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Using long or synthetic division to solve complex indefinite integrals ( AP Calculus)
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Feb 19, 2025
In our Friday morning AP Calculus class, we learned how to use both the long and synthetic division as well as the substitution method to solve indefinite integrals.
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all right so basically so anytime we set
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that up so again we are trying to solve
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this inte by using long division right
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so we set it up so make sure because we
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don't have cuz after X four the next one
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naturally should be X cubed right and
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after X Cub should be x² because we
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don't have any coefficients we have to
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put 0 x Cub + 0 X2 + x - 4 and now we're
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going to go how many times does
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x² goes into X 2 so x² all right and
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then we perform a multiplication x² what
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happens the plus two though no I'm going
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to do that you don't you don't worry
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about the digit here just the the the
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you you put x s in there yeah so x² * X2
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is x 4 right and then X2 * 2 is what 2x2
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all
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right and then now you subtract it but
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you times it you times it you distribute
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this x² * x² is X 4 okay and then X2 * 2
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is
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2x2 right and then you you you subtract
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it so x 4 - x 4 is 0 right and then 0 -
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2x2 is -2X 2 and we can bring down the X
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all right bring that out it's just like
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any other Division and we do the same
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thing wait what did you bring down where
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or the X this you're going to bring down
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4 too so at some point so what about
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what about what about like 0 x² it's
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zero you don't care no we don't worry
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about it we just put it that for the
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sake of keeping theity but we're not
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going to bring that down it's zero is
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just zero no both Z you don't you don't
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care about you don't care about that
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because they have no effect on this
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right so x 4 - x 4 is 0 and then 0 x -
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2x is
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2 even X down right and I'll proceed the
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same way so how many times does x² goes
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into
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2x2 one time it would be -2 right if you
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divide this into that you get -2 left
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okay right you get it yeah so it's -2 so
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we going to go -2 * X2 is -2X 2 -2 * 2
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is4 right you do the same thing oh
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you're timing it okay I times it right
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wa wait explain that last one more time
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I'm going to go -2 *
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X2 -2X 2 and you got two from from
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dividing X2 into -2X
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2 why didn't you also divide by the X
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you don't do it like that you always the
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first one and the first one here all
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right so you always take the first one
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and the first one here okay and the same
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thing so nice -2X - 4 right and now what
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happens is this this cancels out right
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and then you get now what do you have
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you have
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X this is a negative two negatives gives
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you what
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posi so it's x + 4 you bring this next
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down one now so you're left with what
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you're left with X right those oh those
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cancel out those cancel out you left X
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get plus isn't x - 4 it's negative
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and yeah but what
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Happ right
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here just bring it down right and then
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the4 so that's your remainder it's x- 2
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x wait hold up I thought you already
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brought the4 down when you did -2X - 4 I
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didn't bring
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down answer
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be so now now we have x 4 + x - 4 X2 + 2
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should give us now what x 2 - 2 + x over
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x² + Mr can you go over one that has all
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the degree say it again can you go over
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one that has like all the degree not
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like Zer yeah yeah yeah we going to do
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one like that yeah I'm just using the
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one in the package first so now we
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rewrite this whole expression right so
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now we have that X3 4 + x - 4 over X2 +
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2 DX is = to X2 - 2 + uh
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DX + X2 + 2 DX you guys know what to do
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here what's the of this wait how did you
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get X over X2 + 2 that's your remainder
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your remainder is X and this is your
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dividend you always go x x + 2 okay
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right so now be x 3 - 2x and then here
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how do we solve is what do we use here
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use substitution substitution so let U
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equals what
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X plus 2 right oh how do you get how do
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you get 2x oh it's 2x right uhhuh and
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what's the DU uh
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2x DX all right and then what's X
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DX 12 12 du right I'm really concerned
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for Josh man should be screed when he
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comes back 12 right so this will be2 and
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this is what du over U CU you yeah you
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let U become that and then X becomes D =
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2x
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DX and then X DX is
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1
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One log log right so that becomes x - 3
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- 2x + 12 natural log of What U + C and
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we replace U by what x² x + 2 right so
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this is X cub 3 - 2x + 12 natural log of
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X2 +
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2 let C all right so we're going to do a
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couple more of these until we get the
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hang of it let me gra there something in
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the book that I want to work on so let's
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go to the book How come for du you put
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2 what for du I put 2x what's the dtive
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of X2 +
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2 oh yeah yeah 2X
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right so it's D so
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let's why is negative where the second
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yeah why is there's negative why is
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there a negative cuz time yeah so x * X
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is x 4 X2 * 2 2 X2 right and then you
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subtract it when you do long you always
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have to subtract it all right so uh if
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you go to your book so we have a couple
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here that when you got okay when when
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you got uh X2 - 2 2 DX okay so -2 just
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turns in just -2X 2x yeah oh when you
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when you X in front of that's going it
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becom yeah okay all right so let's can I
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eras this yeah let's do some let's the
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wait wait wait hold
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on hold
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up so we going to do something in the
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book here real quick and then that's
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that should help
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us can you do some on the board yeah
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yeah I want to try try out my yeah but
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we have to once you subcribe you have to
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turn off the thing you can't be on the
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video it's it's against
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the
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policy all
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right all right let's do bro
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let me do this one here real quick let's
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see yeah I
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saw yeah so let's say we have
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2X + 7 x - 3 over now let me give you a
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little heads up here
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right
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so so again the the degree of the
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numerator is higher so we're going to
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use the we can use the synthetic or we
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can use the long division right
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synthetic so
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when see we going to use a synthetic
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when uh the degree is one that's not
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Square that's no Square so we're going
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to use a synthetic so I want set that up
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so it's going to be two two and then
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what's the first number here and then uh
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two mhm and then zero no wait is this in
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our seven
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three3 right all right we go down here
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right bring down the two first bring the
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two down so you got four two and two is
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four and then you do that was it's minus
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right no you add in this case you add
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add that 11 11 22 22 19 19 so the
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remainder is what 19 right so if I was
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to rewrite this now again I go one power
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down yeah so it would just be 2x plus
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plus 7 11 oh no plus 11 11 of plus 19
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over what over x - 2 there you go that's
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it that's simple right so now we find
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what anti derivative of all of that
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stuff all right right DX does that make
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sense
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okay go overis oh yeah do the
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long so what's thetive of 2x 2 2 DX 2 DX
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sorry oh the
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anti X 2x over 2 right which is x s anti
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of that 11x 11 and then this will
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be there a whole a whole new problem 19
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* natural log of
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x Okay C all
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right that make sense yeah all right so
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now let's say you were trying to do the
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long division on that right we do the
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same thing x -
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2 and 2x2 + 7 x - 3 we're going to get
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the same answer right so how many times
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does X goes into
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2x2 two two times right time by two and
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you go 2 * X so be 2 x no it's 2 x here
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2 2x * X is 2x 2 right 2x *
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-2 4X -4x right and then you subtract
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right 2x - 2 that's cancel now here you
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have two negatives so that should give
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you what 11x and then you bring down and
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that's three right now how many times is
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X go into
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11x uh one time no you go x 11 times
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right 11 so 11 * X
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11 x 11 * -2 what 22 22 right same thing
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subtract 11 x - 11 x yeah get positive
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19 and then just press 19 we have the
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same answer right 2x + 11 + 19 / x - 2
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so you can both use the synthetic wait
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so so you didn't need to do like long
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division right there you were just doing
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it to prove that you got the same you
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got the same but the problem is when
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this power here is like it's like if
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this exponent here is like higher than
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um one then you use the long division
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long division but that's pretty much
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wait so I'll give you some more to work
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on on your own right and then once you
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get it you can just find the interv oh
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the
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um you not the homework not
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do yeah what's the homework is like
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something 10 uh yeah one to 10 can you
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write on the board
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yeah