Rational Roots Proof

Rational Roots Theorem Proof

In this video, I prove the rational roots theorem, which is a neat way of finding rational roots of polynomials. A little algebra delight to sweeten the day!

[CliffStamp]

Wow, that is such an elegant proof. This is commonly taught in high school, but the proof (aside from a couple of points) is just very algebra.

[garyhuntress]

That was a rock star proof. I've never seen that before (in engineering) and I loved it!

[cendyywarlos]

I love the symmetry in this proof! I could immediately tell where we were going with the a_0q^n term after you'd finished the a_np^n segment

[Illuminous_]

he's so enthusiastic, this eases assimilation

[shayanmoosavi]

Neat and simple proof, although I'm usually terrified by algebra this was easy to follow. I have a question though, do this theorem only work if a_i are rational? If so, is there a more generalized theorem for real or complex coefficients?

[rafadarkside]

It's not a horrible proof. It's very elegant. This theorem is very useful when you combine it with Bézout's theorem.

[SpoonPhysics]

Would love to see some examples of using a Green's function

[miro.s]

You can also get the second proof for divisor 'p' directly from the first form of equation after dividing both sides by 'q'. It can be more intuitive and natural way of thinking during problem solving to go in linear way than multipath way with returns back. Anyway, many matematical proofs follow the second possible path as you show.

[omeryalcinkaya]

Thanks for this elegant and unique proof

[manuelfalzoialcantara]

Dr Peyam, I look at many proofs of this, yours is the best, HIGH DIDACTIc quality!! thank you very much I think to subscribe to you

[eliyasne]

In my opinion, teaching proofs should be mandatory.
I think that proofs are just a superior way of teaching concepts then unfulfilling memorization.
I personally find it much easier (and fun) to study the proof of something then memorising it.
The proof makes sense, if you forget a part of it you can just recreate it yourself. If you forget a part of a memorized formula you can't recreate it.

[plaustrarius]

Awesome so excited to watch thank you Dr. Peyam!!

[yashuppot]

At *5:30* you can also just look at that mod q and mod p and your are done.

[utuberaj]

Superb Dr Peyam- or rather if may call you Dr.P (or Q)😊🙃-. I was also terrified as you about what this theorem was about- and wondered how prove it. You pulled out the solution- which is really elegant Dr.P, except, as the expression goes- you should mind your (p's) and q's"! That is you write your q's as the number "9" Nevertheless Dr.Peyam- your range of Math videos are amzing- and I am re-learning my math (I have a degree in Engineering- but never saw learning this fun).
Thanks a ton👋👌

[daniellejdevlin]

That was an amazing proof!! Thank you!

[TheForever]

keep going Sir nice we are waitinf all amalyisis lectures

[omargaber]

Thank you doctor, make alot vedios like this please and about number theory

[sauravthegreat]

Great video 👍🏻👍🏻 😍😍. I believe you could’ve also elaborated on how if integer roots were mot produced then the answer had to be irrational. If q never divides An, then q has to divide p^n. But since p and q have to have no common factors, it follows that the answer will be an irrational number

[jimnewton]

Does your proof suppose the leading term and constant term are relatively prime?

[rajiv]

Thank you very much sir❤
I have a question that if a number is written in p/q form and p divides a polynomial's constant term and q divides its leading content then is it its rational root?