# 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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### Relationship of the roots of two polynomials both which have the same constant and x term but vary in the x^3 term by a function of the x coefficient [closed]

Given that we know the roots of the first equation is there anyway to use them or the relationship between these two quintic polynomials to discern the roots of the second equation. is there any ...
1 vote
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### The transformation relating two quintic polynomials

I apologise if this a silly question. Is there a transformation relating the following two graphs: $x^5 + -3 \sqrt{\frac{e}{5}}x^3 + ex + f$ $x^5 + ex + f$ if so is there any way to derive it like ...
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### Skolem’s “Theorem 9” and the associated equation

In Diophantine Equations (page 209), Mordell writes: [Skolem] proved that the equation $$x^5+2y^5+4z^5-10xy^3z+10x^2yz^2=1$$ has at most six solutions in integers $x,y,z$, and of these three are ...
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### Root representation using the factorization of polynomials upto degree six

Is there some classical technique (from algebra or analysis) to find the expressions of roots of quintic and sextic using the root expressions of quadratic, cubic and quartic root expressions. For ...
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### The solution to the principal quintic via the Brioschi and Rogers-Ramanujan cfrac $R(q)$

In this interesting post, it quotes a paper that to write down the complete solution (not in radicals) of the general quintic, one would need "a piece of paper as big as a large asteroid". ...
286 views

### Clarifying the meaning of insolvability of quintic

Insolvability of quintic says that no analogue of quadratic formula is available for quintic equations. For quadratic equations, the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ gives you all the ...
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### How large is the gap in Ruffini's 1813 proof that there is no general quintic formula?

I'm reading Ruffini's final attempt at showing there is no general quintic formula which appeared in 1813 see here. (This is a much shorter proof than his first proof of 1799 in his Teoria generale ...
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### Quintic polynomial with only positive roots

The Abel-Ruffini theorem states that the solutions to a general polynomial equation of the form$$x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0$$ have no algebraic expression in terms of the coefficients of ...
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### Integrating $\int \frac{dx}{\sqrt{x^5+ax^3+bx^2+cx+d}}$.

$$\int \frac{dx}{\sqrt{x^5+ax^3+bx^2+cx+d}}$$ I know we have $E,K,$ and $\Pi$ to integrate quartics under the radical, but I wonder what functions quintics yield? Are there any known?
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### Can De Moivre's quintic be extended to other F20 quintics?

For the DeMoivre's quintic $x^5 - 5ax^3 + 5a^2x - b = 0$ exist formulas for the relations between the roots. When two roots, say $x_1$ and $x_2$, are known then one can find the other roots with the ...
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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 ...
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### 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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### 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 ...