Up next in 10
Dividing Monomials: Long And Synthetic Division
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Mar 8, 2025
We’re now moving on to section 2 of chapter 8, and we’ll talk about simplifying monomials and also learn how to perform both long and synthetic division of polynomials. Chapters: 00:00 Introduction 00:41 How to divide a polynomial by a monomial 03:25 How to perform the long division ( with examples) 16:30 How to perform the synthetic division ( with examples)
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0:00
we're moving on we're
0:02
talking dividing pols right we're going
0:06
to talk about that we're going to talk
0:07
about synthetic division we're going to
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talk about long division and this again
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is not going to be that hard now on
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Friday like I said we're going to
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have uh a qu on Section 4748 and 51 I'm
0:21
going to combine those three and I'm
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going to give you one yeah Friday Friday
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yeah so I'm going to give you a review
0:26
on Thursday right so now in this
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section are yeah we're going to divide
0:34
we're going to learn how to divide uh a
0:36
polom with a monomial right so what do
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you do here so you you're going to have
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to split how many terms do I have up top
0:44
three three I
0:46
have one two and three terms right I'm
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trying to divide divide each term by
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what 3 XY so what I'm going to do is I'm
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going to take them one at a time right
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so I'm going to have 6 x 4
1:00
y Cub over 3x y we also going to have
1:05
plus 12 x Cub y^ 2 over 3x y and then
1:12
minus 8 18 x² y over 3x y now we're
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going to simplify them one by one right
1:20
so when I go here right I know three
1:21
goes into six how many
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times two two right so I start a two and
1:27
then let's see what I have now x to the
1:29
four and and x x how do I simplify that
1:32
subtract subtract them so what do you
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get XB X cubed right and then y cub and
1:39
then y y y^ 2 right so that's done here
1:42
so now we go to the next one 12 3 goes
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into 12 how many
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times four right X cub and x x x² y s
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and y y y right so far so
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good okay anybody trouble with this okay
2:02
and then the next one 3 into 18 that is
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6 right and then x² X that is x y y and
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Y that's gone so the final result is 2x
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CU Y 2 + 4x2 y - 6X this is how you
2:19
divide a polom by a monomial right
2:24
you're just going to take them one at a
2:27
time and you just apply every rule that
2:30
we've learned so far right why don't you
2:32
simplify excuse me why don't you
2:34
simplify you can't there's no like terms
2:36
here I can't I can't do anything else
2:38
this is X CU y s so I can't add this up
2:41
right so this this is as far as I can go
2:43
now we're going to learn long division
2:45
right long division is something that
2:47
some people like some people don't like
2:49
I don't know where you stand on it so
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we're going to find out so I'm trying to
2:54
divide X2 + 3x - 40 I'm trying to divide
2:58
that by x - 5 some sometimes you have a
3:00
remainder sometimes you don't have a
3:02
remainder right in this case we're going
3:04
to find out so the key here is this when
3:07
you lay out the uh the divid no this is
3:11
the divisor or the dividend that's the
3:14
dividend dividend right and this is the
3:16
divisor and we have whatever we have
3:18
here right and the remainder so now what
3:20
you do is you want to make sure you
3:22
order them from the highest to the
3:24
lowest exponent right so I can't have 3x
3:28
here and plus X squ I need make sure is
3:30
in a descending order does that make
3:33
sense right also sometimes let's say I'm
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just going to give you a quick example
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we going to go so if let's say I have X
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Cub +
3:42
x -1 and I want to divide it by like
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let's say x + 1 right if I put x + 1
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here I have to make sure I put X Cub + 0
3:53
x2+ x -1 you have to make sure even
3:56
though we do not have x² here you have
3:58
to include 0 x² in your dividend or if
4:02
you don't do that you will mess up your
4:04
order right we're going to get to that
4:06
right now we don't have that yet but
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we're going to get to that I just want
4:08
to introduce it in passing
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right
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yes I'm saying if I'm dividing x + x -1
4:20
by x + 1 right if I want to use the long
4:23
division if I put x + one here I should
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have technically X cub
4:30
+ x -1 here correct but I said we have
4:35
to rearrange them in a decent in order
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right so after X CU what's the next
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exponent that has to that has to come
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one something no Cube two right we don't
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have two so what we do in long division
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is we always include even if you don't
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even though we don't have it here we
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have to include 0x2 + x -1 we do that
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because we do not want to mess up our
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lining right our adjustment here we're
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going to get to that I'm just
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introducing we just setting up the
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problem right now excuse me are we just
5:03
setting up the problem are we solving it
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to yeah we we I'm just showing you
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what's going to happen in the near
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future which is probably in the next
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we're not solving it yet but we you put
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we're just like giving a preview of that
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right now here's how we do use Mr saling
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yes can you send Gavin to the front
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office for early dismissal yes thank you
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okay all right so what we do here is
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this so it says x - 5 right we are
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dividing that into X2 + 3x - 40 so what
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you do is this you always
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take this number this leading uh
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coefficient coefficient right the
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leading coefficient with the leading
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exponent here right and then you going
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to divide that into the first term right
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so I go how many times does X goes into
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x squ two no one one time so what do we
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have what does that give you 1 x right
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so basically x² / X is 1 X you go like
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this
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right so X goes into x² X time right so
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that's what we doing with long division
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so then I go x * X is
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what I don't want to hear 2x again when
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I say x * X right and then x * 5
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is 5X so we we're doing this here right
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and now what we're going to do is we're
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going to subtract that from this right
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we doing long
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what is x -
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X20 right let's go x² - I'm taking it
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out right and then have 3x - - 5x you
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have to be careful here so is that 8X 8
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x right 3 because 3x double negative
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becomes what a positive right 3x - - 5x
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is 8x and then we bring down what - 4 -
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40 so I'm going to do the same thing how
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many times does X goes into 8x
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8 time right X into 8 x is plus what
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plus 8 because it's a positive it is
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plus 8 correct are we good on that now
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what's 8 *
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x 8 x what is 8 *
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5440 right and we do the same thing we
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subtract it 8x - 8 x is
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040 - - 40 that is also zero so x² + 3x
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- 4 / x - 5 gives us what x + 8 it's
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great because the remainder is what zero
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so that works perfectly what if the
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remainder isn't zero so we going to get
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to that I'm glad you asked I think
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that's my next
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thing let's see all right
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now uh yeah that's the next thing that
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we're going to do is something remainder
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right so do you guys want me to work on
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another example like this pleas can you
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just do one more you do one more right
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let's do one more so let's say we have
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x²
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right + 7 x - 30
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/ uh x - 3 right x - 3 so I need to lay
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it out correctly right this is x - 3 is
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this the only way to do it yeah we going
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to No we're going to do the syntatic
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division as well but right now we're
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doing WR division right
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so what do I do first somebody help me
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here okay so you did the X how many
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times it goes okay so it goes one time
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so that work X okay and then
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X okay x times x is X x * -3 um3x all
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right and then we do What minus right so
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it's Z and then it's 10 X 10 X and then
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what um 9 is 30 all right right and then
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uh then you do it again mhm so 10 times
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10 times right because a positive 10 10
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* x 10 10 *
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-3- okay and then we subtract it what we
9:15
get zero so the remainder is zero right
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so X2 + 7 x - 30 x - 3 gives us what x +
9:22
10 x + 10 right that make sense
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yeah H you just going to take practice
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right take practice that's why you do a
9:32
whole practice on me okay we got to keep
9:35
moving we got to keep moving wait don't
9:37
eras that please this that you can eras
9:41
that
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one all
9:44
right now let's watch this here right we
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are dividing watch this here we're
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dividing x² right first listen to uh see
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what what I wrote on the board right so
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times three - x to the1 right you see
10:02
this so how would you solve
10:06
this yeah first you have to um
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distribute thetive one right so how you
10:11
how you going to do that isn't it
10:14
31- o no we can't do that something
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raising the negative what happens to it
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flip it flip it so this becomes what x²
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+ 7 x - 11 and then what over
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- x right you're dividing it does that
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make sense you're dividing it now
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right so cuz why did you because this is
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raised to the ne 1 right when it's
10:41
raised to the ne 1 that means you flip
10:43
it right any number raised to if I have
10:46
X1 is 1 /x oh okay yeah yeah that Mak s
10:51
so that we flip it that's the same thing
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we did here right and now we're going to
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divide this into this but here's what
10:57
we're going to do here if you look look
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at the last example that I did what's
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always the leading coefficient the
11:03
number x x right here we have 3 - x so
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we have to reite the correct
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way x + 3 right so this is going to be
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thank you co it's going to be x + 3
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right divided by
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what
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x - 11 right so now let's
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look how many times does X goes into x²
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one
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think about
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itgax into
11:33
X if you can't remember Negative X right
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if you can't remember just go like this
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if I do this right x x is what I still
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have a negative so this is X right X
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into x² that
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ISX right
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X because I I'm dividing X into x² not
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positive X negative so I still have a
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negative right now what's x * x x s
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what's x * 3 - 3 - 3x right we going to
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subtract again right subtract that x² -
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x² 7 x - -
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3x 10 X 10 X and then guess what I'm
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this one now right 10 x - 11 all right
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mhm now how we time does X go into 10 x
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10 -10 right so - 10 X 10 X -10 * 3 -30
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now we have a case where our remainder
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is not what zero right 10 x - 10 x
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is0 and then -1 - -30 is - 41 no what
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[Music]
12:55
two 19 thank you 19 so now we have a
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remainer of 19 when you have a remainder
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of 19 this is what you do right so to
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write that here's what we're going to do
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because the remainder is not
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zero we're going to say that
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x² + 7 x - 11 3 - x bless you is equal
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to what
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first x - 10 right plus 19 / 3 - x this
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is how you write when there's no
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remainder right we have the division
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gives us this right but because you have
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a remainder it's going to be 19 over the
13:38
divisor right the reason why we don't
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write nothing here when you don't have a
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remainder is because it's zero Z over
13:44
any number is what zero okay do that
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make sense so let's do one more like
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this
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okay you get to
13:55
that tired my brain hurts check your
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brain
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no all right so we have
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x²
14:07
L
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uh 5x + 7
14:16
Time 1 -
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X1 I want to volunteer to help me here
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um the bathroom
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yes all right uh Kay help me look what
14:29
you
14:30
did so what would you do first no I
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don't know how to do it so what are you
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going to do first um you're going to
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put well you're going to under yeah put
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it underneath and that so we get
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x² + 5 x + 7 over what 1 - x and now we
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can divide it right so we can divide we
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going to put it over here 1 - x /
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X2 + 5 x + 7
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flip it we can flip it so that's x + 1
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right make it easy for for us all right
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so what do we do
15:07
now
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X it would be thank you
15:13
X XX is X x *
15:18
1 oh time sorry I
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thought1 1X right want all right now we
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subtract it right this minus this is z
15:30
and then 5x -
15:33
-1x 6 x and then we going to bring down
15:36
this guy here plus 7 right and now we're
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going to do the same thing how many
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times x goes into
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6X -6 right- 6 * X POS 6 x -6 * 1 -6
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right and we subtract again this is gone
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7 - - 6 is
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right so therefore this divid by this
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gives us what it gives
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USX - 6 + what 13
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over- 1 - x or x + 1 whichever want the
16:13
way however you want to write it right
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so this is pretty much a long division
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now we're going to talk about the
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synthetic division synthetic is actually
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fun right it's it's better it's easier
16:24
to write so here's what we're going to
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do to synx division I'm going to show
16:27
you the setup right
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so what you do is this for the synthetic
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division X this is like 2x Cub - 13x 2 +
16:39
26x - 24 right if you pay attention
16:42
everything is in the right order right
16:43
we have the cubic exponent then what the
16:46
square then the X then the nothing right
16:51
so now we divided by x - 4 but because
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this is four right this is x - 4 we
16:56
write four like this and we go like this
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right this is just the synx division you
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just going to have to learn it there's
17:02
no ryal reason we just this is how it is
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so I can't make it any better than that
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so it's just what it is it's four right
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and then we're going to lay out all
17:10
these numbers right we have two just the
17:13
leading coefficient two -3 26 and -4 and
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we draw a line under that right this is
17:22
how we do the synthetic division okay so
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and then the next thing we do is we're
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going to go
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two two 0 right there's nothing here 2 +
17:30
0 is 2 right and then here's what we're
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going to do we're going to go 4 * 2 is
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what 8 we put it here 13 + 8 is uh five
17:43
five right and we going to go 4 * 5 is
17:47
-20 right so 26 -
17:51
20 -6 right no positive 6 you mean 4 * 6
17:56
is 24 24 right right4 + 24 Z right so
18:02
now what you do is this now you when you
18:05
divide this into this what you do with
18:08
the remainder is with what you have left
18:11
is this number here is the L coefficient
18:14
of the next polinomial right because
18:15
this is X Cub this is going to be 2x2
18:19
right so the solution will be
18:21
2x2 - 5x + 6 so basically when you
18:26
divide this into this this is what you
18:28
get right you always go one um exponent
18:32
down once you get your result okay the
18:35
remainder is zero so we're going to do a
18:36
little more of these and then you going
18:37
to get the hangover it the synthetic
18:40
yeah so in the problem four is
18:44
negative does that matter ites yeah
18:47
since since this is x - 4 this is always
18:49
going to be positive if this is x + 4
18:51
this is going to be now what negative
18:53
you just have to switch it I wanted to
18:55
get to that while you jump ahead of me
18:56
but that's is it always going to be 2x2
18:59
like is that what is going to look like
19:01
no if I have 2x 4 yeah we going to do
19:04
some more probably you get it right it
19:05
depends on the the example let me do
19:07
another one here and then we'll see all
19:10
right for example we have uh synthetical
19:13
game we leave this on the board and
19:15
we're going to
19:18
have uh so that's just your answer you
19:20
don't have to do anything else yeah 6X 4
19:23
yeah - 8 x 3r
19:29
+ 12
19:32
x -
19:34
14 / x - 2 right don't you have to add a
19:38
0x2 yeah we have to add it now watch
19:41
this now this is why it's important to
19:43
do the syntic division correctly right
19:45
now if you look at this expression here
19:48
this polom this is x 4 xq what should
19:52
come
19:53
next Square but it's not there right so
19:57
what you do is you not you need to
19:58
rewrite this before you actually set it
19:59
up so we're going to rewrite this as
20:01
what 6 x 4 right - 8 X Plus what 0 x^ 2
20:08
+ 12
20:10
x - 14 / x - 2 now we can set up our
20:15
synthetic division right so if I do this
20:17
little square here what's going to go
20:19
here what number goes here two right and
20:22
then here and then and then Z and then
20:26
12 and then all right we draw a line
20:29
right and we go 6 + 0
20:34
is 6 what's 2 * 6 12
20:38
128 + 12 4 4 right 2 * 4 8 0 + 8 8 2 * 8
20:49
16 12 +
20:51
16
20:53
28 2 * 28 54 56 right and then 14 + 56
21:01
42 right so what's our remainder this
21:04
time 42 is our remainder right so to
21:07
write the result so because this is x 4
21:09
what would this be then X so be 6 X 3r
21:14
plus 4 42 +
21:18
8 plus 2 and then now plus 2 over x - 2
21:26
my man is my man is he's lucky
21:29
I like that right you see that it's just
21:32
like that that's it it's really easy the
21:34
Cent division is so easy right it's
21:37
easier than the long division because
21:38
it's faster it takes like two minutes to
21:40
get
21:41
done can we use it every time what can
21:44
we use it depends on the questions if
21:47
you're learning it's easier but it's
21:48
kind of hard it's like a yeah you just
21:50
have to make sure even with a long
21:52
division if you have a number you still
21:54
have to add the 0x S you still have to
21:57
do it right you still have to do it okay
21:59
now what I want to do now is uh let's
22:02
see
22:04
um I think I want to do this tomorrow
22:07
there's another one here that has
22:09
like your brain is dead yeah okay all
22:13
right tomorrow yeah we can do that
22:16
tomorrow