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.
@@shashwat4920 Oh, in regards to Synthetic Division another thing that one needs to remember is that the number you set out in front, the divider, well, it has to be the NEGATIVE of the constant in your original binomial denominator. In Polynomial Long Division you use the actual denominator as is, then you need to subtract the product across instead of add. So, yeah, there's a few tricks one has to remember.
@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...
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!
The video is mirrored, either with software or with a real mirror. If you saw her doing this live, you'd see her writing with her right hand, and the text would all appear backwards. This becomes a challenge for instructors teaching with this setup, who need to teach about the right hand rule. They need to go against their instincts, and use their left hand to demonstrate it, so it appears properly in the video.
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.
You don't have to use SD. You can also do it using an area model/Polydoku, which I think is faster (less writing) and more conceptual. You're basically viewing division though the lens of multiplication. It works all the time, and I think you are less likely to forget it.
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.
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.
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
Questions: 1. When can I use this? Only cubics? what about quartics or quadratics. 2. does this fully simplify? do I need to keep going or is it done? 3. How does this work?
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.
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.
In the example you gave, there is no rational root. There is one real root, and it is irrational, so this method doesn't work. You'd use Cardano's cubic formula to solve it analytically. You start by some straight-forward pre-processing. Divide through by your a-term, and get: x^3 + 3/2*x^2 + 5/2*x + 7/2 = 0 Then, you shift the graph horizontally with a change-of-variables, so that the b-term is equal to zero. You then get a solution in the form of: t^3 + p*t + q = 0 Once you have it in this form, you calculate the discriminant, capital D: D = p^3/27 + q^2/4 And use it in Cardano's formula, to get t: t = cbrt(-q/2 + sqrt(D)) + cbrt(-q/2 - sqrt(D)) Then undo the shift, to get x: x = t - b/(3*a) To get p and q from your original coefficients: p=(3*a*c - b^2)/(3*a^2) q=(2*b^3 - 9*a*b*c + 27*a^2*d)/(27*a^3) So for us: p = 7/4 q = 5/2 D = 3043/1728 Positive D tells us there is one real & distinct solution, and two complex solutions. Evaluating t: t = cbrt(-5/4 + sqrt(3043/1728)) + cbrt(-5/4 - sqrt(3043/1728)) t = cbrt(-5/4 + sqrt(9128)/72) + cbrt(-5/4 - sqrt(9128)/72) Undo the shift to find x: x = cbrt(-5/4 + sqrt(9128)/72) + cbrt(-5/4 - sqrt(9128)/72) - 1/2 x = 1.4455
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.
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 ❤
The remainder 4 is actually the result of applying 2 to the polynomial like f(2). This is how clever coders apply a value to polynomials without using any power arithmetic.
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
Yes. You would factor your longer polynomial into its roots, so that each one of them would become a linear root term. Even if you had irrational roots, you can do this, although it is more complicated, and probably will be simpler to do polynomial long division. You could also do it with complex roots, in the event that you have an "irreducible" quadratic, but it will probably be a lot simpler to divide by the quadratic with polynomial long division.
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.
