Questions tagged [cubics]
This tag is for questions relating to cubic equations, these are polynomials with $~3^{rd}~$ power terms as the highest order terms.
1,268
questions
0
votes
0
answers
26
views
Are there any ways to convert inverse trigonometric values to radicals?
When we solve a cubic equation $ax^3+bx^2+cx+d=0$, the roots are supposed to be in the form of radicals in real numbers or complex realm. However, if the discriminant is less than 0, the solution is ...
- 99
1
vote
0
answers
97
views
Solving Vieta's jumping problem (1988 IMO problem 6) with a cubic polynomial
I have recently watched this Numberphile video explaining how Zvezdelina Stankova solved the notorious 1988 IMO problem 6 as a student.
(Most of you will know this problem)
Let $x$ and $y$ be ...
- 179
3
votes
0
answers
57
views
Regular heptagon coordinates from a cubic field
Find coordinates for a regular heptagon in 3D Euclidean space where all $3$ components $(x,y,z)$ of all $7$ coordinates are elements of the same cubic field, or prove that it can't be done.
Background:...
- 31
0
votes
1
answer
74
views
Showing that $P$ is unique
Suppose, we have the data points $f_a,f_b$ as well as $f_a',f_b'$. We have a cubic spline $P(x)=c(x-a)^3+d(x-a)^2+e(x-a)+f$ in $[a,b]$ where:$$P(a)=f_a, \, P(b)=f_b,\, P'(a)=f_a' \, \text{ und } P'(b)=...
- 2,097
1
vote
2
answers
70
views
How to determine the order of the real roots of a cubic equation?
This is a self-answered question (I didn't find a reference, and thought of documenting this). Consider the equation
$$
t^3+pt+q=0.
$$
Its discriminant is
$$
\Delta=-(4p^3+27q^2).
$$
Suppose that it ...
- 24.3k
1
vote
3
answers
83
views
How many roots of $x(1-x)^{2}=s$ are there in $(0,1)$?
This is a self-answered question, which is part of answering this related question. Alternative solutions are welcomed.
Let $0<s < \frac{4}{27}$. Prove that the cubic equation $x(1-x)^{2}=s$ has ...
- 24.3k
1
vote
2
answers
70
views
Is this what casus irreducibilis actually is?
Suppose we are working over the complex numbers $\mathbb{C}$. I know that every quadratic equation $x^2 + ax + b = 0$ with complex coefficients $a$ and $b$ has a solution expressible in terms of the ...
- 16.7k
1
vote
2
answers
83
views
Is this quadratic polynomial monotone at solutions of this cubic?
Let $0<s < \frac{4}{27}$. The equation $x(1-x)^{2}=s$ admits exactly two solutions in $(0,1)$: Denote by $a,b$ be these solutions, and suppose that $a<b$.
Does
$$
(1-a)^2+2a^2<(1-b)^2+2b^2
...
- 24.3k
9
votes
0
answers
190
views
On the solvable octic $x^8-44x-33 = 0$ and the tribonacci constant
I had discussed the solvable octic trinomial,
$$x^8-44x-33=0\tag1$$
way back in this old MSE post, but I revisited this inspired by another solvable octic,
$$y^8-y^7+29y^2+29=0\tag2$$
which I also ...
- 49.3k
-4
votes
1
answer
36
views
How is $\xi+2a\eta<0$ an "obvious necessary condition" for $y^3+2y^2(1-2a-\xi)+y(1-4\xi+8a\xi)-2\xi-4a\eta >0$ to be satisfied for positive $y$?
How is $$\xi+2\alpha\eta<0$$ an 'obvious necessary condition' for the inequality
$$y^3+2y^2(1-2\alpha-\xi)+y(1-4\xi+8\alpha\xi)-2\xi-4\alpha\eta >0$$
to be satisfied for positive $y$ (as claimed ...
0
votes
1
answer
181
views
Solving Cubic Systems of Diophantine Equations
What techniques are there for solving systems of Cubic Diophantine equations? I know there is no general purpose technique and looking at some papers it can quickly go over my head even for just a ...
- 3
2
votes
2
answers
50
views
Will the perpendicular bisector between the line connecting two cubic roots of the same arc never intersect its turning point?
The quadratic graph: $$ f(x) = (x+2)(x+1)$$
would have a midpoint between its roots at $x = -1.5$. This line would intersect its turning point.
However the cubic graph: $$ f(x) = (x+1)(x-2)(x+3)$$
...
- 21
0
votes
0
answers
33
views
Cubic Polynomial with leading coefficient 1 satisfying certain condition
I did translation which may differ from original question
The cubic function $f(x)$ with the coefficient of the leading term equal to 1 and when a function $g(x)$ that is continuous on a set ...
- 7,976
0
votes
1
answer
78
views
Further information on the reduction of cubic equations to a system of two conic sections
This question follows on from one I have previously asked, How to separate cubic equations into two conic sections: Deep dive into Omar Khayyam and I now would like some further advice on some ...
- 45
5
votes
1
answer
118
views
Find the value of: $\sqrt[3]{a+b}+\sqrt[3]{b+c}+\sqrt[3]{a+c}$
Let $a,b,c$ be roots of the cubic
$$x^3-x^2-2x+1=0$$
Then, find the value of:
$$\sqrt[3]{a+b}+\sqrt[3]{b+c}+\sqrt[3]{a+c}$$
My attempt.
I used the substitutions $$a+b=x^3, b+c=y^3, a+c=z^3$$
$$x^3+y^...
- 614
2
votes
1
answer
58
views
Spivak, Ch, 25, 2(v): Is there some specific technique to factorize $x^3-x^2-x-2$ or must one guess that 2 is root?
The following problem appears in Ch. 25, "Complex Numbers" of Spivak's Calculus
2 (v) Solve the equation $x^3-x^2-x-2=0$.
Is there some specific technique to factorize this? Must one ...
- 4,030
1
vote
2
answers
78
views
Does an invertible real matrix have a real cubic root
$A$ is an invertible real matrix, is there a real matrix $B$ such that $A=B^3$?
For the complex case, it is not difficult. Over complex numbers, every invertible matrix has a cubic root. In fact, ...
- 2,465
-1
votes
1
answer
75
views
If $ax^3+bx-c$ is divisible by $x^2+bx+c$ then $a,b,c$ are in what kind of progression? (arithmetic/geometric/etc) [closed]
If a polynomial $$f(x)=ax^3+bx-c$$ is divisible by the polynomial $g(x)=x^2+bx+c$, then $a,b,c$ are in ...
$1.$ Arithmetic Progression
$2.$ Geometric Progression
$3.$ Harmonic Progression
$4.$ ...
- 1
0
votes
0
answers
35
views
When we say the odd degree polynomial has odd number of real roots, is there any condition on the coefficients?
I read that if the degree of a polynomial equation is odd then the number of real roots will also be odd.
I took the example of a cubic equation. If it has imaginary roots then that will occur in pair....
- 6,910
2
votes
1
answer
60
views
Stationary points of a cubic function
If t is a positive constant, find the local maximum and minimum values of the function
$f(x) = (3x^2 - 4)\left(x - t + \frac{1}{t}\right)$
and show that the difference between them is $\frac{4}{9}(t + ...
- 33
4
votes
1
answer
82
views
Inverse of a cubic with some constraints
I'm trying to find the inverse of a particular cubic equation over a certain range. I have a partial solution and I'm not sure what's missing to make it complete. I'd like to solve the following ...
- 210
0
votes
2
answers
137
views
Integer solutions of the cubic equation $x^3-a^2bx-1-2ab=0$
Given the equation $x^3-a^2bx-1-2ab=0$. Is there a way to know if any integer solutions exist for $a,b$ integers greater than 1.
I've plotted graphs and tried to brute force it but found no solutions.
...
- 607
1
vote
0
answers
55
views
Counting cubic congruence modulo powers of $3$
I am trying to count the number of solutions to the following congruence relation.
$$x_1^3+x_2^3+x_3^3+\cdots+x_k^3 \equiv 0 \bmod 3^{\lambda}$$
where $0\leq x_i<3^{\lambda}~~\forall ~i \in [1,k]$
...
- 111
0
votes
3
answers
312
views
Solve the equation $x^3-11x^2+38x-40=0$, given that the ratio of two of its roots is $2:1$.
Solve the equation $x^3-11x^2+38x-40=0$, given that the ratio of two of its roots is $2:1$.
By hit and try, I can see that $x=2$ is a root.
Dividing the given cubic by $x-2$, I get a quadratic whose ...
- 6,910
3
votes
2
answers
103
views
Find the total number of roots of $(x^2+x+1)^2+2=(x^2+x+1)(x^2-2x-6)$, belonging to $(-2,4)$.
Find the total number of roots of $(x^2+x+1)^2+2=(x^2+x+1)(x^2-2x-6)$, belonging to $(-2,4)$.
My Attempt:
On rearranging, I get, $(x^2+x+1)(3x+7)+2=0$
Or, $3x^3+10x^2+10x+9=0$
Derivative of the cubic ...
- 6,910
2
votes
1
answer
31
views
Solving Bezier cubic derivative for t
Getting the derivative of the cubic Bezier curve:
$P(t)=P_0(1-t)^3+P_13t(1-t)^2+P_23t^2(1-t)+P_3t^3$
Produces the following:
$P'(t)=3(-P_0-2P_1)+6t(P_0+P_1+P_2)+3t^2(P_3-P_0)$
Assuming P'(t)=0, is ...
- 33
5
votes
1
answer
60
views
Finding a relationship between coefficients to solve a cubic polynomial
From E.J Barbeau's Polynomials, the question states:
Find the relationship between $p$ and $q$ in order that the equation $x^3 + px + q$ may be put into the form $x^4 = (x^2 + ax + b)^2$. Hence, solve ...
- 305
1
vote
1
answer
18
views
please help me solve this question about cubic graphs and their intersection points
I was practicing some maths from the UKMT website's mentoring scheme sheets and I found this question on the sample for sheet 9:
On the same axes, sketch the graphs
$y=(x+1)^3-(x+1)$
$y^2=(x+1)^3-(x+1)...
- 61
4
votes
1
answer
221
views
Is there the continuous real root of the cubic equation and is there a closed formula to present it?
For any cubic equation, $ax^{3}+bx^{2}+cx+d=0$, we know there is always a real root if $a,b,c,d$ are all real. Suppose that $a,b,c,d$ are continuous and real function with respect of $i\in \mathbb{R}$,...
- 91
6
votes
3
answers
347
views
Reducing $ax^6-x^5+x^4+x^3-2x^2+1=0$ to a cubic equation using algebraic substitutions
Use algebraic substitutions and reduce the sextic equation to the cubic equation, where $a$ is a real number:
$$ax^6-x^5+x^4+x^3-2x^2+1=0$$
My attempts.
First, I tried to use the Rational root ...
- 614
2
votes
0
answers
57
views
Deforming roots of a cubic polynomial
This is problem 9-1 from Milnor, dynamics of one complex variable (arxiv).
Let $f_{\alpha}(z) = z + \alpha z^2 + z^3$. Show that $f_{\alpha}$ can be perturbed so that the double fixed point at the ...
- 143
0
votes
0
answers
21
views
Area (quadratics) go negative vs. Volume (cubics) goes to infinity
Consider a typical "fenced garden" problem, which is easily represented by a quadratic, e.g.,
$a(w)=-10w^{2}+100w$ where $w$ is the width (or the wall, if you prefer).
which peaks at $(5,250)...
- 113
0
votes
0
answers
19
views
Is there a graphical intuition behind the Vieta's substitution for reduction of a depressed cubic to a quadratic? [duplicate]
Like for the first substitution from a general cubic equation to a depressed cubic, a substitution is made so that the inflection point of the cubic equation changes to the y axis, so that it ...
- 161
2
votes
1
answer
45
views
Is there an analytic solution to this set of cubic equations?
I have three parameters $(a,b,c)$ that define equations I am looking for an analytical solution.
$$q_1=b/c$$
$$q_2=\frac{(a+b)^3}{3c^2b}$$
$$q_3=\frac{(a+b)^3}{3b^3}$$
I want to solve these equations ...
- 401
3
votes
2
answers
90
views
Unsure how to simplify
I have the following equation:
$$\frac{0.8}{x+1}+\frac{1}{(x+1)^2}+\frac{1.2}{(x+1)^3} = 2.5$$
According to WolframAlpha, it can be simplified as follows, assuming $x$ is real:
$$\frac{1.09047}{x + 1} ...
- 61
3
votes
3
answers
90
views
Let $r,s,t$ are roots of the cubic equation $x^3+bx^2+cx+d=f(x)$ then write down $D=((r-s)(r-t)(s-t))^2$ in terms of $b, c, d$
Let $r,s,t$ are roots of the cubic equation $f(x) = x^3+bx^2+cx+d$ then write down $D=((r-s)(r-t)(s-t))^2$ in terms of $b,c,d$.
Is there any clever way to solve it?
We know $r+s+t=-b, rs+st+rt=c, rst=...
- 8,208
2
votes
3
answers
158
views
Solving Special Case of Cubic Equation
Given a cubic equation of standard form $ax^3 + bx^2 + cx + d = 0$ where $a, b, c, d$ are real numbers, if we know that there is only one real root $x_r$, we know $x_r>0$, and we only care about ...
- 31
3
votes
1
answer
77
views
Real solutions to the depressed cubic equation [closed]
How can I find the allowed domain to this depressed cubic inequality
$$x^3 - 3 x + 2 \cos(\frac{3 \sqrt{3} n}{2}) \geq 0$$
where $n$ is a real non-negative number. Using Cardano's method, I can obtain ...
- 393
1
vote
1
answer
48
views
Finding Lattice points on a Cubic
I want to study the rational points on a cubic. Eventually I found Nagell's algorithm from http://webs.ucm.es/BUCM/mat/doc8354.pdf, but I cannot immediately apply it because I don't know a rational ...
- 744
1
vote
1
answer
70
views
Coloring faces , vertices ,edges of a cube using $3$ colors
We have a classical cube with six faces , eight vertices and twelve edges . We want to color faces ,vertices and edges of this cube using three colors. How many non-equivalent coloring are there ?
...
user1088166
5
votes
0
answers
98
views
Factorization of $x^3-x-1$ over $\mathbb{F}_p$
Factoring quadratic polynomials over finite fields can easily be done by determining if the discriminant is a quadratic residue modulo characteristic in question, and if so, apply the quadratic ...
- 395
-1
votes
2
answers
80
views
is there a cubic or quartic approximation for these data points? [closed]
Is there a cubic or quartic approximation for these data points; $$(0,1000000), (1000000, 100), (10000000, 10)$$ whilst also ensuring that no point from $$1000000 < x < 10000000$$ is greater ...
- 77
0
votes
3
answers
88
views
Finding all possible roots of the equation
Find all possible solutions to the equation $$(x^3-x)+(y^3-y)=z^3-z$$ where $(x,y,z)\gt1$ and $\in\mathbb{Z}$ and not all three of them are equal.
The original question didn't have the last condition ...
- 1
2
votes
1
answer
109
views
Collinearity of the three roots of a cubic equation in the complex plane
The three roots of the equation $x^3+bx+c=0\;(b,c\in\Bbb C,c\ne0)$ in the complex plane are collinear iff$$k\in\Bbb R\land k\le-\frac{27}4$$where $k=\frac{b^3}{c^2}$.
Original post on Math ...
- 487
2
votes
2
answers
139
views
Solving the system $x^3+y= 3x+4$, $2y^3+z = 6y+6$, $3z^3+x=9z+8$
Solve the system $$\begin{equation} \label{equation1}
\begin{split}
x^3+y= 3x+4 \\
2y^3+z = 6y+6 \\
3z^3+x=9z+8
\end{split}
\end{equation}$$
By the theorem of triviality, I assumed $x=y=z=k$ and ...
- 1
-1
votes
1
answer
46
views
Prove that if $a+b\sqrt[3]{2} + c\sqrt[3]{4} = 0$ then $a=b=c=0$, where $a,b,c$ are rational numbers [duplicate]
Trying this for a while and, I found out that if there is another solution $(a_0,b_0,c_0)$, then there is an infinite amount of solutions $(a_0/2,b_0/2,c_0/2), (a_0/4,b_0/4,c_0/4) \ldots$, but I am ...
2
votes
1
answer
128
views
Find roots of a cubic equation
I am following a method to solve a cubic equation, which is given by:
$x^3 + px + c = 0$
For the equation above they set:
$\epsilon = (\frac{c}{2})^2 + (\frac{p}{3})^3 $
and state that if $\epsilon &...
- 175
1
vote
0
answers
37
views
Estimate root scale of a cubic equation
Consider the following cubic dispersion equation of $\omega(k)$ $$\omega^2-\omega_a^2(k)-\frac{\alpha^2k^4}{\omega-\omega_b(k)}=0.$$
$\omega_a=vk,\omega_b=b-v'k$ are two unhybridized dispersions that ...
- 766
1
vote
2
answers
90
views
Is the theory of $\Delta\le0$, true in cubic functions$?$
If $ax^2+\frac{b}{x}\ge c$ $\forall x>0$ where $a>0 \:\: , b>0$ Show that $27ab^2\ge4c^3$
My work:
Let a function $f(x)$ be $$ax^2+\frac{b}{x}\ge c$$ or we can rewrite $f(x)$ as $$ax^3-cx+b\...
user1080568
3
votes
0
answers
85
views
About extreme values of $\{f(x)-x\}^2$ when $f(x)$ is a cubic function.
$t \ge 6$, $t \in \mathbb{R}$
$f(x) = \frac{1}{t}\left( \frac{1}{8}x^3 + \frac{t^2}{8}x+2\right)$
$\{f(x)-x\}^2$ has an extreme value on $x = k$
Sum of such $k = g(t)$
$g(p) = -1$ for some $p \in \...
- 341