# Questions tagged [quartic-equations]

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.

258 questions
Filter by
Sorted by
Tagged with
3answers
167 views

### Is there a quartic or quintic formula? [duplicate]

I know about the quadratic formula, and the cubic formula, so I was wondering if there were any more. My teacher said there was no such thing as a quintic formula, so I was wondering that if there was ...
1answer
55 views

### Solving $6x^4+2x^3+4x^2-6x-3=0$

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 ...
0answers
20 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 ...
2answers
54 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 ...
1answer
70 views

### 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)(...
3answers
155 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: $$x^3(x-2)-x(x-2)+1=0$$ $$x(x-2)(x^2-1)+1=0$$ $$(x+1)(x)(x-1)(x-2)+1=0$$ ...
1answer
60 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 ...
4answers
94 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), ...
1answer
50 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, \...
2answers
172 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 ...
1answer
207 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 $a$, $b$ ...
1answer
18 views

### 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.
0answers
16 views

### 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 $H$...
3answers
84 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 $$(x^2 + 5x + 2)(x^2 + 7x + 9)$$
2answers
51 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. ...
2answers
40 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?
5answers
99 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 ...
1answer
20 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 ...
3answers
78 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 ...
2answers
27 views

### Quartic function sharing three common roots with another function

So the question is "The quartic function f(x) = (x^2+x-20)(x^2+x-2) has three roots in common with the function g(x) = f(x-k), where k is a constant. Find the two possible values of k." So ...
1answer
52 views

### Roots of a quartic form a geometric progression [closed]

Determine all real values of the parameter $a$ for which the equation $$16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0$$has exactly four distinct real roots that form a geometric progression.
0answers
109 views

3answers
522 views

### Find complex roots of quartic function $(3z + 1)(4z + 1)(6z + 1)(12z + 1) = 2$

I found a math problem involving complex number Find all complex number z such that $$(3z + 1)(4z + 1)(6z + 1)(12z + 1) = 2$$ The complex number form is z = a + bi If I multiply all the factor ...