Polynomial division | Polynomial and rational functions | Algebra II | Khan Academy

Polynomial division | Polynomial and rational functions | Algebra II | Khan Academy


In this video, we’re going to
learn to divide polynomials, and sometimes this is called
algebraic long division. But you’ll see what
I’m talking about when we do a few examples. Let’s say I just want to
divide 2x plus 4 and divide it by 2. We’re not really changing
the value. We’re just changing how we’re
going to express the value. So we already know how
to simplify this. We’ve done this in the past. We
could divide the numerator and the denominator by
2, and then this would be equal to what? This would be equal to x plus
2– let me write it this way– it would be equal to, if
you divide this by 2, it becomes an x. You divide the 4 by
2, it becomes a 2. If you divide the 2
by 2, you get a 1. So this is equal to x plus
2, which is pretty straightforward, I think. The other way is you could have
factored a 2 out of here, and then those would
have canceled out. But I’ll also show you how to
do it using algebraic long division, which is a bit of
overkill for this problem. But I just want to show you that
it’s not fundamentally anything new. It’s just a different way of
doing things, but it’s useful for more complicated problems. So you could have also written
this as 2 goes into 2x plus 4 how many times? And you would perform this
the same way you would do traditional long division. You’d say 2– you always start
with the highest degree term. 2 goes into the highest
degree term. You would ignore the 4. 2 goes into 2x how many times? Well, it goes into 2x
x times and you put the x in the x place. x times 2 is 2x. And just like traditional long
division, you now subtract. So 2x plus 4 minus 2x is what? It’s 4, right? And then 2 goes into
4 how many times? It goes into it two times,
a positive two times. Put that in the constants
place. 2 times 2 is 4. You subtract, remainder 0. So this might seem overkill
for what was probably a problem that you already
knew how to do and do it in a few steps. We’re now going to see
that this is a very generalizable process. You can do this really for any
degree polynomial dividing into any other degree
polynomial. Let me show you what
I’m talking about. So let’s say we wanted to divide
x plus 1 into x squared plus 3x plus 6. So what do we do here? So you look at the highest
degree term here, which is an x, and you look at the highest
degree term here, which is an x squared. So you can ignore
everything else. And that really simplifies
the process. You say x goes into x squared
how many times? Well, x squared divided by x is
just x, right? x goes into x squared x times. You put it in the x place. This is the x place right
here or the x to the first power place. So x times x plus 1 is what? x times x is x squared. x times 1 is x, so it’s
x squared plus x. And just like we did over
here, we now subtract. And what do we get? x squared plus 3x plus 6 minus
x squared– let me be very careful– this is minus
x squared plus x. I want to make sure that
negative sign only– it applies to this whole thing. So x squared minus x squared,
those cancel out. 3x, this is going
to be a minus x. Let me put that sign there. So this is minus x squared minus
x, just to be clear. We’re subtracting
the whole thing. 3x minus x is 2x. And then you bring down the 6,
or 6 minus 0 is nothing. So 2x plus 6. Now, you look at the highest
degree term, an x and a 2x. How many times does
x go into 2x? It goes into it two times. 2 times x is 2x. 2 times 1 is 2. So we get 2 times x plus
1 is 2x plus 2. But we’re going to want to
subtract this from this up here, so we’re going
to subtract it. Instead of writing 2x plus 2,
we could just write negative 2x minus 2 and then add them. These guys cancel out. 6 minus 2 is 4. And how many times
does x go into 4? We could just say that’s zero
times, or we could say that 4 is the remainder. So if we wanted to rewrite x
squared plus 3x plus 6 over x plus 1– notice, this is the
same thing as x squared plus 3x plus 6 divided by x plus 1,
this thing divided by this, we can now say that this is
equal to x plus 2. it is equal to x plus 2 plus
the remainder divided by x plus 1 plus 4 over x plus 1. This right here and this right
here are equivalent. And if you wanted to check that,
if you wanted to go from this back to that, what you
could do is multiply this by x plus 1 over x plus 1
and it add the two. So this is the same
thing as x plus 2. And I’m just going to multiply
that times x plus 1 over x plus 1. That’s just multiplying
it by 1. And then to that, add
4 over x plus 1. I did that so I have the same
common denominator. And when you perform this
addition right here, when you multiply these two binomials
and then add the 4 up here, you should you get x squared
plus 3x plus 6. Let’s do another one of these. They’re kind of fun. So let’s say that we have– we
want to simplify x squared plus 5x plus 4 over x plus 4. So once again, we can do our
algebraic long division. We can divide x plus 4 into
x squared plus 5x plus 4. And once again, same
exact process. Look at the highest degree terms
in both of them. x goes into x squared how many times? It goes into it x times. Put it in the x place. This is our x place
right here. X times x is x squared. x times 4 is 4x. And then, of course, we’re
going to want to subtract these from there. So let me just put a negative
sign there. And then these cancel out. 5x minus 4x is x. 4 minus 0 is plus 4. x plus 4, and then you could
even see this coming. You could say x plus 4 goes into
x plus 4 obviously one time, or if you were not looking
at the constant terms, you would completely just
say, well, x goes into x how many times? Well, one time. Plus 1. 1 times x is x. 1 times 4 is 4. We’re going to subtract them
from up here, so it cancels out, so we have no remainder. So this right here simplifies
to– this is equal to x plus 1. And there’s other ways you
could have done this. We could have tried to factor
this numerator. x squared plus 5x plus
4 over x plus 4. This is the same
thing as what? We could have factored this
numerator as x plus 4 times x plus 1. 4 times 1 is 4. 4 plus 1 is 5, all of
that over x plus 4. That cancels out and you’re
left just with x plus 1. Either way would have worked,
but the algebraic long division will always work, even
if you can’t cancel out factors like that, even if
you did have a remainder. In this situation, you didn’t. So this was equal to x plus 1. Let’s do another one of these
just to make sure that you really– because this is
actually a very, very useful skill to have in your toolkit. So let’s say we have
x squared– let me just change it up. Let’s say we had 2x squared–
I could really make these numbers up on the fly. 2x squared minus 20x plus 12
divided by– actually, let’s make it really interesting, just
to show you that it’ll always work. I want to go above quadratic. So let’s say we have 3x to the
third minus 2x squared plus 7x minus 4, and we want to divide
that by x squared plus 1. I just made this up. But we can just do the algebraic
long division to figure out what this is going
to be or what this simplified will be. x squared plus 1 divided into
this thing up here, 3x to the third minus 2x squared
plus 7x minus 4. Once again, look at the
highest degree term. x squared goes into 3x to the
third how many times? Well, it’s going to go
into it 3x times. You multiply 3x times this,
you get 3x to the third. So it’s going to go
into it 3x times. So you have to write the 3x over
here in the x terms. So it’s going to go into it 3x
times, just like that. Now let’s multiply. 3x times x squared is 3x
to the third, right? 3x times squared plus
3x times 1. So we have a 3x over here. I’m making sure to put
it in the x place. And we’re going to want
to subtract them. And what do we have? What do we have when
we do that? These cancel out. We have a minus 2x squared. And then 7x minus– because I
just subtracted 0 from there– 7x minus 3x is plus 4x,
and we have a minus 4. Once again, look at the
highest degree term. x squared and a negative
2x squared. So x squared goes
into negative 2x squared negative 2 times. Negative 2, put it in
the constants place. Negative 2 times x squared
is negative 2x squared. Negative 2 times 1
is negative 2. Now, we’re going to want to
subtract these from there, so let’s multiply them
by negative 1, or those become a positive. These two guys cancel out. 4x minus 0 is– let me switch
colors– 4x minus 0 is 4x. Negative 4 minus negative 2 or
negative 4 plus 2 is equal to negative 2. And then x squared, now it has
a higher degree than 4x, and the highest degree here, so we
view this as the remainder. So this expression we could
rewrite it as being equal to 3x minus 2– that’s the 3x minus
2– plus our remainder 4x minus 2, all of that
over x squared plus 1. Hopefully, you found that
as fun as I did.

100 thoughts on “Polynomial division | Polynomial and rational functions | Algebra II | Khan Academy

  • July 25, 2017 at 5:20 pm
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    It is excellent 🙋 and very understandable
    😄

    Reply
  • August 18, 2017 at 8:14 am
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    Can someone explain why, in the check for the first one, he multiplies (x+2) by (x+1)/(x+1)+4/(x+1)? Why did he add the x+1/x+1? Why not just (x+2)(x+1)+4/x+1??

    Reply
  • September 15, 2017 at 12:44 pm
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    I recommend trying to solve the equations yourself after the first one if you can't get it right check what you forgot to do by continuing to watch the video. It helped me and I got the method in the first watch! 😁

    Reply
  • September 18, 2017 at 2:22 am
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    what if the coefficient in the denominator is greater

    Reply
  • September 26, 2017 at 3:31 pm
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    Thanks Mr.Khan!

    Reply
  • September 27, 2017 at 10:02 pm
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    I dont understand why we can ignore the other terms and only focus on the highest degree term

    Reply
  • October 5, 2017 at 12:08 pm
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    This was in my algebra 1 book. Is it algebra 1 or 2?

    Reply
  • October 8, 2017 at 10:26 am
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    It give me extra knowledge and more confident to face any long division problem. If you don't believe me, ask me any questions from Division Algorithm for polynomial. I'm ready

    Reply
  • October 9, 2017 at 11:46 am
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    Do we even need to learn algebra?😣😣😭😭😭😭😦

    Reply
  • October 11, 2017 at 7:42 pm
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    Great video!

    Reply
  • October 19, 2017 at 6:47 pm
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    Hi ! I'm looking for the software used to make this video if someone know the name

    Reply
  • October 19, 2017 at 10:07 pm
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    2 goes into 2x x times…. what?

    Reply
  • October 24, 2017 at 2:15 pm
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    Still don't get it. This way is harder than the way my professor teaches it D:

    Reply
  • November 8, 2017 at 10:35 am
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    there's a shorter way of doing it, why even bother learning the longer AND more annoying way?

    i'm still questioning when we use this irl

    Reply
  • November 10, 2017 at 11:29 pm
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    I have 12n2+28n+15 divided by 12 help please

    Reply
  • November 21, 2017 at 9:59 am
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    That was fun to you? It was painful 😖

    Reply
  • December 21, 2017 at 1:32 am
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    Ight lets see how this quiz goes tomorrow….*gets a 60

    Reply
  • January 5, 2018 at 8:51 am
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    so if you divide x squared by x, you don't carry over the exponent? same with 2x divided by x? no more variable?

    Reply
  • January 22, 2018 at 11:41 am
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    Seriously you should replace my Math teacher.

    Reply
  • January 23, 2018 at 2:35 am
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    2×2=4-1=3 wiiik mafs

    Reply
  • January 27, 2018 at 3:50 pm
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    X+5/-26=4? pls simplify..

    Reply
  • February 4, 2018 at 8:43 pm
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    Just curious… Where would we see this or use this on real life?

    Reply
  • February 6, 2018 at 10:16 am
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    Pray tell- how does 2 go into 2x "X times"??
    That seems to be a remainder rather than the factorization.

    Reply
  • February 7, 2018 at 6:57 am
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    they are kinda fun
    -Khan academy 2010

    Reply
  • February 9, 2018 at 6:36 pm
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    First

    Reply
  • February 11, 2018 at 3:07 pm
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    Wow you are pretty much better than any teacher!!

    Reply
  • February 13, 2018 at 11:49 pm
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    I don’t know how to do this ahaha FAKE LAUGHS then actually cries 😭😭😭😭

    Reply
  • March 5, 2018 at 11:35 pm
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    Thank you so much 🙂

    Reply
  • March 13, 2018 at 4:23 am
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    Thank you SO MUCH for explaining this is a clear and concise manner.

    Reply
  • March 21, 2018 at 12:18 am
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    Fun…..

    Reply
  • April 10, 2018 at 1:53 pm
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    I’m never ever going to study Math. No matter how hard I try, I’m still super clueless! I guess me and Math were never supposed to be fit for each other xD

    Reply
  • April 15, 2018 at 3:17 pm
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    i didnt understand! have a math test tommorwwwwwwww! cries

    Reply
  • April 19, 2018 at 2:10 am
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    God bless you

    Reply
  • June 2, 2018 at 8:51 am
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    Thank you very much Khan Academy! My Math exam is on Wednesday and this Video has really helped me out A LOT! ThAnks So mUCH!

    Reply
  • June 4, 2018 at 4:12 pm
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    dang u got good mouse skills😍

    Reply
  • June 15, 2018 at 10:41 pm
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    I hate math. The only reason why I am doing this is because my education is important to me.

    Reply
  • June 15, 2018 at 10:56 pm
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    Does it worry anyone to have to perform this on a test. It does me.

    Reply
  • June 19, 2018 at 1:06 pm
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    thank goodness for videos like this. I forgot about this topic and this is my go-to channel from now on.

    Reply
  • June 25, 2018 at 11:56 am
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    x10 y5 divide x4 y2=???

    Reply
  • June 26, 2018 at 2:04 am
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    In the last week example, when you divide the higher degree polynomial by the quadratic equation, why, when you multiply the 1 in, (x+1), with the 2 in, (3x-2), it the two a positive and not negative as illustrated.

    Reply
  • June 28, 2018 at 9:29 am
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    Guys right a variable cant be in a denominator? Why is it valid in this equation?

    Reply
  • July 7, 2018 at 2:01 pm
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    Some people would get really confused, please say that every subtracting time,the must subtracted numbers' signs need to be changed with it's opposite signs, same like you're subtracting a polynomial, the must subtracted numbers' sign must be changed to it's opposite sign.

    Reply
  • July 10, 2018 at 5:50 am
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    THANK YOUUUUUU

    Reply
  • July 18, 2018 at 7:37 am
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    Hi Mr. Khan! I just want to tell you that I got a 100% for my test. I was missing out in class by a lot and I watched your videos and got perfect for my second quiz of the year. I got a 10% on the first test, but after watching your videos I got a perfect score on the second one! THANK YOU SO MUCH… YOU'RE THE BEST!!!!!!!!!!!!!!!!!

    Reply
  • July 20, 2018 at 5:41 pm
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    Isn't this the guy who explained what the Federal funds rate is?

    Reply
  • July 22, 2018 at 6:51 am
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    thank you

    Reply
  • July 22, 2018 at 6:56 am
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    12 minute well spent thanks.

    Reply
  • July 22, 2018 at 10:26 am
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    Thank youuu. 💜

    Reply
  • August 9, 2018 at 11:03 am
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    ahhh… the first khan academy video i have ever watched

    Reply
  • August 13, 2018 at 7:25 am
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    :(( I still dont get the long division thing
    Geez I guess Im gonna fail my quiz again

    Reply
  • August 23, 2018 at 10:46 pm
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    Your 12 min video explains this more clearer than my teacher talking about it for 30 min

    Reply
  • August 29, 2018 at 4:25 am
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    This is very simple, but the way you write it confuses me.

    Reply
  • September 18, 2018 at 6:19 pm
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    What if the 3rd example was divided the opposite way…

    Reply
  • September 20, 2018 at 12:24 am
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    for a student who has a teacher who seemelesly breezes through the material, this is a life-saver .
    Thank you!

    Reply
  • October 7, 2018 at 11:45 am
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    This is hurting my brain

    Reply
  • October 11, 2018 at 2:30 pm
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    This method worked very well!

    Reply
  • October 11, 2018 at 6:32 pm
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    Mind-blown. It really helped that you showed it from the very beginning in simpler terms and then show what you found after by putting it back into the original, especially cos i was commmppllleetteelllyyy unaware of that division method until today…i just put the remainders next to the digit (which i tried to apply to this and failed miserably) xD
    I spent a good portion of today in confusion and mild despair at the first bloody chapter!, feel a bit better now that i've understood this v_v'

    Reply
  • October 14, 2018 at 10:27 am
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    Better than teachers though teachers are lazy

    Reply
  • October 16, 2018 at 1:51 pm
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    You are a miracle

    Reply
  • October 30, 2018 at 3:44 pm
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    This guy makes math way easier.

    Reply
  • November 6, 2018 at 12:48 pm
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    Wow i learned more in this video than in 1 Year of math class

    Reply
  • November 8, 2018 at 8:20 pm
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    Wow. Fun.

    Reply
  • November 21, 2018 at 5:32 am
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    “How many times does 2 go into 2x?” X!! That’s right… wat.

    Reply
  • December 12, 2018 at 12:56 am
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    "lets do another one of these. They're kind of fun!" ~rewinds video to make sure I heard that right~

    Reply
  • December 16, 2018 at 11:10 am
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    Has a divide of cancel terms.

    Reply
  • January 7, 2019 at 3:26 pm
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    They did surgery on a grape

    Reply
  • January 8, 2019 at 10:33 am
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    "hopefully you found that as fun as I did"

    Reply
  • January 14, 2019 at 3:51 am
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    still don't get it

    Reply
  • January 20, 2019 at 4:24 pm
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    Great help thanks a lot

    Reply
  • January 20, 2019 at 7:18 pm
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    OMG I watched this video many times before I understood. THANKS SOOO MUCH

    Reply
  • January 23, 2019 at 6:19 pm
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    It has been 15 years since I've done algebraic long division and this video helped me remember it. Thanks!

    Reply
  • January 24, 2019 at 2:09 pm
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    😱😵

    Reply
  • January 29, 2019 at 1:52 am
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    I GET IT NOW THANKS SAL!!!

    Reply
  • February 2, 2019 at 4:16 pm
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    Very good khan academy you saved my final exams

    Reply
  • February 4, 2019 at 4:21 am
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    This really helped me in my homework because i was having trouble with it and my teacher explained it weirdly

    Reply
  • February 7, 2019 at 1:55 pm
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    But what if you don't have to use long division?

    Reply
  • February 12, 2019 at 1:47 am
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    My schoool just became a 1 to 1 school and now we use this alll the time

    Reply
  • February 17, 2019 at 6:45 pm
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    God bless Khan Academy.

    Reply
  • March 4, 2019 at 2:08 am
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    This made no sense

    Reply
  • March 19, 2019 at 4:11 am
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    His writing is so nice oml mine is just a mess

    Reply
  • April 9, 2019 at 12:04 pm
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    is this also known as division of algebraic expressions?

    Reply
  • April 9, 2019 at 12:04 pm
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    is this also known as division of algebraic expressions?

    Reply
  • May 21, 2019 at 9:36 pm
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    This won't help me pass the NJSLA

    Reply
  • May 26, 2019 at 10:12 pm
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    thanks sam, it was very helpful!!

    Reply
  • June 10, 2019 at 2:54 pm
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    The divide subtract multiply version is a lot easier than trying to find factors even when factors exist.

    Just gets messy.

    Reply
  • June 15, 2019 at 1:21 pm
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    Let's do another one of these

    It's kind of fun

    Reply
  • June 18, 2019 at 7:34 pm
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    I don't understand this at all

    Reply
  • July 12, 2019 at 5:15 pm
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    3:20 thank you for clarifying the minus thing, it made all the difference!

    Reply
  • September 5, 2019 at 5:52 pm
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    at 2.42 you look at the highest degree term, and its the "x": how do you know its the "x"? The value of the "x" could be less than 1, no ???

    Reply
  • September 7, 2019 at 4:54 am
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    What If you’re doing division with non common variables like (u+du)/(v+dv)

    Reply
  • September 24, 2019 at 5:52 pm
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    When Sal Khan’s video can’t help you, no one can🥺😢😭.

    Reply
  • September 30, 2019 at 7:28 pm
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    My professor teaches things a different way rather than the vids I watch on YouTube. Waahhh😭

    Reply
  • October 6, 2019 at 1:03 pm
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    Took me 30 min to try and work this out with worked examples, no use. This video has been extremely helpful, thank you.

    Reply
  • October 9, 2019 at 11:00 pm
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    i would say the explaining is too fast. couldnt keep up even just writing it down. gon play it at 0.5 speed now lol

    Reply
  • October 11, 2019 at 3:37 pm
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    Why did you skip the -2xsquared and place the -3x underneath the +7x, in the last problem? I was taught differently.

    Reply
  • October 16, 2019 at 12:21 am
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    Really helps

    Reply
  • October 17, 2019 at 12:37 am
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    You have changed my life

    Reply
  • November 14, 2019 at 3:55 am
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    The second problem was looking like the Irish flag.

    Reply
  • November 20, 2019 at 1:31 pm
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    They Said That Is 6 minus 0 is nothing the rational functions still making into a long division for algebra

    Reply

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