# Synthetic Division… How? (NancyPi)

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Published on September 19, 2018 by

For my video on how to do LONG DIVISION of polynomials instead, jump to: https://youtu.be/RPXMBIFG_W4
For my video on FACTORING to solve for the zeros of a polynomial, jump to: https://youtu.be/Z5MnP9da4EM

Follow Nancy on Instagram: https://instagram.com/nancypi

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: http://nancypi.com

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• Nancy for cpt marvel

• Yesss

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

• You’re so cool😄

• 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!

• 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. 💛

• 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.

• Good luck 👍 with school

• Thanks again!

• thank god u reuploaded this!!!

• She can’t see us though – right?

• rarely used in my opinion

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

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

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

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

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

• Thank you Nancy!

• Hi nancy 🙂

• 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!

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

• Like the video and Nancy. Dislike synthetic div 😖

• you are good teacher thank u

• Synthetic — so its not real division ?
LOL

• 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.

• It’s a real division:

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

• Have you written any book on maths?

• 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..

• Wow, thanks!

• She’s back?!!😃😃

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

• Please make a video on conic section

• 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 ❤

• Lord have mercy, I love math.

• This video was made right on time.

• 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.

• This video is gold thank you 😍😍

• Thanks!

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

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

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

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

• We doing that rn

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

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

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

• Didnt you have a channel called Math BFFS or something?

• 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.

• Thank you ♡

• Lovely and nice way to teach and explain. very helpful

• Thank You!

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

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

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

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

• Boi plz this joint algebra 2

Ik cause i got test tomorrow 😰😰😰

• I’m 14 and I’m doing this

• Mck Harris I’m doing this is algebra 1 lmao

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

• 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.

• Lion Alesso oh. I’m the quiet kid

• She’s better than my math teacher

• You’re so great, thank you so much!

• Thank you

• Thank you so much you Nancy are super savior

• I want to learn how can write board.

• Tysm 😘

• 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

• k

• is she writing everything backwards

• 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.

• 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.

• 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

• 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.

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

• You are a life saver!

• YOU ARE CLEVER AS WELL AS BEAUTIFUL

• thx for the help 🙂

• This really helped me so much.

• Nice

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

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

• Nice

• thank u

• Mam you teach as beautiful as u look

• 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

• 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

• brilliant…

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

• 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!

• 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.

• i respect you

• ❤️

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

• 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.

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

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

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

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

• im learning this and I am super confused

• U r so beautiful

• What a pretty and smart teacher, wow! 😍

• This is so much easier than what my teacher is teaching

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

• this is sooo awesome

love your videos ma’am

love from india

• I love the videos

• Love your lessons! Thank you….

• Thanks so much, You always do a GREAT JOB!

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

• Queen

• 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.

• Good explanations. Clear videos.

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

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

• 😭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😅🖐🙏

• Love you teacher