Synthetic Division... How? (NancyPi)

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.

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.

[pennyl]

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

[kaylabrown]

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

[bruitzgaby]

this girl is the only person who can teach me math

[Hersche]

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

[Coreyahno]

Oh how I miss the days when math was this easy! I’m taking Differential Equations now and synthetic division just showed up again. I learned it in pre-calculus but didn’t use it a single time in Calculus 1, Calculus 2, or Calculus 3. Totally forgot how to do it, so thanks for the great explanation! Important reminder to make sure you understand as much as possible in your math classes because you never know when you’ll be expected to remember something you learned years ago.

[leorael]

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

[stevenblack]

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

[faiefairyfriend]

still watching these in 2021. You are a lifesaver!!!💚💚💚

[charliemorris]

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.

[Rockpod]

Was confused cuz I missed a day of school when this was taught but watching this made me understand it clearly. Thank you!

[Staravora]

My textbook be using scientific space magic terms to explain how to do stuff when really it's as simple as this. Amazing.

[samanthat.]

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. 💛

[davidwaldronwellnesscoach]

Thank you! Why are math teachers and textbooks unable to communicate clearly? You are a life saver

[timothyhowie]

Nancy, you are a BRILLIANT TEACHER! 😅👍🙏

[darshangovender]

Thanks for these you helped me through my first semester calculus course

[daniyalzain]

thanks so much now got a help for my exam.❤
wish to have you as a teacher.

[skpodtar]

Madam today I watched your video on synthetic division of quadratic equations. it is realy very. Interesting.I feel extremely satisfied tu watch your videos on maths. thanks s lot madam.Realy u are one in lakhs tu teach mathematics so brilliantly.Hats off u madam

[leestenson]

@NancyPi, great explanation on how to complete synthetic division but still have not seen anyone discuss the applications of synthetic polynomial division to Computer Science (Cryptology specifically), systems engineering (Control Systems design), and so many other areas. This is the real problem with HS & some college level mathematics...it is taught devoid of application and most kids wonder..why the hell do I need to know this? Most students are not going be engineers, computer scientists, mathematicians, or physicists so why are they learning info that they will NEVER use or in many cases forget by the time it is required in typical STEM tracks in College or grad school ???? Just an old engineer and OR Analyst trying to help his HS daughter learn Pre-Calc and CALCULUS...

[sofialorenzini]

i finished my math homework for the night but oh my god she’s just so relaxing

[The_Bible_Camp_Victim]

good stuff
should be getting more publicity, and I'm just amazed at how she can write inverted letters, my brain would fry itself if I'd tried.