Solving a Quintic Equation

We solve the equation x^5 - 25x^4 + 250x^3 - 1249.9x^2 + 3124x - 3122.5 = 0.

00:00 Intro
00:14 Observation 1
02:00 Observation 2
03:36 Simplifying
06:01 Finding solutions

[AT-zrtv]

Your exercises are always a fun treat. I haven't done school math in 20 years and this is a great way to toy with mathematics without the burden of having to re-learn too many theorems.
Question: could you make a video explaining how you come up with such exercises? Do you go backwards from a solution, do you borrow from books / friends, do you just invent whatever and hope you can solve it? Thanks again for you content!

[henryginn]

This is a really nice problem, I love how natural the route to solving it is.

It reminds me of the problem “factorise 2491”. I have some friends who solved that in seconds, but I find people who might be good at maths but not really passionate about it have a hard time with it.

[anastasissfyrides]

Very useful insight, and the train of thought is very sound, thank you

[johnporter]

I love how quickly he expanded (a+b)^5

[alexandermorozov]

Вам очень, очень повезло, что коэффициенты многочлена совпали с формулой произведения разности 5-й степени.
~
You are very, very lucky that the coefficients of the polynomial coincided with the formula of the product of the difference of the 5th degree.

Very well chosen example that clearly conveys the point. To the experienced the the strategy is the natural one since one sees that the equation is "close to" (x-5)^5.

[MF-qpyr]

Nice problem and well-explained solution.

[holyshit]

What about numerical methods
For polynomial roots I like two approaches
Eigenvalues approach
Here QR method with shifts will be useful
(QR method should work on also on the chosen block of matrix)
Choosing shifts correctly may cause troubles
Baistow method
Deflation with quadratic trinomial leads us to the system of nonlinear equations
I suggest to replace Newton's method for solving system of nonlinear equations
because it may case problems with convergence

[danmimis]

There's that saying: "Go with your first instinct". First assumption I made because of 1249.9 and 3122.5: there can be no integer solutions. There goes my "first instinct" ...

[hypebeastuchiha]

I like how you take high school maths and turn it into something new and interesting
I regret deciding not to get a maths degree but this lets me join in on the fun 😄

[dqjfrue]

0:01: Don't "1249*9" and "3122*5" mean multiplication?

[azzteke]

Your decimal point looks like a multiplication sign.

[holyshit]

If you guess that x=5 is a double root the rest is easy
If you know something about polynomial GCD you can get rid of possible repeated roots

[peterstromberg]

Even I would not come up with the "trick" in the video. Looks more like a gambling with numbers to come up with this.

[ianfowler]

So I'm thinking that if you have a multiple root to a polynomial equation then it must be real (tangent to the x-axis). i.e. you can't have a double, triple, ... complex root. Is this because w, w^2, w^3, ... will all give a single root? Powers of w (up to n-1) will always place you on a different point on the unit circle.

[Jkauppa]

how about taking derivatives, to sequentially find the zero derivative point, down to degree of one equation