Questions tagged [quintic-equations]

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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4
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1answer
126 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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0answers
64 views

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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3answers
53 views

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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2answers
23 views

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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0answers
60 views

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 Newtons identities) to solve a depressed quartic equation , where the coefficients are essentialy that of a cubic . We know ...
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2answers
149 views

Find a closed form to the solution of $\sqrt{2-\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2-x}}}}}=x$

Hi I try to solve the following nested radical : $$\sqrt{2-\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2-x}}}}}=x$$ Miraculously the related polynomials is a quintic .More precisely : $$ x^5 - x^4 - 4 x^3 + 3 x^...
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22 views

The roots of quintic polynomial

I have some region $[a,b],$ and the polynomial $P(x)$ in it $(\deg P(x)=5)$: Given these conditions $P(a)\cdot P(b)\lt0$ $P'(x)\gt 0$ or $P'(x)\lt 0$ when $x\in (a,b)$ $P'(a) = P'(b) = 0$ I know ...
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2answers
93 views

Series representation of polynomial root

Consider the quintic: $$(1+q)x^5-(2+3q)x^4+(1+3q)x^3-x^2+2x-1=0$$ for $q=0$, this equation has a triple root: $$(1-x)^3(x^2+x+1)=0 \implies x=1.$$ Kopal (1959) then states that, for small $q$, a ...
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2answers
76 views

Given a polynomial with roots $a, b, c, d, e$, find the polynomial whose roots are $abc, abd, abe, …$

Let $p(x)=x^5-4x^4+3x^3-2x^2+5x+1$ and say $a, b, c, d, e$ are the roots of $p$. Find the polynomial whose roots are $abc, abd, abe, acd, ace, ade, bcd, bce, bde, cde$. By Viete's theorem we just ...
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33 views

Are there any examples of using Bring radicals to solve a quintic equation?

It says here on Wikipedia that the general quintic equation can be solved if it is reduced to the principal quintic form, then to the Bring–Jerrard normal form. It gives a good overview and after ...
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3answers
113 views

There is no solution in radicals to general polynomial equations of degree five.

I have read that there is no formula for the quintic equation. Tow questions: 1- What does that even mean? By the fundamental theorem of algebra, any equation of degree n has n solutions in the ...
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1answer
75 views

Special case quintic equation analytic solution

I've met this quintic equation in my research: $$x^5 - \frac{k}{k-1} \cdot x^3 +\frac{r}{k-1}=0$$ with the additional conditions: $$k>1; \quad 0<r<1; \quad 0<x<1$$ My background in math ...
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2answers
100 views

How to factor $x^6-4x^4+2x^3+1$ by hand?

I generated this polynomial after playing around with the golden ratio. I first observed that (using various properties of $\phi$), $\phi^3+\phi^{-3}=4\phi-2$. This equation has no significance at all,...
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2answers
168 views

How can I solve, for example, $n^5-625n+1632=0$ for $n$ if it is solvable?

I understand that some quintics in Bring-Jerrard form are solvable but first one must identify a solvable group for it or any quintic. I don't know how to identify a group for this equation or how to ...
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1answer
42 views

how should Cayley's resolvent be used in order to see if a given quintic equation is solvable or not?

Hi. I chose a simple equation to work with: $x^5+x^4+x^3+x^2+x+1=0$ (which is solvable and $x=-1$). the amounts of $p$, $q$, $r$ and $s$ where $\frac{3}{5}$, $\frac{14}{25}$, $\frac{87}{125}$ and $\...
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2answers
262 views

Is there alternative factoring of a quintic equation?

In a paper here the author appears to be able to factor a Bring-Jerrard quintic making $$P=2mn(m^2-n^2)(m^2+n^2)=2m^5n-2mn^5\\ \implies n^5-m^4n+\frac{P}{2m}=0 \rightarrow x^5+px+q=0$$ become $$(x^3+...
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0answers
81 views

Solving a family of quintics

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 ...
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4answers
584 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 ...
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0answers
153 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-...
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1answer
100 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^...
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0answers
40 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\...
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0answers
67 views

Why does transcendentals appear in the proof for Insolvability of Qunitic?

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 ...
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1answer
140 views

How to factorize the quintic polynomial $2x^5 + 6x^4 + 7x^3 + 21x^2 + 5x + 15$? [closed]

The polynomial is $$2x^5 + 6x^4 + 7x^3 + 21x^2 + 5x + 15$$ I want to find out the easiest way I can do factorize. Please show me the steps.
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1answer
44 views

How to solve a quintic polynomial using elliptic functions with Mathematica?

I followed the exact steps from this forum post to solve quintic polynomials of the form: $x^5 - x + d$ But I got a different answer in number form from Mathematica than the original quintic ...
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0answers
60 views

Is it possible to solve this reduced quintic equation?

Since there is no general formula for quintic equations, is there for a formula for quintic equations of the type: $$ax^5+bx^3+cx+d=0$$ or is it at least possible to solve: $$16x^5-20x^3+5x-q=0$$ in ...
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39 views

quintic residue formula

let $\zeta_5$ a $5$-th primitive root of unity, and $\pi$ a prime in $\mathbb{Z}[\zeta_5]$ above $p$, I need to simplify the quintic residue formula: $\zeta_5^i(1+\zeta_5)^j \equiv X^5 (mod \pi$) with ...
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0answers
60 views

Converting a root of a principal quintic to a root of the corresponding general quintic

I'm working on an implementation of a numeric solver for quintic equations and have a question regarding usage of the principal form. Say you have some quintic equation in general form. For example, ...
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0answers
24 views

Where exactly Riemann surfaces and monodromy groups appear in this video?

This is regarding the proof of the unsolvability of the quintic equation (Abel-Ruffini theorem) which is explained really well in this short video. I also understand that the original proof by ...
3
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1answer
153 views

Arnold's proof for the insolvability of the quintic

I am trying to understand Arnold's proof for the insolvability of the quintic from the manuscript: https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf which is actually well written. ...
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0answers
75 views

When is the Galois group of a quintic not $S_5$ if we use this particular method?

I think that I am misunderstanding something fundamental about the technique used to decide if higher order polynomials are solvable by radicals using Galois theory. If we have a cubic it's not to bad ...
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0answers
54 views

How to solve for imaginary roots of a quartic that cannot be factored and has no real roots?

Given the equation $4x^5 + 3x^4 + 2x^3 + x^2 + x - 11 = 0$, how do I find all complex roots? Fundamental theorem of algebra states that since this is a 5th degree polynomial, there are $5$ roots. I ...
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1answer
41 views

Is quintic equations have anything to do with their coefficients?

I was reading about quintic equations and this came up: In the early nineteenth century, Paolo Ruffini (1765–1822) and Niels Henrik Abel (1802–1829) proved that no such general formulas ...
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0answers
54 views

Geometrical shape of the solutions of the quintic equation in 3d

I understand the geometric meaning of the solutions of the cubics and quartics in the plane using combinations of equilateral stars with 3 and 4 arms, respectively. I heard that quintic equations can ...
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1answer
98 views

Quintic polynomial with three real roots

I want to get a quintic polynomial $f(X) \in \mathbb{Q}[X]$ whose Galois group $\mathrm{Gal}(L/\mathbb{Q}) \cong S_5$ where $L$ is the splitting field of $f(X)$. One of strategies to get it is ...
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1answer
74 views

What do I need to learn, to understand Galois' solution of the quintic?

I studied maths to degree level many years ago, but never got a "gut feel" for any of the "pure maths" side, because the way it was taught didn't fit at all with how I learn. I focused on numerical ...
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4answers
283 views

Factor $x^5-x+15$

It's possible to factor $x^5-x+15$. WolframAlpha gives the answer of: $$(x^2+x+3)(x^3-x^2-2x+5)$$ According to the wikipedia article on quintic functions, the general form $x^5-x+a$ is factorable ...
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1answer
96 views

Equation with Galois group twisted $S_{3}$

I note that a Galois group is not just a Galois group. Let $r_{1}$, $r_{2}$, $r_{3}$, $r_{4}$, denote the roots of a quartic equation. Then $x^4-5x^2+6$ has Galois group $Z_{2}^2$, where the ...
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1answer
143 views

Number of real roots of a quintic equation

Find the number of connected components of the set $$\left\{x\in \mathbb R : x^3\left(x^2+5x-\frac{65}{3}\right)>70x^2-300x-297\right\}$$ under the usual topology on $\mathbb R$. We solve the ...
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1answer
135 views

Condition for Quintic Reciprocity

Let $p=x^2+11x-1=1\pmod 5$ be a prime. Show that $x$ is a quintic residue $\pmod p$. It holds for $x<200$ and should hold for all such $x$. Any proof ideas? Thanks in advance.
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2answers
222 views

Quintic reciprocity conjecture

Let $p=x^4 + 25x^2 + 125$ be a prime. Prove that $2$ is a quintic residue $\pmod p$, and therefore $y^5=2\pmod p$ is solvable. A similar example was first conjectured by Euler: If $p=x^2 + 27$ is a ...
5
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1answer
173 views

Show: If some $x_i \neq x_1$ occurs as $\sigma(x_1)$ for a $\sigma \in$ Gal$ (E:F)$ then each $x_i\neq x_1$ occurs as $\sigma(x_1)$

I'm currently studying Galois Theory (or am trying to do so), and since almost two weeks I try over and over again to solve a particular question from a textbook (J. Stillwell, Elements of Algebra, p. ...
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2answers
59 views

Is $x^5-10x^3+20x= 8.58368$ solvable?

This quintic has 5 real roots, how do we find out if it is solvable and ,in that case, how to solve it? Is there a generally valid numeric approach?
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2answers
191 views

Are there some methods to solve the quintic equation?

I'd seen it on wikipedia that no analytical solutions for quintic equation. However, I would like to ask are there some methods to solve it if we just consider the odd power terms shown as below: $ax^...
3
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5answers
137 views

Confirmation of Proof: $f(x)= x^5 - x - 16$ has at most three real roots.

I've been struggling to find a way to solve this question for a while now. I can prove that there is at least one real root using IVT but have no idea how to prove that there can exist more. ...
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2answers
179 views

Find all real solutions of the equation: $x^5+5x^3+5x+2017=0$

I joined the math contest $1$ week ago ( In Azerbaijan). There were $6$ questions. Unfortunately, I could solve $1$ question correctly.I know I can not write all the questions here, because It is ...
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0answers
75 views

Is the $t^5+15t+12=0$ equation congruent with the Galois group?

Is the quintic equation: $t^5+15t+12=0$ congruent with the Galois group? And is it true that Lagrange tells us that there are other methods, which that we do not know for the quintic equation?
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2answers
473 views

Why is there no solution with radicals for the quintic equation $x^5-x+1=0$? [closed]

I'm looking for solution for this equation $$ x^5-x+1=0. $$ I know that there is no solution with radicals. But, I can not find possible solutions (in MSE or internet resource). I know Abel-...
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1answer
48 views

Finding the sum of the squares of the roots of a quintic

I would like to solve the following problem. $a, b, c, d, e$ are the roots of $P(x) = x^5 − 160x − 128 = 0$. Compute $a^5+b^5+c^5+d^5+e^5$. The first thing that came to mind was expand $(a+b+c+d+e)...
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2answers
346 views

Determine the Horizontal tangents of f(x)

What are the steps used to find horizontal tangents of a function f(x) as defined: $$ f(x)=(x+1)^2 (2x-3)^3 $$ I know we have to calculate the derivative of the function, but how do we do that? (...
4
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1answer
641 views

Explanation of the Tschirnhausen transformation

I am studying the resolution of the quintic equations, which involves the so-called Tschirnhausen transform. The idea is to cancel the fourth and third degree coefficients by a change of variable of ...