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.

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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 ...
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1 vote
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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 ...
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3 votes
0 answers
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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:...
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1 answer
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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)=...
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1 vote
2 answers
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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 ...
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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 ...
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1 vote
2 answers
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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 ...
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1 vote
2 answers
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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 ...
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9 votes
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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 ...
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1 answer
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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 ...
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1 answer
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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 ...
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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: $$ f(x) = (x+1)(x-2)(x+3)$$ ...
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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 ...
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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 ...
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^...
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 ...
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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, ...
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1 answer
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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 $4.$ ...
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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....
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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 + ...
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 ...
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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. ...
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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 $0\leq x_i<3^{\lambda}~~\forall ~i \in [1,k]$ ...
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 ...
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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 ...
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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 ...
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 ...
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1 vote
1 answer
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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)...
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 $i\in \mathbb{R}$,...
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 ...
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 ...
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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)...
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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 ...
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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 ...
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} ...
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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=...
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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 ...
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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 ...
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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 ...
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1 vote
1 answer
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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 ? ...
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5 votes
0 answers
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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 ...
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-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 ...
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0 votes
3 answers
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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 ...
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 ...
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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 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 &...
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 ...
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\...
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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 \...
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