# Questions tagged [quintics]

Questions about polynomials with degree $5$. There is no general algebraic solution to these equations as proven by the Abel-Ruffini theorem, although some quintics are solvable.

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### Factoring the quintic $n^5-16n^4+95n^3-260n^2+324n-144$

I was attempting to solve $n^5-16n^4+95n^3-260n^2+324n-144=0$ but then realised I didn't know how to. How would one go about factoring such a quintic and solve for n? I know that the factored form is ...
1 vote
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### Radical representation of $\cos\frac {2\pi}{11}$

I want to find radical representation of $\cos\dfrac {2\pi}{11}$. My attempt Consider the 11th root of unity: \begin{aligned} ω&= e^{2πi/11}\\ ω^n&=ω^{n\bmod 11} \end{aligned} From Euler'...
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### Can a general quintic be solved using Inverse Beta Regularized function?

Tyma Gaidash has recently posted solutions to some quintics in terms of Inverse Beta Regularized function. He also found the closed form for the equation $\cos x=x$ using the same Inverse Beta ...
46 views

### Can roots of quintic polynomials be a solution of radicals? [closed]

Assuming integer co-efficients, does there exist a solution to a quintic polynomial that is a solution of radicals? I understand that there is no general formula to solve any quintic, but that doesn't ...
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### Is the polynomial $x^5+5x-10$ is solvable by radicals over $\Bbb Q$?

I have tried many ways.This polynomial has exactly one real and four complex roots and it is in Bring-Jerrard form, the solvability of which is about finding $\varepsilon, e$ and $c$ to equate some ...
566 views

### Solving quintic equations of the form $x^5-x+A=0$

I was on Wolfram Alpha exploring quintic equations that were unsolvable using radicals. Specifically, I was looking at quintics of the form $x^5-x+A=0$ for nonzero integers $A$. I noticed that the ...
101 views

### Solutions to quintic equation

Abel's Theorem shows that there is no general formula that gives the solution to $x^5+Ax^4+Bx^3+Cx^2+Dx+E=0$ with radicals. My math is not sufficient to understand Abel's theorem. Considering ...
349 views

### Topological proof for the unsolvability of the quintic

Sorry, but in order to ask the question, you will have to view this video http://drorbn.net/dbnvp/AKT-140314.php. Here a topological proof for the unsolvability of the quintic is given, based on ideas ...
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### Loops through zero in Arnold's proof of the insolvability of the quintic

I have a question regarding Arnold's proof of the insolvability of the quintic (https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf). Here are my questions (but I'll give more context below)...
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### Find quintic roots of $x^ 5 = 1$

How many quintic roots, i.e. fifth roots, does the number 1 have? $$x^ 5 = 1$$ What are their values and how can one find them?
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### Proving equation has solution for every $c ≥ 0$

Task: Proof that the equation $x^5 − x = c$ has a solution for every $c \ge 0$ in the interval $[0, \infty)$. No idea where to start, anyone have any suggestions? Kind regards Anthony
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### Why couldn't Euler extend his method of solving a quartic to solve a quintic polynomial?

Euler creates a structure/identity (which are now generalised and known as Newton's identities) to solve a depressed quartic equation, where the coefficients are essentialy that of a cubic. We know by ...
208 views

Is it possible to find all roots of the family of polynomials $1+(-1)^df\cdot t^d+\sum\limits_{j=1}^{d-1}(-1)^{j+d}{d\choose j}t^{j-1}=0$ for $d\ge2, d\in \mathbb{N}$, $f\in \mathbb{C}$ and $|f-1|\le ... 7 votes 4 answers 689 views ### How do I solve the quintic$n^5-m^4n+\frac{P}{2m}=0$for$n$? I want to solve the following equation for$n$in terms of$P$and$m$. $$n^5-m^4n+\frac{P}{2m}=0$$ I've bought and read many books, including "Beyond The Quartic Equation" but I've either missed ... 4 votes 0 answers 174 views ### How does symmetry help solve an equation? From what I've read, I think I have a symmetric quintic polynomial equation. I've spent months trying to solve it for one of it's variables$(x)$. It seems convertable to Bring-Jerrard form: $$2y^5x-... 1 vote 1 answer 146 views ### What is Wolfram Alpha telling me about a quintic solution? I'm trying to solve a quintic equation for y in terms of x and z where x,y,z\in\mathbb{N} and correct combinations of z,x will yield a \textbf{positive} integer y.$$2 x^5 y + 10 x^4 y^... 1 vote 0 answers 74 views ### Problem with Dummit's Article on Solving Solvable Quintics "$(7)\space\space\space[x^2 + (T_1 + T_2\Delta)x+(T_3+T_4\Delta)][x^2 + (T_1 - T_2\Delta)x+(T_3-T_4\Delta)](8.0)\space\space\space l_0 = (a_0 + a_1\theta + a_2\theta^{2} + a_3\theta^{3} + a_4\...
I'm currently studying Galois Theory using Fraleigh's abstract algebra. when proving the insolvability of the quintic, the book takes five independent transcendental elements over $\mathbb{Q}$ and ...