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

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


In this expression, we’re
dividing this third degree polynomial by this
first degree polynomial. And we could simplify
this by using traditional algebraic
long division. But what we’re going
to cover in this video is a slightly
different technique, and we call it
synthetic division. And synthetic division
is going to seem like a little bit of voodoo
in the context of this video. In the next few
videos we’re going to think about why it actually
makes sense, why you actually get the same result as
traditional algebraic long division. My personal tastes are not
to like synthetic division because it is very,
very, very algorithmic. I prefer to do traditional
algebraic long division. But I think you’ll see that
this has some advantages. It can be faster. And it also uses a lot
less space on your paper. So let’s actually perform
this synthetic division. Let’s actually simplify
this expression. Before we start, there’s
two important things to keep in mind. We’re doing, kind of,
the most basic form of synthetic division. And to do this most basic
algorithm, this most basic process, you have
to look for two things in this
bottom expression. The first is that it has to
be a polynomial of degree 1. So you have just an x here. You don’t have an x
squared, an x to the third, an x to the fourth or
anything like that. The other thing is, is that
the coefficient here is a 1. There are ways to do it if
the coefficient was different, but then our synthetic
division, we’ll have to add a little bit of
bells and whistles to it. So in general, what
I’m going to show you now will work if
you have something of the form x plus or
minus something else. So with that said,
let’s actually perform the synthetic division. So the first thing
I’m going to do is write all the coefficients
for this polynomial that’s in the numerator. So let’s write all of them. So we have a 3. We have a 4, that’s
a positive 4. We have a negative 2. And a negative 1. And you’ll see different people
draw different types of signs here depending on how they’re
doing synthetic division. But this is the
most traditional. And you want to leave
some space right here for another row of numbers. So that’s why I’ve gone
all the way down here. And then we look
at the denominator. And in particular, we’re
going to look whatever x plus or minus is,
right over here. So we’ll look at, right over
here, we have a positive 4. Instead of writing a positive 4,
we write the negative of that. So we write the negative,
which would be negative 4. And now we are
all set up, and we are ready to perform
our synthetic division. And it’s going to
seem like voodoo. In future videos, we’ll
explain why this works. So first, this first
coefficient, we literally just bring it straight down. And so you put the 3 there. Then you multiply what you
have here times the negative 4. So you multiply it
times the negative 4. 3 times negative
4 is negative 12. Then you add the 4
to the negative 12. 4 plus negative
12 is negative 8. Then you multiply negative
8 times the negative 4. I think you see the pattern. Negative 8 times negative
4 is positive 32. Now we add negative
2 plus positive 32. That gives us positive 30. Then you multiply the positive
30 times the negative 4. And that gives you negative 120. And then you add the negative
1 plus the negative 120. And you end up with
a negative 121. Now the last thing
you do is say, well, I have one term here. And in this plain,
vanilla, simple version of synthetic division, we’re
only dealing, actually, when you have x plus
or minus something. So you’re only going
to have one term there. So you separate out one term
from the right, just like that. And we essentially
have our answer, even though it
seems like voodoo. So to simplify this, you get,
and you could have a drum roll right over here,
this right over here, it’s going to be
a constant term. You could think of it
as a degree 0 term. This is going to be an x term. And this is going to
be an x squared term. You can kind of just
build up from here, saying this first one is
going to be a constant. Then this is going to be an
x term, then an x squared. If we had more you’d have
an x to the third, an x to the fourth, so
on and so forth. So this is going to be equal
to 3x squared minus 8x plus 30. And this right over
here you can view as the remainder, so minus
121 over the x plus 4. This didn’t divide perfectly. So over the x plus 4. Another way you could have
done it, you could have said, this is the remainder. So I’m going to have a
negative 121 over x plus 4. And this is going to be plus
30 minus 8x plus 3x squared. So hopefully that
makes some sense. I’ll do another example
in the next video. And then we’ll think about
why this actually works.

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

  • August 1, 2012 at 9:26 pm
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    What sorcery is this?

    Reply
  • August 2, 2012 at 1:15 am
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    aka polynomial division. you just need to understand fractions.

    Reply
  • August 2, 2012 at 2:02 am
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    LOL I can't stop laughing, that's exactly what I thought.

    Reply
  • August 8, 2012 at 1:46 pm
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    I found my son watching this video and when I got to him his brains was scattered all over his desk. Now im on the run from the cops. Dont let your kids watch this by themselves you will be sorry!

    Reply
  • August 18, 2012 at 12:40 am
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    Oh honey… I'm in 12th grade… it'll be back to get you when you least expect it

    Reply
  • August 30, 2012 at 5:17 am
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    Interesting technique. Seemingly synthetic division is traditional division that has been "flipped" around. Everything(even the division bar itself) is turned around. Multiplication is performed by the opposite of the denominator and instead of subtraction, addition is used in the synthetic version and the corresponding variables are assigned to the results, making the calculations faster and not as messy as the traditional method.

    Reply
  • October 24, 2012 at 10:49 pm
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    i dont understand the whole move one to the right

    Reply
  • October 30, 2012 at 8:39 pm
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    does he have a video of the extra bells and whistles required to do this when the coefficient of your divisor isn't 1?

    Reply
  • November 9, 2012 at 9:52 pm
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    Something I'm already better than you at πŸ˜›

    Reply
  • November 9, 2012 at 9:52 pm
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    Yay, cookies are way better than synthetic division.

    Reply
  • November 11, 2012 at 9:43 pm
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    that is what i think when i listen to my teacher

    Reply
  • November 15, 2012 at 11:29 pm
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    this is so much easier than long division with polynomials

    Reply
  • January 7, 2013 at 3:09 am
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    Oh Asians.

    Reply
  • January 8, 2013 at 1:57 am
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    Math? Never heard of it.

    Reply
  • January 25, 2013 at 4:14 am
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    Holy shit, Mind Blown.

    Reply
  • January 25, 2013 at 5:20 pm
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    That's not algebra, that's voodoo magic.

    Reply
  • March 10, 2013 at 7:45 pm
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    Are you a wizard?

    Reply
  • April 15, 2013 at 2:20 pm
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    Thanks for sharing…

    Reply
  • April 15, 2013 at 2:20 pm
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    Hahaa sure is πŸ˜‰

    Reply
  • May 30, 2013 at 1:07 am
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    You must be Asian. And I must be racist…

    Reply
  • September 2, 2013 at 4:54 pm
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    Nice.

    Reply
  • October 14, 2013 at 3:27 am
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    1:58 lmao

    Reply
  • October 17, 2013 at 12:50 am
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    My mind just got fucking blown.

    Reply
  • October 18, 2013 at 12:48 pm
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    thumbs up if your in seventh grade.

    Reply
  • October 24, 2013 at 6:35 am
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    middle schoolers up in here sayin they're doin this, meanwhile I'm in college having to do this DX

    Reply
  • October 28, 2013 at 2:25 am
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    Cool

    Reply
  • January 15, 2014 at 1:43 am
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    11th grader 😳…. I feel dumb

    Reply
  • February 27, 2014 at 1:33 am
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    tooo long

    Reply
  • March 31, 2014 at 8:19 pm
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    Thanks! I'm in 9th and I'm in algebra 2 honors, and my teacher doesn't describe the lesson well this helps a lot! subscribe you make it easyer to understand! Precal and algebra 1 for collage here i come! (Nxt year) ;-; ~tear

    Reply
  • July 22, 2014 at 11:16 pm
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    He makes me laugh sometimes "it may seem like voooodoooo" πŸ˜›

    Reply
  • July 24, 2014 at 12:10 pm
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    I appreciate this guy for making these vids, but I can't stand the way he talks… he comes off as a tool…

    Reply
  • October 20, 2014 at 2:50 am
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    you da man thanks

    Reply
  • November 28, 2014 at 8:11 pm
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    Nvm I understand but what do you do when the x isn't an x. Say its 9x

    Reply
  • January 23, 2015 at 5:12 am
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    Amazing video; thank you very much

    Reply
  • November 25, 2015 at 6:38 pm
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    thank youπŸ‘ŒπŸΌ

    Reply
  • February 24, 2016 at 8:24 pm
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    This is sorcery

    Reply
  • March 8, 2016 at 2:04 am
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    Wasn't paying attention in class today so I missed this lesson in class… Thanks!

    Reply
  • May 8, 2016 at 7:06 pm
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    Dude. You're a freaking legend. Your help is much appreciated. Keep it up!

    Reply
  • May 30, 2016 at 9:53 pm
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    when exams are coming up and u need to be refreshed

    Reply
  • June 20, 2016 at 11:34 am
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    Hey, I don't understand you said there would be square 3 but where is the x coming from?, should I just follow that without any reason? please help.

    Reply
  • August 31, 2016 at 12:17 am
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    VOODOO MAGIC

    Reply
  • September 4, 2016 at 1:46 pm
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    Dude! what method are you using?

    Reply
  • September 14, 2016 at 12:52 am
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    I find this video offensive I'm an active voodoo conjurer.

    Reply
  • September 29, 2016 at 6:47 pm
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    Extremely helpful, thanks a bunch

    Reply
  • October 9, 2016 at 11:06 am
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    do synthetic division with numerical coeficient
    do it if its no voodo magic!!!
    #dooksrR'us

    Reply
  • October 15, 2016 at 3:38 pm
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    10 questions 1. Are you sure you cant do it with coefficient? do you know binary?

    Reply
  • November 30, 2016 at 7:19 pm
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    Thanks for the tutorial, you explained it better than my teacher did. It really is voodoo!

    Reply
  • December 7, 2016 at 1:26 am
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    what happens if your remainder is zero

    Reply
  • December 13, 2016 at 2:41 am
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    today is my exam lol thbx

    Reply
  • December 17, 2016 at 1:26 am
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    I love learning math at home because i can say "Oh shitttttt" and "HELL YEAHHH" whenever i get something.

    Reply
  • January 13, 2017 at 2:10 pm
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    Why do you put the opposite sign. Like if it's -2 why do you put 2? Thanks! πŸ™‚

    Reply
  • February 20, 2017 at 5:45 pm
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    Thank you

    Reply
  • March 29, 2017 at 2:54 am
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    Dude thanks! you helped.

    Reply
  • April 28, 2017 at 2:55 pm
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    I've always heard, "Set (x + 4) = 0" But same concept, same result using the negative of 4 ( -4 )

    Reply
  • August 18, 2017 at 8:23 am
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    Sal Khan is the math god in human form.

    Reply
  • September 22, 2017 at 1:01 am
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    don't give more if one hundred twenty explanation

    Reply
  • September 29, 2017 at 3:36 am
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    You are math jesus

    Reply
  • October 10, 2017 at 10:21 pm
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    I prefer traditional long division but I had a problem that it didn't work for. Please tell me if i am making simple mistakes or synthetic only works for this problem. PROBLEM: (xβ΄βˆ’3xΒ³+x-4)(x+1)

    Reply
  • October 18, 2017 at 2:44 pm
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    pure magic

    Reply
  • October 19, 2017 at 4:19 am
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    for the bottom equation, the one that your dividing by all your doing is finding the zero of it/roots so basically X+4 is X+4=0, that's why it is -4 and if you graph y=x+4 the root aka x-int is -4 same if you divide the polynomial by x^2-1, you find the zeros which are 1 and -1 and use it for the synthetic division(same place the -4 was put in the videos example)

    Reply
  • October 27, 2017 at 12:51 am
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    Question.. why do you change the +4 on the bottom to a negative in dividing?

    Reply
  • November 6, 2017 at 3:54 pm
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    Thank you very much 😊😊😊

    Reply
  • November 8, 2017 at 10:43 am
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    F R I K I N V O O D O O

    Reply
  • November 8, 2017 at 11:34 am
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    IT IS VOOOOOODOOOOOO!!

    Reply
  • November 21, 2017 at 1:51 am
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    Thank you sooo much.It help alot

    Reply
  • January 2, 2018 at 9:02 pm
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    Ah, thanks so much! Kahn Academy is awesome!

    Reply
  • January 15, 2018 at 5:17 am
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    This video is very helpful especially for me because we are just in 8th grade but we are already studying about thism

    Reply
  • January 23, 2018 at 12:47 am
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    You have saved my life!

    Reply
  • February 9, 2018 at 6:50 pm
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    first

    Reply
  • March 2, 2018 at 4:50 am
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    So the last term is always the remainder?

    Reply
  • May 16, 2018 at 6:47 pm
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    i love ur language…cool

    Reply
  • June 16, 2018 at 2:34 pm
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    I hope that this video is more helpful then the other video.

    Reply
  • June 21, 2018 at 10:18 am
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    I am so stress about our test tomorrow because i don't know how to do it.

    Reply
  • July 16, 2018 at 11:53 am
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    why 3x squared? isn't supposed to be 3x cube?

    Reply
  • August 9, 2018 at 5:55 am
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    At the start what if it's already negative do we change it into positive

    Reply
  • August 21, 2018 at 11:25 am
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    Thank you

    Reply
  • August 22, 2018 at 3:38 pm
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    WOOWW BUT THE NEGATIVE IS CONFUSING ME

    Reply
  • September 8, 2018 at 2:44 pm
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    My class is for 40 minutes. I do not understand a thing my teacher is saying but you taught it to me in 6 minutes. Thank you

    Reply
  • October 2, 2018 at 12:44 pm
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    yow! thanks so much it will soon help in my ft tomorrow because i got zero on my 2 ft hopefully it will develop my score

    Reply
  • October 30, 2018 at 5:00 pm
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    taking my exams later and still watching this tho! much needed!!!

    Reply
  • November 13, 2018 at 9:29 am
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    Starts at 1:39

    Reply
  • November 20, 2018 at 2:20 pm
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    Who watched this video bcuz u have Quiz for Calculas tomorrow

    Reply
  • November 30, 2018 at 4:49 pm
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    I just love how I can learn in five minutes what takes a normal teacher 30 minutes to teach thanks

    Reply
  • December 9, 2018 at 6:03 pm
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    thank you very much very helpful!

    Reply
  • December 13, 2018 at 11:44 pm
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    I have an exam tomorrow and finals next week :’) IM DOOMED JVKCK K K

    Reply
  • December 20, 2018 at 5:24 pm
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    But why is the 4 negative?

    Reply
  • January 31, 2019 at 12:00 am
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    FINALLY I UNDERSTAND HOW THIS WORKS!

    Reply
  • February 4, 2019 at 5:14 am
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    Im an 8th grader and we are doing this in Pre Ap Algebra 1 i didnt get it when my teacher explained it so so but this helped a lot thanks

    Reply
  • March 17, 2019 at 2:14 am
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    This has saved me from having the most miserable night of precalc in my life. Many thanks!

    Reply
  • April 10, 2019 at 1:08 pm
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    What happens if the bottom is 2x +8?

    Reply
  • June 5, 2019 at 3:38 pm
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    HAVING THE BOARD BLACK AND THE WRITING SO MINUTE MAKES IT A BIT AWKWARD…DON'T YOU THINK SO!???

    Reply
  • July 8, 2019 at 1:02 pm
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    Really nice….sir

    Reply
  • July 28, 2019 at 8:35 pm
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    I have one question sal at 2:28 is there a reason why u have to take the negative?

    Reply
  • September 28, 2019 at 2:24 pm
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    Buuuut how about dividing with polynomials of degree greater than 1?
    Every vid only shows for deg (g)= 1….:|

    Reply
  • October 18, 2019 at 1:08 am
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    -1 -(-120) = -1 + 120 = -119

    Reply
  • October 21, 2019 at 1:39 am
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    lol people need to stop bragging about what age they're learning this at

    Reply
  • November 5, 2019 at 8:59 pm
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    Thank u so much….i have an exam today..

    Reply
  • November 13, 2019 at 1:42 am
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    ooooooooooooooooooooooooooooooooooooooooooooh lit

    Reply
  • January 27, 2020 at 7:36 pm
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    All my math teachers got confused when I used synthetic division. I just felt the "traditional" way took up way too much space.

    Reply

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