what the polynomial is just by the
09:25 numbers . All right now , I'm not going to
09:28 show you how to do it yet , but we
09:29 go through this process that I'm gonna show you in
09:31 just a minute . And what you're gonna end up
09:32 with is with some numbers down here . Uh Right
09:36 . And then you're gonna have a line here and
09:39 then you're gonna have some numbers down below . Yes
09:41 , I'm not I'm not expecting you to understand this
09:44 yet . I'm just showing you how it is set
09:46 up . So what happens is when you do the
09:48 process , which is very simple . What you end
09:51 up with is the first three numbers that you get
09:53 , or I should say , let me let me
09:54 back up and say this way , the last number
09:56 that you get this is the remainder notice I got
10:02 a negative for their the remainder in our long division
10:04 was actually a negative for all of the numbers to
10:07 the left of this are basically the whole number part
10:10 . So this is the remainder , This is the
10:12 rest of the guy here and this means two x
10:16 squared minus x minus three . So notice this polynomial
10:20 is what we got the two X squared minus one
10:22 minus three . That's what we got up here .
10:24 The remainder in long division is way down at the
10:26 bottom and then you have to reconstruct the answer as
10:29 we've done many times in synthetic division . When you
10:32 do the process , the very last number you get
10:34 on the right is the remainder and then the numbers
10:36 to the left represent the whole the whole number part
10:39 of the polynomial . The rest of the answer ,
10:40 basically which we can reconstruct here . All right .
10:44 Um And then of course I've already said it to
10:48 you before but I'll just kind of like pointed out
10:51 100% clear this represents two X cubed minus seven X
10:57 squared plus zero X plus five . That's what that
11:00 represents . And then this represents x minus three .
11:03 So if you have X minus three , you put
11:05 a positive three that you're dividing by . If you
11:07 end up divided by X plus five , then you
11:09 have to put a negative five . So whatever the
11:11 sign is that you're dividing by . You put the
11:16 don't expect you to know anything about how synthetic division
11:18 works yet . I haven't taught you that all I'm
11:20 showing you is that when you write all the numbers
11:22 down , this is what they mean . The number
11:24 on the outside of the little houses what you're dividing
11:26 by . But you have to use the opposite sign
11:29 . The numbers on the inside of the house is
11:31 what you're dividing into . And then the numbers on
11:33 the bottom represent the remainder , along with the rest
11:36 of the answer . As we have done in long
11:37 division . All right . So now it's time to
11:42 you see where I want to do this . Mm
11:46 I think now it's time to actually do this problem
11:52 , right ? And show you where it comes from
11:54 . So I'm gonna rewrite everything in purple here .
11:56 I'm not gonna write the answer down . I'm just
11:57 gonna rewrite the problem statement here in purple . Uh
12:00 as if we're dividing these polynomial together are into one
12:04 another . So I have a three outside the house
12:06 . And then I had a to a negative 70
12:09 and a five . And you draw like an upside
12:12 down division , simple sort of right ? Here's how
12:14 you do it . You write this problem statement down
12:17 , and then you draw a line there and then
12:20 you draw a little arrow . I do this anyway
12:21 . And you need to drop down the first number
12:24 . Okay , the first thing I want to show
12:26 you actually I forgot to mention it to you is
12:29 one very very important thing . I'm sorry I didn't
12:31 mention this before when you're doing long division . Obviously
12:35 everything written on here is important . But when you're
12:37 doing this subtraction and you're working through this black part
12:40 here , you see how the twos are subtracting .
12:42 The negative ones are subtracting , the negative threes are
12:44 subtracting and so on . The most important numbers .
12:48 Um Are these numbers boxed in red ? Um You
12:52 have uh you have a negative six . Let me
12:55 just double check myself . You have a negative six
12:58 . You have a three . Uh and you have
12:59 a nine here . Why are these the most important
13:01 numbers ? Because these are the numbers that you get
13:04 basically uh when you when you do the multiplication down
13:08 and then you need to subtract . Right ? So
13:10 these numbers give you the next line Uh in the
13:14 process . So notice that the -6 , the three
13:16 and the nine are also present in the synthetic division
13:19 here , but they're opposite signs , right ? So
13:22 whereas when you were divided by X -3 , we
13:26 had to subtract all of these polynomial . And that's
13:28 why the subtraction was really cumbersome . When we instead
13:31 of putting X -3 , when we write this as
13:33 a positive three , then what happens is these numbers
13:36 there become opposite signs of what they were here .
13:39 So we don't have to do any subtraction anymore .
13:41 We end up adding . In fact , you can
13:43 kind of see I haven't really shown you the process
13:46 yet , but you can kind of see right here
13:47 that the negative 67 plus six gives me this this
13:50 plus this gives me this and this plus this gives
13:52 me this . So one of the main main advantages
13:55 of synthetic division is that you no longer have to
13:57 subtract . You can just add things together . Which
13:59 is much much easier to do . Yeah . All
14:03 right . So let's get to the process . We
14:05 drag the first number down . That's our starting point
14:08 , right ? And then what we wanna do ,
14:11 I'm not gonna do this for every problem . But
14:12 just to explain it to you , I'm gonna draw
14:13 a little arrow through here , right ? And in
14:17 this era I'm gonna write three times two . Because
14:19 what I'm doing is I'm taking three times two and
14:22 I'm gonna write it in this position which is six
14:25 . That's what this means three times to arrives in
14:27 the sixth position . Then I just take negative seven
14:33 . Plus six negative seven plus six is negative one
14:36 . Right ? Once this number is in place ,
14:39 then I'm gonna again kind of carve through the one
14:42 position here like this and say three times the negative
14:45 +13 times the negative one which is negative three ,
14:49 which goes in the next position right here . And
14:53 then I add these together . Zero plus a negative
14:56 three is negative three . And then once I have
14:59 this in position , then the same thing happens here
15:02 . I go through here and it's gonna be three
15:05 times a negative three , Which is -9 Like this
15:10 . And then I add these guys together and it's
15:12 -4 . So then I have arrived at the answer
15:15 notice that's what I showed you here . I said
15:16 I just didn't put all these arrows everywhere . I
15:18 said you wrote the problem statement down there's some intermediate
15:21 number six , negative three negative nine . That's where
15:23 they come from . And then you have the answers
15:26 on the bottom . The very last number here is
15:29 the uh remainder . And then everything before that is
15:33 the actual polynomial . Now , when you're writing the
15:36 answer down from synthetic vision , you know , this
15:39 is an X cubed polynomial because this is a constant
15:42 term , This is an X term , This is
15:44 an X squared term , This is an X cubed
15:46 term . So because this is an X cubed term
15:49 , you know that this has to be a two
15:50 X squared term . You know , this has to
15:53 be an X term . You know , this has
15:55 to be a constant term and then this is the
15:57 remainder which is negative four over whatever I divided by
16:01 , which we know is x minus three because it's
16:03 opposite signs what we put up there . How do
16:06 you know that ? The answer you right , is
16:07 actually squared here . It's because when you look back
16:10 to the long division problem , that's why I wanted
16:13 to do it first . We're dividing into a cubic
16:17 and we're dividing by x minus something . So you
16:19 always start have a answer that's gonna be one degree
16:22 lower than what you're dividing into . When you're dividing
16:25 by X minus something , right ? You can see
16:28 the answer was X squared when we're dividing into an
16:31 X cube . So when we're doing synthetic division ,
16:33 because we always have to divide by X plus number
16:36 X minus and number the answer you get is always
16:39 gonna be one degree lower than what you're dividing into
16:42 . Okay , so here's the procedure , you write
16:45 down the numbers of the polynomial , dividing into .
16:47 You write down what you're dividing by but you have
16:49 to put a different sign , the opposite sign of
16:52 what you're dividing by . We were divided by X
16:53 -3 . So we put a three there . We
16:56 dropped the two down three times two is six .
16:58 Okay great add these together we get a one .
17:00 Okay , three times negative one is negative three .
17:03 Great . We add these , we get negative 33
17:05 times a negative three is negative nine . We add
17:08 these , we get negative four . All the numbers
17:12 have to do any subtractions . The reason we didn't
17:15 have to do any subtractions is because we write the
17:17 problem while we're dividing by we basically change the sign
17:21 instead of X -3 . We divide by X-plus three
17:24 . That ends up with the effect of making all
17:26 of these numbers opposite in sign to the long division
17:30 . So we don't have to subtract them anymore .
17:31 We can just add them . That's the reason why
17:33 it works . But procedurally you just multiply add .
17:40 here . This is the remainder divided by what we're
17:42 dividing by . This is the fractional part of the
17:45 remainder of the remainder and then this is the the
17:47 whole part of the polynomial answer . All right .
17:51 Now that you have the procedure in place , we
17:54 can go crank through two more problems that are gonna
17:58 be really , really fast . Now let's say that
18:00 you're doing three x cubed minus five X squared plus
18:05 x minus two . And we're dividing that by x
18:08 minus two . Now , if you were doing long
18:11 division , you would have to write this whole thing
18:13 out and then do all the stuff with the subtraction
18:16 . And it would be basically take the whole board
18:17 , right ? But since we're doing long synthetic division
18:21 , we don't do that . We take the numbers
18:23 . We take away the numbers and strip away all
18:25 the variables . And we write it under this house
18:27 here . So three is the first number negative .
18:30 Five is the next 11 is the third one and
18:33 negative two is the last one . Now we have
18:36 cube square first power , no power . So we're
18:38 not missing any terms . So we don't need any
18:41 zeros here . But if we were missing , for
18:43 instance the X squared term , if it was just
18:45 three X cubed plus X minus two and this one
18:47 were gone , then we would have to put a
18:49 zero here would be +3012 You have to pad the
18:52 zero same as long division . Then you put an
18:55 upside down house and you're dividing by X -2 .
18:58 But you don't put a -2 here , you always
19:00 flip the side . You can think of it like
19:05 subtract . So when you're dividing by something you just
19:07 flip the sign of whatever it is and notice it
19:09 has to be X minus something or X plus something
19:12 in order to even do this at all . Yeah
19:15 . All right now we're ready to do a very
19:17 simple process . We take the three . We draw
19:19 a little era . We drop it down three times
19:22 two gives me six . No need to draw those
19:24 arrows . Now we know what we're doing three times
19:26 two is six . Then we add these negative five
19:29 plus six is just the number one , Then two
19:31 times 1 is two . Then we add these to
19:35 to give us three , then two times three is
19:37 six . Then we add these to give me a
19:40 positive four . Let me double check myself 3134 Now
19:44 I think you can agree that this process is way
19:48 more cumbersome and complicated than this process . So again
19:53 , just one more time from the top . Drop
19:54 the first number down three times two is six .
20:00 a 33 times two is six . Ad get a
20:03 four and you always know that this number at the
20:06 end is the remainder , everything else before it is
20:10 the rest of the answer basically . So the way
20:13 you're going to write it is since you know that
20:15 this is a cubic your this represents the cube term
20:18 , the X squared term , the X term and
20:20 the negative two term . Then the answer here has
20:22 to be three X squared plus X plus three because
20:25 it has to be one degree less than what you
20:28 were dividing into . And then you have the four
20:31 as a remainder divided by the x minus two .
20:34 This is the whole answer . Three X squared plus
20:37 x plus three Plus fraction for over X -2 .
20:42 That's all done . Synthetic division is a really ,
20:46 really huge time saver . And there's also other other
20:49 uses of synthetic division . We're gonna discuss a little
20:51 bit later . Right now we're just doing it just
20:53 to divide polynomial . But there's actually some other uses
20:56 that we can use this technique for a little bit
20:58 later . Now what we're gonna do is our final
21:01 problem just to get one more bit of practice .
21:03 Let's say that we have the polynomial we want to
21:06 divide is as follows , X cubed Plus three ,
21:10 x squared minus two times x minus six . And
21:14 we're gonna divide all that by X plus three .
21:18 So you have to ask yourself , am I divided
21:20 by x minus a number or X plus a number
21:22 ? I am divided by X plus the number .
21:24 So , I'm allowed to use synthetic division . If
21:26 I were allowed . If this problem were different .
21:29 If it were all this junk divided by X squared
21:32 plus three , then you can't do it . You
21:34 can't use synthetic division . You have to go back
21:36 and use the long division method . That works for
21:38 everything . Okay ? If you're dividing by X cubed
21:42 plus three three X minus two , you can't do
21:45 it . It only works when it's X plus a
21:47 number or X minus the number . All right .
21:50 So then what we're gonna do is write down the
21:52 relevant information . This is a cube , a square
21:55 of first power . And in the no power we
21:56 don't have any missing terms . So it's just gonna
21:59 be one , three negative to negative six . Draw
22:03 my upside down a little house and then we're divided
22:06 by X . Plus three . You put the opposite
22:08 sign of whatever you're dividing by their . That is
22:11 really what allows us to do addition into this attraction
22:13 . So that's the way to remember it . Whatever
22:15 you divide by just change the sign of it .
22:17 And then you draw your horizontal line and you drop
22:21 the number one down here and then you say three
22:23 times negative three times the one is negative three .
22:26 I add these when I get a zero And then
22:29 three times 0 is zero . Uh And I add
22:32 these and I get a negative two and then the
22:34 three times negative two is positive six . I add
22:37 those and I get a zero . So I get
22:39 10 negative two . And zero . Now notice the
22:41 interesting thing here is the remainder is actually equal to
22:45 zero . So this is one of those examples where
22:47 if I had done this long division way , I
22:50 would get all the way down to the bottom .
22:51 I would do my final subtraction . I would get
22:53 a zero . So I would not have a fractional
22:55 part because it would be the fractional part will be
22:58 zero over over this , which is zero . So
23:02 I can write the answer down even easier than before
23:04 . I know that it's a cubic dividing by .
23:08 So this has to be X squared one X squared
23:11 plus three . X . I hope so . I'm
23:13 writing the wrong thing down . I'm sorry about that
23:16 . I'm sorry about that . What I have here
23:19 is an answer is one X squared plus zero x
23:23 minus two . This is the answer . Or you
23:25 can just write it as X squared minus two ,
23:28 X squared minus two . So the answer is X
23:30 squared minus two . All right . So that's the
23:33 process of synthetic division . It makes our long division
23:35 problems much much easier to solve . The only thing
23:38 we have to do is remember that when we're dividing
23:40 by whatever we're dividing by , it can only be
23:43 X plus the number , or X minus the number
23:47 front of the synthetic division symbol . After that .
23:50 This process is really , really fast , right ?
23:53 And we have other uses for it . We have
23:54 other theorems that use synthetic division . I'm gonna show
23:56 you a little bit later , but for right now
23:59 use it to solve your long division problems . Whenever
24:02 you're dividing by X minus the number , or X
24:04 plus a number , solve all of these problems .
24:06 Follow me on to the next lesson . We're gonna
24:07 get a little bit more practice with synthetic division .
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