Fundamental Theorem of Algebra - Numberphile

Catch David on the Numberphile podcast: • A Proof in the Drawer ...
Part one on odd polynomials: • Odd Equations - Number...
More links & stuff in full description below ↓↓↓
A nice extra bit about complex numbers: • Complex Numbers are Aw...
Professor David Eisenbud is an algebraic geometer (and director of the Mathematical Sciences Research Institute at Berkeley)
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[@numberphile]

Catch David on the Numberphile podcast: kzbin.info/www/bejne/b6qUc3qso7msh6M

[@FourthDerivative]

"You can tell it's an important theorem because it has a name. And you can tell it's a *very* important theorem because it has a *pompous* name." -James Grime

[@tim60312]

Which video is this from?

[@ragnkja]

@@tim60312 “1 is not prime”, I think.

[@photonicpizza1466]

@@tim60312 "1 is not prime" like the other person said, the quote is referring to the fundamental theorem of arithmetic

[@PC_Simo]

Maybe Pythagoras’ Theorem also deserves a pompous name. 🤔

[@fuckmyego]

"do you have any more of this nice paper, Brady?" Completely lost it.

[@ditzfough]

Omg!! Funny

[@jacderida]

"It makes me happy because I'm an algebraic geometer so it gives me something interesting to do." This guy is totally awesome!

[@dnddmdb642]

There's a few different videos with this professor and he seems to be the king of understatement, lol.

[@SquirrelASMR]

@@dnddmdb642 hahah "most prime numbers are odd"

[@SilverArro]

This guy has such a relaxing voice; I could listen to him lecture on math all day.

[@njabulomahlalela2912]

me too lol

[@benjaminlehman3221]

His voice is too relaxing for me. If I listened to is lecture I might get very tired and fall asleep.

[@Alexagrigorieff]

I've actually had a professor in university with extremely soporific voice (means it puts you to sleep).

[@savonliquide7677]

he's not soporific at all, when you listen to him!

[@savonliquide7677]

he's very confident and clear, it's a chance for youg student, they can understand the proof of such an important theoreme. Congratulation to him!

[@wouldntyaliktono]

Holy shit this guy's voice. If mathematics ever goes belly up, he's got a career in audio books for sure.

[@rudnickulous]

wouldntyaliktono Bob Ross

[@TheMilwaukeeProtocol]

+wouldntyaliktono Don't joke about it going belly up, because everything else is going that way. :-( Case in point: the only type of education with job openings anymore in my country is Special Education. ...But if you weren't joking, then disregard my request.

[@AcornFox]

Rhett Rudnicki The Joy of Mathematics

[@cameronspalding9792]

I agree

[@dansanger5340]

I'm ashamed to say that I was a math major in college and had never seen that proof before now. It was quite beautiful.

[@ALeafOnTheWind42]

It's because to be technical with the proof, you're using algebraic topology, which is something not generally taught in undergrad (at least in my experience). It is a great proof, though

[@becomepostal]

Dan Sanger it’s more an explanation than a proof.

[@marquez2390]

It wasn't a proof though it was more of a visual concept.

[@SilverLining1]

@@ALeafOnTheWind42 You need to use algebraic topology? I don't believe this. Can't you just use limits and the intermediate value theorem?

[@ALeafOnTheWind42]

@@SilverLining1 You don't necessarily need algebraic topology because there are other proofs that don't use it (if I recall correctly, as it's been years since I watched this video, the actual proof they were getting at in this video does require algebraic topology). That said, no, you can't use the IVT for the simple reason that some polynomials with real coefficients only have complex roots. As a simple example, take f(x) = x^2 + 1. We can't use the IVT to find a root because for all real values of x, f(x) is positive. The solutions in this case are, of course i and -i, so we need to go to the complex plane in order to get the solutions, and the IVT simply doesn't exist in the complex plane. There are generalizations of the IVT to other metric spaces, but that's already getting into point-set topology at the very least.

[@MadaxeMunkeee]

That was the most concise, amazing explanation of the Fundamental Theorem of Algebra I've ever seen. I've seen a fair number of proofs of it before, but none of them have been as clear as this one. Thanks for sharing, and wow - mind blown.

[@Brien831]

@@welid9772 yea I have seen the Gauss proof and it is a lot. My linear algebra professor scared us first semester students with it back then. This is not even a summary of the gauss proof. I dont believe for a second op has seen the actual gauss proof.

[@MrAksupriatna]

@@welid9772 Where is the proof? Is it available in this channel?

[@urmi-e-azar1776]

@@Brien831 If I'm not mistaken, this is Gauss' fundamental idea. To make this precise without using algebraic topology is absolutely bound to be morbidly technical. Indeed, a somewhat equivalent of Intermediate Value Theorem for the plane would be Brower's fixed point theorem - which is a fairly involved result from algebraic topology. If you could refer me to an English translation of Gauss' PhD thesis, by the way, I would be grateful.

[@HowToBasic]

X = Egg

[@lilpump2183]

y÷x+7 (7-7×3+4-x) x=2 y=6

[@nestorv7627]

HowToBasic nurd

[@sphinx1659]

HowToBasic 😂😂😂

[@SkywardPvP]

HowToBasic wow didnt expect you here.

[@dwither6594]

HowToBasic wtf why are you here 0_0

[@ChessNetwork]

Glad to not be the only one. 8:27

[@52bicycles]

Jerry watches Numberphile. All these years of looking for some form of validation... finally my search comes to a close.

[@tigervision7]

Jerry! :D glad youre a fellow numberphile :)

[@junka22]

love your videos man

[@devagarwal1591]

Am I solving a chess puzzle or a math problem. Confused!

[@RachelsSweetie]

ChessNetwork, I love the way you walk through your thinking about a chess game, just like Prof Eisenbud took us through this theorem.

[@NuisanceMan]

Yes, the circle shrinks down and gets very close to the red c...but it never quite reaches it. And why is that? Because only Moses could cross the red c.

[@JohnnyDoeDoeDoe]

Haha, love this

[@Patalenski]

Well, Chuck Norris can jump it over! :-)

[@davecrupel2817]

lololololol

[@kleinfaf]

yeah... they come around, but they never come close to

[@TosiakiS]

No it does reach it, because the function is continuous.

[@EternalBooda]

I had a lot of things I needed to get done today. I got the laundry done. The rest of the day was Numberphile.

[@Clint945]

+EternalBooda There are far worse ways to procrastinate. At least this way you're learning something interesting.

[@Aruthicon]

For me, I just take my machete, my grappling hook, etc, and I go on an adventure to Wikipedia.

[@lazzerbear]

I have no idea how my school makes math boring. They must work very hard at it.

[@1998wiwi]

The difference between learning by choice and learning by necessity

[@SkillUpMobileGaming]

They make math *SCARY* not boring. Not all teachers (some are really great!) but a whole lot of teachers do this. This is because of shitty teachers, plain and simple. Everything I learn at school from my teachers I can learn so much better elsewhere. Even math tutors are better at teaching than actual teachers nowadays.

[@christianrodriguez823]

The art of mathematics is lost through school, mainly due to standardized testing and a lack of knowledgeable teachers. My favorite book is A Mathematician’s Lament by Paul Lockhart, he discusses what is wrong with the math education system and how students should really be exposed to mathematics so it’s not boring.

[@tommydashed4205]

Because some math is boring. You have to know the boring bits to understand the mindblowing ones.

[@bwatspro]

Because they think calculating like machines is equal to doing math. And using computer, degrades understanding of math, when in fact, its quite the opposite.

[@rlamacraft]

Quite possibly the best Numberphile video ever! Perfect length, really interesting and pushing the boundaries of my knowledge! Couldn't ask for more, other than more like this…

[@jacoboribilik3253]

yes I know, the lenght of the video is some power of Pi

[@caiorimoli3466]

Man, I have PhD in Biophysics and this is the most beautiful approach I've seen to prove a mathematical theorem in my life. It is such a physicist way to see things, playing with scales to get an approximate feeling of what is going on. So much different approach from all the mathematical strict formalism that I've seen from my mathematics professors at the university. The coolest thing, in my opinion, was that the main constraint he used to work with was the polynomial continuity. Because he knew the polynomial is a smooth complex surface/curve, each term of the polynomial is most likely to work at different ranges in the image-plane f(x). This is so beautiful. I've never thought from that perspective, although I've been indirectly working with that all the time. Thank you!

[@DudeGlenn]

Came for the Fundamental Theorem of Algebra Stayed for the Paper Change.

[@NuisanceMan]

+Glenn Beeson (BeesonatotX) Actually, I wish they would change either the paper or the markers to something more compatible with my spine.

[@AkshaySinghJamwal]

Hahaha those "Paper Change" stills with the elevator music always crack me up.

[@pwhqngl0evzeg7z37]

@Vasbrasileiro Haha! I came to see if anyone else noticed!

[@element4element4]

Interesting proof of the fundamental theorem of algebra. As a physicist using topology a lot in my work, I am always amazed by simple topological proofs of basic mathematical theorems.

[@hanniffydinn6019]

element4element4 truth is numbers are higher dimensional objects, so true maths is algebraic geometry. That is everything is geometry in higher dimensions. Because reality is actually 11 dimensions. The number line and 2d plane is a gross simplification of reality.

Hanniffy Dinn 11 dimensions?! Could you elaborate on that?

[@hanniffydinn6019]

Viktor Rucký it's all clear from string theory which needs 11 dimensions and describes reality. Search for documentaries on string theory and quantum physics.

[@jameswood7207]

string theory is a THEORY and still unproven-and you shouldn't recite is as a fact.

[@ganifraterdogan1062]

@@jameswood7207 Can you tell me what a theory is?

[@666Tomato666]

9:20 a real Professor: "The formula is trivial and left as an exercise for the viewer" ;)

[@emlmm88]

666Tomato666 It's certainly trivial if you're allowed Liouville's theorem from analysis.

[@michaelwoodhams7866]

One of my undergraduate maths professors told us of an exam he had once had to sit. It was a single question: Proove the fundamental theorem of algebra as many ways as you can.

[@ronnies07]

8:29 - "If you remember your high school trigonometry" I would if it was offered. Standardized testing exists here. The results of which determine school funding, staffing levels, and whether the principal gets fired. Many schools are dropping anything that is not explicitly tested for in order to focus on classes that improve their scores. The cursory glance of the ratio of two given sides being equal to the sine/cosine/tangent satisfies the states trig requirement and that's about all we get. We might even spend 10 minutes and calculate an angle. Once. If I want a proper trig class here I have to enroll in the local community college for something and bomb the placement exam so I can take a remedial.

[@SeanBenson23]

This is the most accurate thing I've ever read. Sad.

[@andrewxc1335]

8:30 - I taught the Trigonometric multiplication rule this year in terms of Euler's Formula: r*e^(i*theta). It went really well.

[@Macgki]

You definitely should make some videos about Gödel and his incompleteness theorem.

[@3niknicholson]

they started making one....

[@darkmage07070777]

13:40 - The moment where the lightbulb went off for me and I could see how this all was supposed to work. Excellent video, as usual!

[@walt4670]

"Somehow this circle shrinks down slowly in some very uneven way" this made me think of Ricci Flow... but I don't know enough to say exactly why or how that applies to this as this theorem is on a complex plane and RF is a partial differential equation.

[@fatpie2.0]

I want to see more of Professor Eisenbud! Soothing and informative

[@robobrain10000]

That made alot of sense. thnx for the video brady.

[@numberphile]

you're welcome - Prof Eisenbud deserves the thanks!

[@steveoti6434]

True .

[@cassandrakarshner4157]

+Numberphile for dndndnnmmmjuiuhhbhjjjhbbjjjjj

[@TrojanVirus]

at around 4:40 he says that f(x)=x^2, right, so since a function is what you do to a variable. Right? Then, why did he get the function of i, when it is f(x) so it should be function of any number on the x axis, not the y, and i is on the y axis, so i do not understand this.

[@Egovip635]

the name of the variable for the function can be any random symbol. so f(x)=f(a)=f(b)... The x is just used twice with different meaning.

[@izakj5094]

I'm just learning this in school and I got goose bumps watching the video. Math is love, math is life.

[@TommiHimberg]

The beauty of mathematics - fundamental theorem of algebra. Excellent video!

Et ole sattunut saman luokan tilastotieteen peruskurssia verkosta löytämään? Pääsääntöisestihän ne tylsyyteen tappavia hyvin lyhyellä altistuksella...

[@TheSentientCloud]

Beauty of mathematics is certainly seen in fractals.

[@TommiHimberg]

Hmm, Tero Kankaanperä en ole niin katsellut, että osaisin suoraan suositella, mutta Courserassa on kyllä paljon matskua ja myös tilastotieteen kursseja, mutta ne voi olla aika kuivasti tehtyjä.

[@Cleisthenes2]

'This is the fundamental theorem of algebra because it's the basic connection between algebra and geometry. Roots are points somewhere, so they're geometric objects, and polynomials are algebraic objects, so this is the connection that makes algebraic geometry work.'

[@MoonlightPassions]

Really enjoyed this, currently studying mathematical methods at school so I feel I could actually understand how this theorem works to a degree. You guys seriously make maths fun :)

[@adenpower249]

You're in the Australian system?

[@mBlazer2]

Got to love the paper change interlude.

[@landonkryger]

I hope we can see a video about quaternions sometime. They're like super complex numbers.

[@JayTemple]

I just saw a link to one in a sidebar. You got your wish!

[@justinsankar1164]

@@newkid9807 ?

[@Bransaurus]

Why don't you guys put the Hello Internet podcast onto youtube? Mainly because I usually just put a bunch of videos into my watch later playlist and let it play while I do work. Yes I'm saying going to another website to listen to a podcast is too much work.

[@numberphile]

It is on KZbin, but one season (10 episodes) behind what is on the main site, iTunes, etc. kzbin.info

[@vdeave]

On the Hello Internet subject, the guy in the video says he didn't remember his trig identities, and he works in maths...how useful would learning a language be?

[@battleforevermore]

Numberphile and the channel also has the bonus reel of the newest episodes. presumably done to F**K with our heads.

[@RINB3R]

+vdeave you'd be impressed by how many mathematicians don't know the trig rules, simply because exponentials are much easier to use and trig functions are very VERY closely related to exponentials.

[@NuisanceMan]

+RINB3R Wow, that makes me feel much better. I always hated all those annoying little trig rules.

[@Faxter313]

9:10 It's actually also really easy to multiply complex numbers mathmatically, by transferring them from karthesian to polar form first. (I learned math in German and I have no idea if those words make any sense to English speakers). By doing that you acutally see why you can multiply the lengths and add the arguments: |z1| * e^(î*theta1) * |z2| * e1(i*theta2)

[@nychold]

Makes sense to me, though the spelling of Karthesian made me think of Carthage before Cartesian. Don't know why...

[@PeterGeras]

Yes, but the polar form of complex numbers isn't immediately clear.

[@SuperTheguy1234]

what you said is exactly what we learn in america (at least for me). the only difference is that we spell it Cartesian, not karthesian, but that is only one word.

[@Keldor314]

I thought of a completely different approach during the video. Observe that when you raise z to any power, you map the complex plane to the complex plane, with no holes. Observe that the same thing happens for addition and multiplication by a constant. This means that each of these three operations will map some point on the plane to the origin. Since every polynomial can be written as a tree-like composite function of these three elementry operations, it follows by induction that the entire polynomial must map some point to the origin.

[@newkid9807]

Wrong!

[@newkid9807]

You must prove that with composition there remains no holes.

[@lejink]

This man is a very good teacher

[@Metal_Icarus]

i searched you up after hearing about your math stuff. this video was the first one i have seen of your channel. As a person who is 14h into pre-calc prep to go to college as an adult, the first 51 seconds provided more context and intuitive concepts about the expressions than I have been provided so far.

[@oilbay54]

I studied this theorem nearly 45 years ago. But this explanation is just excellent. Thank you very much.

[@PunmasterSTP]

That was a beautiful illustration, especially at the very end where he swooped in and explained root multiplicity as well!

[@slotmoon]

Olha que coisa mais linda, mais cheia de graça...

[@EtzEchad]

I never saw this explained better. Thanks!

[@oO_ox_O]

Really great explanation, one of my favorite Numberphile videos now! Maybe someone could make an interactive tool that also shows it the way he described it (with the circles) for an arbitrary polynomial (of positive degree). BTW, how many fundamental theorems were covered yet? I certainly remember arithmetic one being even covered several times, maybe you could do a playlist of it Numberphile (on the other hand with so many videos it would take quite some work).

[@NeedsEvidence]

9:25 "PAPER CHANGE" - best moment (almost like "Welcome to the middle of the movie!")

[@kwstaskartas9488]

Great insight and intuition behind of a difficult theorem. Amazing.

[@brianpso]

I just loved the little things you did in this video, like the "paper change" part, or the part where the professor was joking about how to spell right. The whole video was greatly informative as always, but these little parts caught up my attention. Thanks Brady!

[@ThrawnTheater]

I've watched many Numberphile videos, and this is the first time that I've seen him. There should be a play list for all of Eisenbud's videos btw.

[@splendidteaching]

Thing of beauty. Thanks for creating. Girl from Ipanema music a nice touch during the paper change :)

[@spieagentl]

That was a really nice video and explanation of the proof, thanks!

[@cepson]

I would love for someone to do a video investigating the point when The Girl from Ipanema became the go-to song for short interludes.

[@Felipe-sw8wp]

"That's right except for you're saying that I could find them. I'm gonna show you they exist, I'm not gonna find them for you" lol

[@emlmm88]

Yay, David Eisenbud! Love your commutative algebra book!

[@john7787]

This was really awesome and very intuitive! It's a shame my school didn't teach me this! And the n-root part is also a nice bonus! Thanks Brady!

[@ricardgavalda6135]

The "aha!" moment promised in 13:10 comes at 13:40 and it is really "AHA!!". Fabulous! Thanks!

[@cyber-commie4447]

Unbelievably intuitive explanation. Thank you sir.

[@willdeary630]

I have to hand it to Brady, he really is good at listening to the comments and presenting it in such a way to keep as many people possible happy.

[@samuelromero1763]

I remember being in algebra 2 and just not getting how or why we could draw this imaginary axis and have this new set of numbers. But now I see it as an axiom. Mathematicians didn’t need to justify the creation of imaginary numbers, just let them exist. And by letting them exist you end up being able to solve many problems that you couldn’t before. As a high school student this idea seemed so foreign to me; that you could just make up something and if it solved your problems or made something make for sense then just let it be and accept it.

[@tannerpittman]

"Paper change" was brilliant. Loved it.

[@beverleyhill2199]

do bezout's theorem after this :) you'll have to talk about multiplicity and the projective plane in order to properly state it though. but it's one of the best theorems in mathematics IMO

[@newkid9807]

Your opinion lacks value

[@ianmichael5768]

This is seriously underrated. Calming and profound.

[@hugge123456]

smooth girl from ipanema right there ;)

[@hugge123456]

humble brag?

[@battleforevermore]

hugge123456 BRADY.. stop freebooting, at least mention the music in the credits brah!

[@obeythoven5221]

hugge123456 Why are you here ? Aren't you in your room crying like any other Brazilian ?

[@HanBurritoz]

I was wondering why the music at the paper-change seems so familiar.

[@nychold]

I thought that was the Girl from Ipanema. I even caught myself singing it when the video came back on.

[@StewartUSAF]

That is a beautifully elegant layman's explanation of the proof of the FToA.

[@trafalgarla]

I wonder if Brady will ever do a video on the philosophy of mathematics?

[@SirNickyboy]

Just had to give a presentation to other students about this theorem two weeks ago. I proofed it in another way, but I didn't gave such an intuitive idea. Would've been great if this video had been out there two weeks ago :D

[@thangpham4196]

I love your channel so much, maths look much easier than I was taught so far. I'm turning to love math instead of being scary.

[@tacopacopotato6619]

I like that in the edit you kept the spelling mistake and correction. More like that please

[@smoothtriston6203]

This guy's haircut is a complex function.

[@SuperMind92]

World of complex numbers is so different, thank you sir, for making me understand complex numbers in a better way, than I was before.....

[@MBogdos96]

Maybe you should have employed limits to help explain it a bit better, that whole circle story isn't very understandable without the maybe to back it up. Or maybe just like other times, a link on the description to a video with all the math in depth

[@attilakiss8585]

It is not about limits, but it is about the property of continuity (or smoothness). Though precise math def of continuity requires limits, for this level it is way enough as he explained, using the "common sense" of continuity or smoothness.

[@MrThegster123]

I really like that intro! It was like "previously on"

[@thom_wye]

possibly a future algebraic geometer watching right now! algebraic geometry ftw! :D LOL, loved the paper change :D and the proof is fantastic!

[@emlmm88]

Thom Wye gaming Out of interest, what area of algebraic geometry did you end up pursuing?

[@xanokothe]

I found this video awesome! I worked with numerical methods in college, where I transformed algorithms into programs. Some methods I did not understand how do they work, but seem to use the same rules

[@swapnilmaiti3974]

The proof was amazing !!

[@newkid9807]

Stinky swapnil!

[@Mahmood42978]

This was one of the coolest things I learned about in Intro Complex Analysis.

[@DjVortex-w]

Is f(x)=1 not a polynomial? If it is, what are its zeros? If it isn't, how is polynomial defined, then?

[@marisolmaciel2184]

WarpRulez k

[@KnakuanaRka]

Since its one term would be represented as 1x^0, that is a zeroth-degree polynomial, which by the theorem would have zero solutions, as is proper.

[@valeriobertoncello1809]

Well you could write it as f(x)=x^0, therefore a 0-degree polinomial with n=0 solutions

[@elias_toivanen]

It is a polynomial of order zero. It has zero roots ;-)

[@trangium]

It is a polynomial of degree 0, therefore it has 0 solutions.

[@13makk]

Thanks for asking the right questions Brady.

[@Istlok]

make video about 4th+ dimensions

[@aamol3696]

Fantastic video Brady, I was absolutely intrigued!

[@SchumiUCD]

Next week on ABC: "A team of mathematicians at numberphile have proved the fundamental theorem of algebra".

[@hobomanchild2504]

If I close my eyes and listen from a distance it's almost like Mr. Burns is giving me a lecture; I love it.

[@michelef406]

"You know, for a mathematician, he did not have enough imagination. But he has become a poet and now he is fine."

[@cparks1000000]

It's nice to see Eisenbud on screen. His book on Commutative Algebra is a beast (in the best possible way)!

[@CarlosToscanoOchoa]

Could I solve partially Collatz conjecture before Numberphile post a video about it? I think so.

[@reimannx33]

And I think my cat would have come up with General theory of relativity even if Einstein did not formulate it.

[@reimannx33]

Carlos Toscano Ochoa ....and I think my cat would have come up with General theory of relativity even if Einstein would not have formulated it....

[@a690ac52ed7]

I doubt it. Special Relativity, maybe. But knowing cats as I do they haven't the patience to learn the maths for General Relativity.

[@j.stephenpearson4348]

The Paper Change Intermission made me laugh out loud -- great gag, Brady!

[@OwenPrescott]

You should do competitions to win some fancy brown paper, maybe with the written equations on them. :P

[@ZipplyZane]

They already sell them, so making them a prize of some sort wouldn't be too hard...

[@Trunks47r786]

This proof was simply exquisite. I've heard about the Fundamental Theorem of Algebra from Gauss before but to actually see the proof...Thank you very much Numberphile.

[@MozartJunior22]

YES! Some hardcore mathematics, just as I like it.

[@mattikauppinen459]

So hardcore and stuff, with only graphical representation, no limits or any actual maths involved. He showed that we have a polynomial (should be "non-constant polynomial") of degree n and it has "at least" 1 (actually its exactly n) complex roots. But no actual maths presented

[@CastelDawn]

you must be SO popular with the ladies

[@MozartJunior22]

You must be Sherlock Holmes

[@junka22]

this is an important part of engineering maths, try electrical engineering maybe

[@MozartJunior22]

Teemu Vilkman I take my other comment back, YOU are Sherlock Holmes. It's like you know.

[@Joona527]

Oh, that 'Paper Change' intermission made my day. Keep up the good work, Brady!

[@AlexMcshred6505plus]

This is all very well done and all but I'm being somewhat distracted by how much the professor looks like Sean Connery in Indiana Jones and the last crusade.

[@mathmagicproductions431]

Gave me a better understanding of why we get n roots, thank you

[@omjoeandsteve]

don't you actually only need a 3 dimensional graph to do both axes with complex planes?

[@amandadube156]

no, because both x and y need an imaginary number line. hence, why x and y both had their own planes in the demonstration: together, they add up to 4 dimensions.

[@omjoeandsteve]

oh i get it

[@glmathgrant]

To be more precise, a polynomial of degree n with one variable, x, and whose highest-degree term (x^n) has coefficient 1, can be written uniquely as the product of n linear terms of the form x-a, where a is a zero of the polynomial. Some of these zeroes may be the same; x^2-2x+1 has the unique zero 1, but in a sense, it counts twice, because the factorization has two x-1 terms.

[@frankharr9466]

Oooooooooooh, now I get it. It took me like three or four tries. But I was baking at the time, so I have an excuse.

[@OrbusS]

A thorough explanation of the theorem. The kind of explanation I wish I could come up with just like that.

[@fasle2]

"If you remember your high school trig- I certainly don't..." There is hope for me! XD

[@flumpyhumpy]

I was very happy when a lecturer of mine said "and here we'd use the cosine rule... I don't remember it... You could hold a gun to my head and I wouldn't remember the cosine rule".

[@mah38900]

My college algebra teacher really hammered this home to us. I don't think I could ever forget it. In fact this guy reminds me a lot of that teacher, even looks like him.

[@josephdavis3349]

There's a much simpler way to prove The fundamental theorem of algebra using properties of functions and their inverses. Although, perhaps that involves to much hand-waving about the actual properties of inverses?

[@ChristianRosenhagen]

The degree d of the polynom must be at least 1. eg. the polynom f(x) = c, has no root for c != 0. Mayby already mentioned in the tons of comments...

[@Anamnesia]

Apparently Newton has a method to find a Root... hehehehe... :P

[@0x8055]

newton-raphson for 1-D gauss-newton for n-D check on wikipedia

[@0x8055]

ObeseYeti help me then plzzz?

[@0x8055]

just saw on urban dictionary. fuck right? im brazilian sorry. never saw root used in this way. thx!

[@Anamnesia]

Yes. Apologies... Childhood humour! :P A colloquial term for 'sex' is "root".

[@nychold]

Too bad he died a virgin.

[@IshayuG]

That is a friggin' phenomenal explanation.

[@shadowmagician21]

"Do you have more of this nice paper, brady?"