Questions tagged [quartics]

Equations that can be written in the form $p(x) = 0$ for a univariate polynomial $p$ of degree $4$ or $p(X_1, \ldots, X_r) = 0$ for a multivariate polynomial $p$ of total degree $4$. Questions that use this tag should usually also have the polynomial tag.

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How to get the general term for a quartic sequence really need help [closed]

Is it possible to find the general term for a quartic sequence and if so how do you do it? The sequence I am using is 1,9,36,100,225,441, 784, 1296, 2025, 3025 I am only interested in finding the ...
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1 vote
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Is it true that when a root of a 4th degree polynomial is natural, the radicals inside the formula are always rational?

Given the formula for the 4th degree polynomial, is it true that a root is a natural only when all the radicals inside the formula are rational numbers? Edit1: The coeficients are whole numbers. https:...
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0 votes
1 answer
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Mean of Cubic and Quartic forms of Gaussians

I am trying to calculate the following means: $$ E[ (x-\mu_k)b^T(x-\mu_l)(x-\mu_l)^T ] $$ $$ E[ (x-\mu_k)(x-\mu_k)^TA(x-\mu_l)(x-\mu_l)^T ] $$ Where x is some multivariate gaussian random variable. I ...
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1 vote
1 answer
77 views

System of Two equations with two unknowns of degree four

I am wondering if there is a direct way to solve exactly a system of two equations of this shape (the A to I are constants): $Axy + Bxy^2 + Cx^2y + Dx^2y^2 + Ex^2 + Fy^2 + Gx + Hy + I=0$ this problem ...
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2 answers
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Finding the roots of $x^4+x^3+2x^2+x-109=0$

The given equation is $$x^4+x^3+2x^2+x-109=0$$ I was thinking of using substitution method say $x^4=t^2$, but I wasn't sure what to do for the other terms. Using general quadratic theory we can find ...
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1 vote
1 answer
140 views

Roots of a fourth degree polynomial.

I am trying to solve this fourth-degree equation: $\omega^4+iB_3\omega^3+B_2\omega^2+iB_1\omega+B_0=0$, where coefficients $B_{0,1,2,3}$ are real, and $i$ is the imaginary number. The numerical values ...
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1 vote
1 answer
60 views

Factorizing a quartic expression to show that it is a perfect square. [duplicate]

Show that $\frac{a^4+b^4+(a+b)^4}{2}$ is a perfect square. I tried this, $$\frac{a^4+b^4+(a+b)^4}{2}$$ $$\frac{a^4+b^4+(a^2+b^2+2ab)^2}{2}$$ $$\frac{2a^4+2b^4+4a^2b^2+2(a^2b^2+2a^3b+2ab^3)}{2}$$ $$a^...
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2 votes
1 answer
61 views

Why does Klein's quartic curve have 24 heptagonal faces/56 triangular faces as a tiling?

I first learned about Klein's quartic from John Baez' blog: https://math.ucr.edu/home/baez/klein.html In it, he claims that if we want to wrap up the heptagonal tiling of the hyperbolic plane by "...
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2 votes
3 answers
83 views

Is the $\sqrt{\alpha+2y}$ in solving quartic equations with Ferrari method real?

Assume I want to solve $$ u^4 + \alpha u^2 + \beta u + \gamma = 0 $$ (only real-valued solutions are needed) for real $\alpha, \beta, \gamma \in \mathbb R$, $\beta \ne 0$ and assume I have already ...
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2 votes
1 answer
76 views

Quadratic form for quartics

This is an example of a quadratic form $$x_1 x_2 + x_1x_3=(x_1,x_2,x_3)\begin{pmatrix}0&\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&0&0\\\frac{1}{2}&0&0\end{pmatrix}\begin{pmatrix}x_1\...
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0 votes
0 answers
131 views

How to solve $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ if $x_{1,2} = x_0 \pm \sqrt{R^2 - (y - y_0)^2}$?

I'm trying to find the intersection points of circle and conic section curve. So, I'm solving the system of equations: $$ \begin{cases}Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \\ (x - x_0)^2 + (y - y_0)^2 =...
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  • 189
2 votes
2 answers
107 views

Solve the equation $x^4-8x^3+23x^2-30x+15=0$

Solve the equation $$x^4-8x^3+23x^2-30x+15=0$$ As $x=0$ is obviously not a solution, we can consider $x\ne0$, so I have tried to divide both sides by $x^2$ to get $$x^2-8x+23-\dfrac{30}{x}+\dfrac{15}{...
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  • 131
1 vote
1 answer
74 views

Which two roots of this quartic polynomial lie within the unit disk?

I have the following quartic polynomial, $$f(z) = z^4-rz^3+rsz^2-rz+1$$, where $r\in\mathbb{R}$ and $s\in\mathbb{C}$. Since this polynomial is palindromic, the roots can be easily computed. They are ...
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2 votes
0 answers
36 views

How to minimize quartic function for regularizing commutativity?

My objective is regularizing a function over $X\in \mathbb{R}^{I\times J}$ such that its covariance is jointly diagonalizable with a positive semi-definite matrix $A \in \mathbb{R}^{I\times I}$. For ...
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4 votes
2 answers
106 views

How can one solve polynomial equations with constant terms that have a high number of factors?

Today I was given this question on a test: $x^4 - 20x^3 - 20x^2 + 1500x - 9000 = 0$. Find the value(s) for $x$. I know how to solve these types of equations. I must record the positive and negative ...
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2 votes
4 answers
113 views

How to prove $X^{4}+X^{3}+X^{2}+X+1$ is irreductible in $\mathbb{F}_{2}$

How to prove $X^{4}+X^{3}+X^{2}+X+1$ is irreducible over $\mathbb{F}_{2}$. The main 2 "weapons" I have at my disposal is the Eisenstein criteria and reduction criteria but neither seem to ...
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0 answers
26 views

Condition for positive quartic polynomial with limited domain

I have the following quartic polynomial $$ p(x) = a x^4 + b x^3 + c x^2 + d x + e $$ Do you know any necessary and sufficient condition to ensure $p(x) \geq 0$ for any $x \in [0,1]$. I ready found ...
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0 answers
72 views

On expression of resolvent cubic of a quartic

Is it true that there is more than one resolvent cubic associated with a quartic, or are they equivalent? For example for a (reduced)quartic equation, i.e., $a_1=0$: $$x^4+a_1 x^3+ a_2 x^2+ a_3 x +a_4=...
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0 votes
0 answers
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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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2 votes
2 answers
59 views

$S$ , $P$ are sum and product of real roots of $x^4-7x^2-5=0$ what is the value of $2P^2-3SP+2S$?

If $S$ and $P$ are sum and product of real roots of $x^4-7x^2-5=0$ what is the value of $2P^2-3SP+2S$ ? $1)59-7\sqrt{69}\qquad\qquad2)7+\sqrt{69}\qquad\qquad3)50\qquad\qquad4)59+7\sqrt{69}$ I solved ...
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3 votes
5 answers
193 views

How to get roots in the form of a quadratic from the quartic $x^4 - 4x^2 + 16$?

So I need to factor this function into quadratics: $$x^4 - 4x^2 + 16$$ I know that there are only complex solutions to this question, however, it is still possible to obtain quadratic factors without ...
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3 votes
2 answers
245 views

Solving a quadratic equation problem with two variables

This is a post of two three problems regarding the method to solve bivariate quadratic equations. In brief, How does the elimination happen here. Or, how is the elimination used? (Update, I know how ...
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0 votes
2 answers
55 views

Solve a quartic function with three unknowns and two given roots.

I'd like to know how I could solve the following quartic function: $p(z)=2z^4+az^3+bz^2+cz+3$ given that it $2$ and $i$ should be part of their roots. I thought I should maybe be trying to turn this ...
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5 votes
3 answers
153 views

Resolve: $4\sin(2x)+4\cos(x)-5=0$

The first thing that comes to mind is to substitute $\sin(2x)=2\sin(x)\cos(x)$ and so we have: \begin{align*} 8\sin(x)\cos(x)+4\cos(x)-5=0 \end{align*} But after that I can't see what other identity ...
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0 votes
0 answers
72 views

Where Lagrange resolvent appears in solution of equations?

I saw method of Ferrari for solution of degree $4$ (with simple case, not involving cubic term): $x^4+px^2+qx+r=0.$ Method is as follows: complete the square from terms $x^4$ and $px^2$: $$x^4+2px^2+p^...
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0 votes
1 answer
67 views

Area enclosed by bean curve [closed]

I just found this interesting article on Wolfram Mathworld. https://mathworld.wolfram.com/BeanCurve.html I am interested in the following implicit equation: $(x^{2}+y^{2})^2=a(x^{3}+y^{3})$ (The curve ...
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  • 31
4 votes
2 answers
263 views

If $x^4 + ax^3 + 3x^2 +bx +1\geq0$, $\forall x\in \mathbb{R}$, find maximum of $a^2+b^2$

Given that $x^4 + ax^3 + 3x^2 +bx +1$ is always greater than equal to $0$ for all $x$ belongs to $\mathbb R$, find $\max(a^2 + b^2)$. What I did was to show that that above expression is equivalent ...
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2 votes
3 answers
136 views

Solving the equation $2x^2-\left(4x-7\right)\sqrt{x-1}-5x+4=0$

Solve the equation $$2x^2-\left(4x-7\right)\sqrt{x-1}-5x+4=0$$ Here is my work: $$2x^2-5x+4=(4x-7)\sqrt{x-1}$$ $$2(x-1)^2-(x-1)+1=(4(x-1)-3)\sqrt{x-1}$$ Using the substitution $\sqrt{x-1}=t$, $$2t^4-...
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  • 5,521
0 votes
1 answer
35 views

Functions from numerical solutions

Here is the context, I have numerically solved a cubic equation which is an irreducible case. Here is the polynomial I have solved: $P(x)=x^{3}+(1.45-2i\omega)x^{2}+(0.64 \lambda-w^{2}-1.9i\omega+1.06)...
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1 vote
0 answers
93 views

Solving the quartic polynomials

I have three quartic polynomials: $$1)\ x^4-(1+r+s+rs)x^2-2rsx+rs=0, s\geq r\geq 1,$$ $$2)\ x^4-(2q+qs+s)x^2-2qsx+qs=0,q\geq 2, s\geq 1,$$ $$3)\ x^4-(6+4s)x^2-8sx+4s=0,s\geq 2.$$ $q,r,$ and $s$ are ...
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4 votes
3 answers
254 views

If $\alpha$ be a multiple root of the order 3 of the equation $x⁴+bx²+cx+d=0$ then $α= 8d/3c​$

I found this statement online here and I think this statement is false.I would first prove a general statement and then I would use it to find $\alpha$ in order to discard the statement given in ...
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2 votes
2 answers
69 views

Let $f(x)=\int_0^x(4t^4-at^3)dt$ and $g(x)$ be a quadratic polynomial satisfying $g(0)+6=g'(0)-c=g''(0)+2b=0$, where $a,b,c$ are positive real numbers

Let $f(x)=\int_0^x(4t^4-at^3)dt$ and $g(x)$ be a quadratic polynomial satisfying $g(0)+6=g'(0)-c=g''(0)+2b=0$, where $a,b,c$ are positive real numbers. If $y=g(x)$ and $y=f'(x)$ intersects in $4$ ...
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  • 5,218
0 votes
1 answer
73 views

Solving $6x^4+2x^3+4x^2-6x-3=0$ [closed]

I'm having some trouble solving for $x$ in the following quartic equation. $$ 0=6x^4+2x^3+4x^2-6x-3 $$ Do you have any suggestions on how I should go about solving this equation? I tried using the ...
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3 votes
1 answer
92 views

Lagrange quartic resolvent $x_1+ix_2-x_3-ix_4$

Suppose we want to solve the "reduced" quartic equation $x^4+px^2+qx+r=0$ by means of Lagrange resolvent. I denote the roots by roots $x_1, x_2, x_3, x_4$; we have $x_1+x_2+x_3+x_4=0$. In ...
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2 votes
2 answers
66 views

$f(f(x))=a^3\left(x^2-(2+b)x+2b-\frac2a\right)\left(x^2-(2+b)x+2b-\frac ba\right)$, $a\ne0$ has exactly one real zeroes $5$.

Let $f(x)=a(x-2)(x-b)$, where $a,b\in R$ and $a\ne0$. Also, $f(f(x))=a^3\left(x^2-(2+b)x+2b-\frac2a\right)\left(x^2-(2+b)x+2b-\frac ba\right)$, $a\ne0$ has exactly one real zeroes $5$. Find the minima ...
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0 votes
1 answer
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Are these quartics polynomials having a real quadratic factor with complex conjugate roots?

I just want some help on some of these equations below to confirm whether they are polynomials having a real quadratic factor with complex conjugate roots? \begin{align} p(x)&=\frac{1}{3}(2x-6x+1)(...
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  • 3
2 votes
4 answers
183 views

How to solve $x^4-2x^3-x^2+2x+1=0$?

How to solve $x^4-2x^3-x^2+2x+1=0$? Answer given is: $$\frac{1+\sqrt5}{2}$$ I tried solving it by taking common factors: $$x^3(x-2)-x(x-2)+1=0 $$ $$x(x-2)(x^2-1)+1=0 $$ $$(x+1)(x)(x-1)(x-2)+1=0$$ ...
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  • 5,218
3 votes
1 answer
80 views

Solving complex quartic equation $z^5=1$

While finding the 5th roots of unity $z^5=1$, I arrived at the following $$(z-1)(z^4+z^3+z^2+z+1)=0$$ Now, I am well aware that I can arrive at the roots by using the fact that each root is separated ...
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2 votes
4 answers
295 views

Prove $X^4-2X^2+4$ is irreducible in $\mathbb{Q}[X]$

For some problem from my Galois Theory course, I need to prove that the polynomial $X^4-2X^2+4$ is irreducible in $\mathbb{Q}[X]$. I know it has no roots in $\mathbb{Q}$ (by rational root theorem), ...
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  • 3,420
0 votes
1 answer
52 views

Proving that $x + y + z < 3abcd$

Suppose $a,b,c,d$ are real numbers greater than $1.$ Given that \begin{align*} a + b + c + d &= -x \\ ab + ac + ad + bc + bd + cd &= y \\ abc + abd + acd + bcd &= -z \\ abcd &= 858, \...
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3 votes
2 answers
220 views

Limit of Newton's Method on polynomial $Ax^4 + Bx^3+ Cx^2 + Dx + E$?

So if you took the function $f(x) = Ax^4 + Bx^3+ Cx^2 + Dx + E$ and did Newton's Method repeatedly, you would get a sequence that converges to at most $4$ roots. I was wondering what would happen if ...
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9 votes
1 answer
249 views

Factorization $x^4+px^3+qx^2+r x +s=(x^2+a x +b)(x^2+\bar a x +\bar b)$

Question: Under what condition, does the quartic polynomial with rational coefficients $p$, $q$, $r$ and $s$ factorizes as $$x^4+px^3+qx^2+r x +s= (x^2+a x +b)(x^2+\bar a x +\bar b) $$ with $a$, $b$ ...
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0 votes
1 answer
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Complex quartic factorisation

Answer: I'm really not sure how to factorise. I understand that z.z* gives 2Re(z) but it's still not clear to me how it works.
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  • 51
1 vote
0 answers
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Diagonalizing 3-by-3 and 4-by-4 matrices using Givens rotations (solving 3rd and 4rth order polynomial equations)

The question is inspired by physics applications, where we are often interested in diagonalizing a Hamiltonian (a Hermitian matrix) by a unitary transformation: $$ S^\dagger H S = \Lambda, $$ where $H$...
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2 votes
3 answers
90 views

Factoring $x^4 + 12x^3 + 46x^2 + 59x + 18$

How do I factorize the following? $$x^4 + 12x^3 + 46x^2 + 59x + 18$$ I've tried looking for a root by trial and error to no avail. The answer is $$(x^2 + 5x + 2)(x^2 + 7x + 9)$$
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-1 votes
2 answers
54 views

Theory Of Equations : Prove that the roots are real [closed]

Prove that the roots of the equation $1/(x-1) +2/(x-2) +3/(x-3) =x$ ? Is real I have deduced by taking the recipocals and cross multiplying and its $4x^3-18x^2+10x+11 = 0$ not able to solve further. ...
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0 votes
2 answers
69 views

Determine a possible quartic polynomial equation such that $f(x) > 0$ for $-4 < x < -2$ and $3 < x < 7$

In typical high school fashion, nowhere in the curriculum was there a question about creating a polynomial equation. Yet here it is in the exam practice questions. Little help please?
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0 votes
5 answers
107 views

How to fully factorise $2x^4+7x^3+4x^2-4x$? [closed]

How to fully factorise $2x^4+7x^3+4x^2-4x$? I'm struggling to factorise polynomials like this one. I'm not sure how to best approach this problem. I've tried using the remainder and factor theorems ...
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0 votes
1 answer
60 views

Determining instantaneous rates of changes for quartic functions

I have 5 quartic functions that were found with quartic regression. Each function models a country's relationship with rotavirus vaccination rates (x) against years (y, years are integers and not real ...
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0 votes
3 answers
229 views

Solving a Quartic Function

A user named 'Uzdawi' from another post asked a question about how to solve the quartic function of One of the responses included an answer from the user 'Peđa Terzić', which is as follows: Could ...
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