How Do I Find the Roots of a Quintic Equation? (Help Me Out !!!)

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[mohankumaro]

Yes. Can you explain Galio's theorem.
Please make a video on explaining Einstein 's "On the elecrodynamics of moving bodies" in detail may be in 2 or 3 episodes.

[SobTim-euxu]

Another day
Another loved video from your channel)

[Nikioko]

First, it helps to bring the equation into its normal form, i.e. the highest polynomial having the factor a = 1.
Then try to find obvious solutions and factor out the binomial (x minus that solution). If you are lucky, you factor out so many solutions that you end up with the product of a couple of binomials and a quadratic term at highest, which you can factorize as well.
So, let's see:

x⁵ − x = 0
x · (x⁴ − 1) = 0
x · (x² + 1) · (x² − 1) = 0
x · (x + i) · (x − i) · (x + 1) · (x − 1) = 0
x₁ = 0 ∨ x₂ = −i ∨ x₃ = i ∨ x₄ = −1 ∨ x₅ = 1

x⁵ − 2x³ + x = 0
x · (x⁴ − 2x² + 1) = 0
x · (x² − 1)² = 0
x · [(x + 1) · (x − 1)]² = 0
x · (x + 1)² · (x − 1)² = 0
x₁ = 0 ∨ x₂, ₃ = −1 ∨ x₄, ₅ = 1

x⁵ − 10x³ + 15x − 7 = 0
For this 5th degree polynomial, the method obviously fails, as there is no obvious solution to simplify the equation.

[thedude]

I'd love to see a video about Newton's method.

[Satisfiyingvideo-uupw]

Thank you sir for accepting my request and make video about unsolvablity of quintic formula. I am a student from Bangladesh in south asia. In your country internet was stop untill today. Students and people.are protesting Against the dictator government. More than 200 people are killed by army and police. So Govt took a complete shout down. I usually watch your every video and i suggest my friend to subscribe you. But i cannot watch your video for internet shut down. Please pray for my country 😢😢. Btw are you from italy?

[victork]

I really wish to see the formula for exact roots of quintick polynomial! I mean, i believe there are such thing, but we need new algebra to epxress it in finite form... Or could there be only infinite siries for certain galoa groups of some polynomials)