Synthetic Division… How? (NancyPi)

Published on September 19, 2018 by

MIT grad shows how to do synthetic division, a shortcut for long division. It's a fast way of dividing polynomials, if you're dividing one polynomial by a linear expression like x+1 or x-3. To skip ahead: 1) For how to know WHICH NUMBERS to put in the corner of the division box and the first row, skip to time 0:22. 2) For the synthetic division STEPS of carrying down coefficients, multiplying, and adding, skip to 1:12. 3) For how to WRITE the FINAL ANSWER and how to WRITE the REMAINDER (if the last number you get is not zero, there is a remainder), skip to 3:37. 4) For what to do if you have a MISSING TERM and putting a zero as its placeholder, skip to 5:04. Nancy formerly of MathBFF explains the steps.

For my video on how to do LONG DIVISION of polynomials instead, jump to:
For my video on FACTORING to solve for the zeros of a polynomial, jump to:

Follow Nancy on Instagram:

SYNTHETIC DIVISION is often faster than long division for dividing polynomials, if the polynomial you're dividing by is "x plus a number" or "x minus a number". It can be used to factor a polynomial and to find the zeros or roots of a polynomial.

HOW TO DO synthetic division of polynomials:
First draw a little corner symbol. Inside the corner, write the number that makes your denominator equal to zero. If your denominator is "x+1", you would write -1 in the corner. Then write the coefficients of your top polynomial in a row, to the right of the corner symbol. Leave space for another row and draw a horizontal line beneath that. Here are the steps to repeat for synthetic division, to carry down coefficient numbers, multiply, and add:

1) Drop down the first coefficient number.
2) Multiply this number by the corner constant.
3) Write the product in the second column, second row spot.
4) Add the two numbers in the second column, and write the sum below the line in the second column.
5) Repeat those steps until you have written a number in the final column spot below the line.
6) Write your new polynomial answer using this bottom row of numbers. The first number is the leading coefficient of your polynomial and is the coefficient of an x-term that is one degree less than your original polynomial. The second number is the coefficient of the second term, etc. For example, if the numbers are 1 4 3 2, the new polynomial is 1x^2 + 4x + 3 + 2/(x+1). Notice the last number is part of the remainder, so you write that number, 2, over the original divisor (x+1) as a fraction, for the remainder term.

Sometimes there is no remainder in synthetic division examples. If the last number you get is zero, in your final division numbers, then there is no remainder. Your final answer will just have a polynomial and no fraction term added at the end. Synthetic division is a way of factoring polynomials: if there is no remainder when you finish, you have found a true factor of the original polynomial. In the example above, if the remainder had been zero instead, the original polynomial would factor into (x+1)(x^2 + 4x + 3).

Note: also watch out for a missing term in your original polynomial: if there is no x-squared term, or no x term, etc, then in order to get the right answer for synthetic division, you will need to make sure to put a zero number in your first row of coefficient numbers, as a placeholder for that missing term.

For more rational function and rational expression math examples in my algebra, algebra 2, precalculus, and college algebra mathematics videos, and for more practice problems, check out:

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  • amty 4 years ago

    Nancy for cpt marvel

    • D3lete 4 years ago


  • NMEofdaST8 4 years ago

    Wow nice throwback I should think about taking clac 2. Great video as always

  • Ishrat Naseer 4 years ago

    You’re so cool😄

  • Wran Ther 4 years ago

    Short and sweet synthetic division for late evening learning. Hmmm up next could be real sleep. Thank you for a wonderful synthetic night cap Nancy!

  • Sam T. 4 years ago

    gah, I used to always forget how to do this. 😅 My teachers would say this was easier than the long way, but both seemed pretty tricky. The missing term thing did usually trip me up lol. Thanks for the refresher! I really like how you draw it out and explain it so it’s easy to understand. 💛

    • Leo Volont 4 years ago

      Hi Sam, Yeah, in regards to either Synthetic Polynomial division or Long Polynomial Division, the Synthetic Division is easier, but only if you remember how to do it. The Long Division is almost just like regular division, it just gets clunky because as you multiply through, you have to subtract, and to subtract, you have to change those signs, and so your work gets messy.

    • TXU bby 3 years ago

      Good luck 👍 with school

  • Abel Lopez 4 years ago

    Thanks again!

  • Zurviver _ 4 years ago

    thank god u reuploaded this!!!

  • Chris Kennedy 4 years ago

    She can’t see us though – right?

  • Nomon Munir 4 years ago

    rarely used in my opinion

  • rav ernot 4 years ago

    How does this work/when is it practical? Thanks for the refresher.

  • OrphanPaper 4 years ago

    yes it is ,,, friction or segmented reinforced wave function is all ways a bit of a blast

  • Mücahit Yeksoy 4 years ago

    Hard ask education please

  • AZTECA REALM 4 years ago

    Just letting you know I married you on my first year of school so we are married ok

  • Lizard-Truth 4 years ago

    Reminds me of horners method…might be a nice video topic 🙂 Still pleasantly surprised to see your activity!

  • Chris Preston 4 years ago

    Great tutorial, any chance you would do one covering Fourier series please

  • Antonio Mele 4 years ago

    Thank you Nancy!

  • Joey Nel Bongabong 4 years ago

    Hi nancy 🙂

  • Nicole L 4 years ago

    Hey Nancy, do you think in addition to this you could show how to use synthetic when the number is going to be a fraction? For example when the denominator is 3x-2? You’re videos are extremely helpful for my students! Thank you so much!

  • Math and Logic 4 years ago

    Only part of your videos I don’t like is the end where you “sympathize” with people not liking math.

  • Mr Ch33seHed 4 years ago

    Like the video and Nancy. Dislike synthetic div 😖

  • Mark 4 years ago

    you are good teacher thank u

  • first name 4 years ago

    Synthetic — so its not real division ?

    • Leo Volont 4 years ago

      Yes, Synthetic Polynomial Division is different, and easier than Polynomial Long Division. On Long Division, instead of just using the constant, you use the entire binomial denominator, and instead of Adding throughout the steps, you have to subtract, and you have to be careful to keep your X columns aligned by their powers… a lot of stuff to pay attention to and far easier to screw up, but when you are done, you have your polynomial quotient sitting pretty as a picture on top. So, yeah, it is good to know both, because Long Division is alot like Elementary School Division, and Synthetic division you need to remember those basic Nancy Rules which you will tend to forget. so, yeah, Synthetic is Easier if you can remember how to do it.

    • Дарья Волкова 3 years ago

      It’s a real division:

  • LIFE GETS US 4 years ago

    can you do a video explaining IVT and EVT???

  • Mohsin Jawaid 4 years ago

    Have you written any book on maths?

  • Pieces Hsk 4 years ago

    Please don’t vanish again from YouTube i really missed you.. and your previous channel videos really helped me in my exams and become a reason that I scored 97/100 in my finals..

  • MaxineAndrew 4 years ago

    Wow, thanks!

  • pin pon 4 years ago

    She’s back?!!😃😃

  • Christopher Bulger 4 years ago

    Can you please make a video on rational integrals that have irreducible denominators using partial fractions?! Thanks Nancy!!

  • HARSHIT SINGH 4 years ago

    Please make a video on conic section

  • le kiddo 4 years ago

    please make video for First Order Differentrial Equation. There are 5 types which are; seperable variable, linear, exact equation, homogenous and Bernoulli equation. I hope you read my comment & Im enjoy your video bcs I understand. Love from Malaysia ❤

  • Cody Brown 4 years ago

    Lord have mercy, I love math.

  • Elton Lucien 4 years ago

    This video was made right on time.

  • THABISO NKOSI 4 years ago

    hi Nancy thanks a lot for helping me in the past semester may you please again help me on Laplace and inverse transformations, unit-step functions and Ordinary Differential equations thank you.

  • The Cammy Show 4 years ago

    This video is gold thank you 😍😍

  • Saad Alshemrani 4 years ago


  • Ahmed Mzhr 4 years ago

    Wow that ain’t tought at my school
    Thanks nancy u helped me ❤️❤️

  • Leo Rael 4 years ago

    omg how is it that you post exactly what I need exactly when I need it???? Nancy your amazing

  • Yifu Li 4 years ago

    helped a lot, thank you!!! please keep doing videos like this.

  • Chaitanya Varu 4 years ago

    can you please tell how to find factors of any higher order polynomial using synthetic devision

    • Voklo 4 years ago

      We doing that rn

    • Chaitanya Varu 4 years ago

      @Voklo here its divided by another polynomial,I what if its just a polynomial
      example) x^2 + 3x +2=(x+2)(x+1)

  • Dr Crawdaddy 4 years ago

    Can you do stuff on 3d vectors? Parametric equations and what not?

  • Hersche9 4 years ago

    Wow that was a very quick, but effective method and taught in a precise way. Amazing!!

  • Cambo Da Producer 4 years ago

    Didnt you have a channel called Math BFFS or something?

  • T Player 4 years ago

    you are marvelous. You took your time in explaining every step. I’ve watched the videos with other people and they speed through everything without fully explaining why they’re doing what they’re doing. Keep up the great work beautiful.

  • furqan jamal 4 years ago

    Thank you ♡

  • Fred Davor 4 years ago

    Lovely and nice way to teach and explain. very helpful

  • Alex Ruff 4 years ago

    Thank You!

  • Dektoonics Inc. 4 years ago

    For the placeholder thing,why would it be there if you put it into standard form

  • Mck Harris 4 years ago

    Everyone is saying this is calculus……i’m doing this in algebra 2….plz save me i’m dying

    • Egg 3 years ago

      @ButterPlayz_YT Yeah, my algebra skills are pretty strong. I actually can’t wait for calculus next semester, lol.

    • omar alkhemeiri 3 years ago

      Aaliyah Feacher i think this is like pre calculus or something like that

    • Supreme 3 years ago

      Boi plz this joint algebra 2

      Ik cause i got test tomorrow 😰😰😰

    • BigBaller 3 years ago

      I’m 14 and I’m doing this

    • Yvonne Karsten 3 years ago

      Mck Harris I’m doing this is algebra 1 lmao

  • Kayla Lillian 4 years ago

    Imagine having her as a teacher oh my god ❣️😊😊 explains it so well

    • Lion Alesso 3 years ago

      I’ve been always lucky. Usually, when during chit-chat others don’t pay attention and the next day they blame the teacher. I observed it not only once.

    • Kayla Lillian 3 years ago

      Lion Alesso oh. I’m the quiet kid

  • Julia Geib 4 years ago

    She’s better than my math teacher

  • Dennis Sanchez 4 years ago

    You’re so great, thank you so much!

  • Ellen Vankreift 4 years ago

    Thank you

  • Turki 4 years ago

    Thank you so much you Nancy are super savior

  • Ali Fuat Dirier 4 years ago

    I want to learn how can write board.

  • Reham Ali 4 years ago

    Tysm 😘

  • As ki vines 4 years ago

    plz tell me that if there is remainder remaining greater than zero and equation for power 5 so how will we solve further please tell me

    • As ki vines 4 years ago


  • Erica X H 4 years ago

    is she writing everything backwards

  • AfterGlowMusic 4 years ago

    what would you do if, per say, your problem was 3x cubed – 5x squared + x – 2 without it being a fraction? Im a freshman(age 15) in algebra 2 and none of the ones ive done or seen have had a denominator.
    also mine hardly ever start with a number in front of the x cubed. for me its just x cubed then the rest of the problem, not like that matters much but still.
    ok 1 last thing. Has anyone else been taught that the last number after doing the S division MUST be a zero, because thats how i was taught and its kinda confusing me.

    • Leo Volont 4 years ago

      Hi Afterglow, Yes, Synthetic Division is used to TEST for possible Zeros in Cubic Polynomials, that unless you get a zero then your Test Constant is not a Zero of the Cubic you are playing with.

  • Mohd Yousuf Khan 4 years ago

    hay nancy i wanna know how to solve the equation if there is no such divisor…such as 2x^3+3x^2+5x+7=0

    • Leo Volont 4 years ago

      Hi divine, from 5 months ago. wouldn’t your 2x^3+3x^2+5x+7 be the dividend (the numerator), because your divisor needs to be a binomial in 1 degree. Now in Polynomial Long Division you can have a divisor in more than the first degree.

  • internetçi çocuk 4 years ago

    Thank you so much!!! Really helpful. I have a final tomorrow!!

  • Maegan Jackson 4 years ago

    You are a life saver!

  • siragan 4 years ago


  • JayJoe Tenry 4 years ago

    thx for the help 🙂

  • Camden 4 years ago

    This really helped me so much.

  • DOKter EzE DAnK 4 years ago


  • Emily Schatz 4 years ago

    How does she write backwards like that? From her point of view she should have to write backwards so we can see it forwards…

    • evzv 3 years ago

      She’s an MIT grad… or she’s just right-handed and flipped the video image in post to make the writing forward.

  • Kisan Purohit 4 years ago


  • Nancy Sulaiman 4 years ago

    thank u

  • Govind Gupta 4 years ago

    Mam you teach as beautiful as u look

  • Steven Black 4 years ago

    Thanks for making these videos. You really help me to get through my maths course. Writing exam next Tues. Your videos helped me more than I can articulate. God bless you and your family

  • Justin Spencer 4 years ago

    Express the polynomial in degree 1,
    ((3x-5)x+1)x-2 => then substitute 2 to ‘x’

    this is the underlying truth behind synthetic division

  • mono man 4 years ago


  • Lolll 4 years ago

    is there another method besides factoring by grouping and this to find the roots?

  • Abdul Rehamn 4 years ago

    Hey Nancy,
    I would like to make a content suggestion over here, not because i think I’m any smarter than you or something…..but because i feel that your videos don’t really communicate the beauty and joy of solving maths. in that part i will like you to provide more reasons as to why we solve things the way we do and provide context to why each step is carried out. i understand that the proofs for some things might be too complex but there are simple proofs out there as well that are lesser known to people, i think that your channel can provide people with these proofs so that they are able to appreciate the simplicity of maths and not be afraid of it. also why do we need solutions, what do solutions mean and whats the importance of making any mathematical simplification or step. these are question that when answered leave a deep level of clarity about something in peoples mind! atleast thats how it worked for me! i used to run from maths!

    • Leo Volont 4 years ago

      Hi Abdul, Well, a lot of these kids are being rushed through the system. there heads are spinning. They are so hurried that they don’t take the time to step back to even see what they are really doing. But us old men can relax and take our time and see how cool this stuff is.

  • Yianni Plagos 3 years ago

    i respect you

  • Gabriel Caesar 3 years ago


  • Ray Cowan 3 years ago

    I know how you write backwards. I’ll say it without saying it; you’re not left handed. I like the technique.

  • Charlie Morris 3 years ago

    You, Nancypi, explain everything batter than that advanced algebra instructor that I have.

    He is a bit of a smart ass.

    Thank you for posting your mathematical videos.

  • Ginger Hugges 3 years ago

    Is there a way to use synthetic when dividing by longer polynomials than x-2? Like ones including cubes and squares

  • Penny L 3 years ago

    the problems are hard and all, but i’m more impressed about how neatly she writes backwards

  • Miguel Magaña 3 years ago

    Many thanks for this video and all your videos. You make math understandable 😊

  • Brendan O'Rourk 3 years ago

    Is it just me or is she writing that shit backwards for us to see it correctly?

  • Xitlaly Ramirez 3 years ago

    im learning this and I am super confused

  • Ghulam Rahim 3 years ago

    U r so beautiful

  • Alfonso Neil Jimenez Casallas 3 years ago

    What a pretty and smart teacher, wow! 😍

  • 2 Is Better Than 1 3 years ago

    This is so much easier than what my teacher is teaching

  • Amarmend Amaka 3 years ago

    ooooooh i have found my crush …………..

  • Apollo MANIA 3 years ago

    this is sooo awesome

    love your videos ma’am

    love from india

  • Ilya Hajiaghayi 3 years ago

    I love the videos

  • Debora De Boer 3 years ago

    Love your lessons! Thank you….

  • Elijah jean-leon 3 years ago

    Thanks so much, You always do a GREAT JOB!

  • the egg father 3 years ago

    My normal writing looks like chicken scratch and you can write backwards, I’m really the definition of inferior

  • Aziz Kash 3 years ago


  • Joseph Shaff 3 years ago

    Higher power numerator slant asymptote. For a slant asymptote the remainder goes to zero ? And then the divisor must be a monomial can it be a higher power ? There is a requirement upon it.

  • Lion Alesso 3 years ago

    Good explanations. Clear videos.

  • Tyler Page 3 years ago

    Synthetic Division or Long Division? For Synthetic Division -> Divisor must be in the form “x – c”; where c must be a constant

  • ertas zorlu 3 years ago

    Hello NancyPie… do you want to hang out with me….

  • Bright Young Brains 3 years ago

    😭Oh my word! I just learnt how synthetic division works, I only have Long Division and other methods on my channel! This is so much easier. THANK YOU SO MUCH! I really appreciate it plus thanks for all the help during first year😅🖐🙏

  • waqar awan 3 years ago

    Love you teacher

Comments are closed.