# 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 ...
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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### 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 ...
1 vote
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### 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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### 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 ...
1 vote
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### 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 ...
1 vote
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### 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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### 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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### 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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### 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 ... 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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### 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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### 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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### 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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### 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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### 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 ...