Why You Can't Solve Every Equation (and Why That's Awesome)

Solving polynomial equations by radicals impossible (and also, mostly useless!). However, some amusing math arises from this problem. Check out this quick intro, and subscribe for future videos on this topic

00:00:00 Polynomial equations.
00:00:13 Some examples
00:00:46 Tell me your reaction to this part (comment below!)
00:01:57 The search for a general formula
00:02:50 Why impossibility is not a failure?
00:03:32 Why we cannot solve quintics?
00:04:48 The invariant behind quintics
00:05:43 Where do we go from here?

This video was inspired by The Academy of Useless Ideas study group on Galois theory. Would you like to learn more? Join or discord server for more info!

[gregorymorse]

Not solvable by radicals != unsolvable. Using Bring radicals, it seems any high order polynomial equation may be solvable.

[andrebarrett]

Of course you can solve any equation. Just define a constant that represents the solution. We already did it with sqrt(-1)

[modolief]

0:40 — I liked your questions here about solving various equations in their specific number systems, and came up with a question in return:
Can you think of a number system in which the equation (xy - yx)^2 = -4 has a solution?

[eggchipsnbeans]

Very interesting. I never got anywhere with Galois Theory, just did a little basic group theory but I recall Dr Roger Webster's History of Mathematics lectures at Sheffield University in the 1980s.
The story of Galois is so good (well, less so for him). How he failed to prepare properly for university entrance exams and failed them. How he sent of his work to Poisson and Cauchy (I think) but they didn't know what he was on about . How he joined the army but stood up for a young woman being abused by another soldier and was challenged to a duel. How he sat up the night before writing down all his discoveries and was killed the next day.

I teach boys and they love to know that a great young mathematician died in a duel.

[JorgeTorresH]

Very interesting! I'll be waiting for the follow-up

[abhijiths]

are there exact solutions for quintic polynomials and above if we use more functions than radicals?

[Misteribel]

Ifind this fundamental algebra definition more useful and closer to what you're getting at: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots.

[blue_blue-]

Very much my interests: Useless ideas😃

[s.o.m.e.o.n.e.]

Hey, if complex numbers are enough for any answer to a polynomial, what about for example quaternions then? How are they at all useful and for what? I've heard they're used to describe 3d rotation. And on the topic of obscure extensions of comonly thought things, what about polynomials but with tetration (for example 2x+x+1=0). Or repeated tetration, and so on. I don't have much math knowledge so don't feel feel the need to answer if the explanation is too long. Either way, thanks for the video.

[nbooth]

Any method for finding solutions to arbitrary precision with polynomial time complexity should count as a general solution in my opinion.

[Qermaq]

Um... 5:05 x^4 - 11x^2 + 18 is factored (x + 3)(x = 3)(x + sqrt(2))(x - sqrt(2)).In fact, never mind, I think you did that on purpose simply to fuck with me. Brilliant.

[TheOiseau]

It seems you made a mistake with the polynomials presented at 5:01. If you want the top one to have the four factors shown, with both the square root of 2 and the square root of 3, the bottom one has to be x^4 – 5x^2 + 6. Also, one of the signs is wrong.

[garysimpson]

Newton Raphson Method: Here, hold my beer. If there is a real root of course.

[lucaswiese]

x⁵ - x + = 1
x = -1
(-1)⁵ - (-1) + 1
(-1)⁵ = -1
-1 - (-1) = 0
0 + 1 = ...
Oh frick, i thought i was onto smth

[Fire_Axus]

how is a theory that takes more than a lifetime to understand awesome?

[cooltaylor]

x is approximately 1.1673

There are also several complex solutions.

Don't give me any credit though, I asked Wolfram Alpha

[Qermaq]

If you can maintain this dry, disassociated, almost-bored-but-not-really delivery, this can really work!

[j.r.]

After seeing "Cannot be solved" in the video caption, you can imagine my disappointment upon entering the equation x^5 - x + 1 = 0 into Wolfram Alpha and getting all five roots (one real: -1.1673). Regardless of the substance of this video, it appears that its presentation is simply clickbait. If what you advertise doesn't match what you provide, viewers have been cheated.

Also, what's with the weird "Gorlack" stuff?