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 ...
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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 ...
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-3 votes
1 answer
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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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3 votes
2 answers
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Finding the number of roots of $f(z) = z^5 + z^3 + 3z + 1$ in the unit disk

Suppose we have $$f(z) = z^5 + z^3 + 3z + 1$$ Find how many roots this function has in the open unit disc $\{z : |z| < 1\}$. Here's what I think about it: I tried to split $f$ into two functions $...
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Is there a systematic method to decompose this?

Suppose that $$x^5+ax^4+bx^3+cx^2+dx+e=0$$ with coef.. all integers, decompose as follows $$(x^2+Ax+B)(x^3+Cx^2+Dx+E)=0$$ which leads to the following system $$ \begin{cases} C+A = a\\ ...
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1 vote
1 answer
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Can a quintic equation be solved without needing a "piece of paper the size of a large asteroid"(!)?

I remember from a long time ago reading a paper regarding the solution of quintic polynomial equations using hypergeometric functions. In particular, the methods are based around the solution of the ...
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2 votes
2 answers
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What would the field corresponding to a Galois group of $S_5$ look like?

I'm wondering what other tools there are aside from radicals can be used to extend fields in the context of solving polynomials. Since $S_5$ isn't solvable, constructing a field with a Galois group of ...
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Are general degree 5 polynomials solvable only with only elementary functions and the Lambert W function?

I know it is not solvable in terms of radicals, and I don't know if it is known if it can be solved with only elementary functions of any kind. But can it be solved using only elementary functions and ...
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2 votes
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Is there a proof showing super roots and super logarithms won't lead to a solution for the quintic?

So I am learning about tetrations and I just learned that tetrations are not elementary functions. When I heard that I remembered back to the statement that there is no general solution to the quintic ...
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A better way to find the radius of the complex roots of $(z + 1)^5 = 32z^5$?

I was gnawing on this problem today: All the complex roots of $(z + 1)^5 = 32z^5,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle. I solved this by first dividing ...
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9 votes
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The most generic radicals-solvable quintic

It's well known that it is impossible to solve a generic quintic equation in terms of radicals involving its coefficients. However: what's the "most generic" quintic equation that is still ...
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4 votes
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If a real number can be expressed in terms of complex solutions of quintics, can it be expressed in terms of real solutions of quintics?

I'm considering a notion of quintic-constructible numbers. This is a follow-up from the cubic case. Suppose a real number $\alpha$ is contained in a tower of fields $$\alpha\in\mathbb F_m\supset\...
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Real solution to a quintic equation

If $a$ is a real root of the equation $x^5-x^3+x-2=0$ such that $[a^6]=3k$. Find k. Here $[x]$ denotes the GIF. My try: First I plugged in $1$ in the equation. I got $-1$ which is close to $0$ so the ...
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0 answers
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Can a Quintic Polynomial Be Solved?

I am well aware of the fact that a general quintic polynomial cannot be solved in radicals. However, is there a known way to obtain a formula with other functions (e.g. infinitely nested radicals)?
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Expression of a unique real root of a quartic and a quintic

Is it possible to derive the formula for the unique root of a quartic equation[given in rational expression] $T(x)$ in the interval $]-1,1[$, knowing that the quartic has in fact two real roots and ...
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5 votes
1 answer
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Arnold's Topological Proof for the Insolvability of the Quintic

It seems similar questions have been asked about these notes by Leo Goldmakher (linked below) which give a topological proof by Arnold of the insolvability of the quintic by radicals. I think I ...
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4 votes
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Solving $x^5-x-a=0$ by inverse function theorem?

If you can find a differential equation of the form $$y'=F(y)$$ for a quintic $y$, its inverse would be expressed as $$x=\text{const.}+\int\frac{dy}{F(y)}.$$ Differentiating any quintic yields a ...
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1 vote
2 answers
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$\alpha,\beta$ are roots of the equation $3x^2-(m-2)x+(m-5)=0$ such that $\alpha^5+\beta^5=33$. Find the value of $m$.

$\alpha,\beta$ are roots of the equation $3x^2-(m-2)x+(m-5)=0$ such that $\alpha^5+\beta^5=33$. Find the value of $m$. $$\alpha+\beta=\frac{m-2}3$$ Squaring and cubing it one by one and then ...
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1 vote
1 answer
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Sketchable parts of infinite formulas for solutions of some quintic equations

According to Abel-Ruffini Theorem and Galois Theory, you cannot solve the general quintic equation by radicals - i.e., it is impossible to express its solutions by a finite number of additions, ...
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On Dummit's paper

I have been recently reading Dummit's paper on solvability of quintic polynomials. At certain point, he mentiones that all results are valid "over any field $K$ of characteristic different from 2 ...
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3 answers
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Find all roots of a quintic polynomial given 1?

Let's say I have solved for one of the roots $a$ of a quintic equation using Newton's method. I read somewhere that you can "simply" divide the equation by $x - a$ to get a quartic and solve ...
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1 vote
1 answer
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Is there any existing standard name for primes of the following kind in mathematics?

Given a prime $p$ all primes $q$ such that $$q \bmod p = 1\text{ and }2^{(q-1)/p} \bmod q = 1$$Do primes $q$ have any standard name ?
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1 vote
1 answer
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Help Solving Multivariable Quintic Function

I am developing a game, and I am working out some of the equations for one of the features of it. Essentially, I am adding a way for a character's skills to be inherited when they die, and this ...
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0 votes
1 answer
118 views

Quintic Equation

I got problem written as this, For solvable quintic $$ \frac{32}{7} \cos^5 x - \frac{32}{7}\cos^3 x-\frac47 \cos^2 x + \frac{22}{7}\cos x - 1 = 0, $$ show that one of the cosine function is $\cos x = ...
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4 votes
1 answer
263 views

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 ...
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10 votes
2 answers
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 ...
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2 votes
1 answer
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 ...
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4 votes
1 answer
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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3 votes
0 answers
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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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3 answers
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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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2 answers
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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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0 answers
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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 ...
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2 votes
3 answers
208 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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0 votes
0 answers
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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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5 votes
2 answers
147 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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1 vote
2 answers
259 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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1 vote
0 answers
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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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3 answers
621 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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  • 1
0 votes
1 answer
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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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3 votes
2 answers
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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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-1 votes
2 answers
207 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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1 vote
1 answer
233 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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3 votes
4 answers
356 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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0 votes
0 answers
138 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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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 ...
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  • 5,066
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-...
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  • 5,066
1 vote
1 answer
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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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  • 5,066
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\...
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3 votes
0 answers
77 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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