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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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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5 votes
1 answer
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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 ...
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-1 votes
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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 ...
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2 votes
1 answer
90 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 &...
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1 vote
0 answers
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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 ...
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2 votes
2 answers
85 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
79 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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4 votes
2 answers
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What does $f(x)f(-x) \le 0$ mean when $f(x)$ is a cubic function?

Like the title, what does $\forall x\in \mathbb{R}, f(x)f(-x) \le 0$ say about $f(x)$, when $f(x)$ is a cubic function? The book says $f(x)$ has to be the odd function, but I can't figure out rigorous ...
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2 votes
1 answer
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On Réalis’s solution of the “cubic Markov equation”

I am interested in the Diophantine equation $$X^3+Y^3+Z^3=3XYZ$$ and its solutions. In Nouv. Corr. Math. 1879 (page 7), Réalis claims that, other than the trivial solution $x=y=z$, the complete ...
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0 votes
1 answer
72 views

Calculating the roots of a polynomial with parameters

I'm dealing with a modelization problem and i've arrived at one point where i need to calculate the roots of a cubic polynomial without specifing their value: $$ Px^{3}-(Pb+RT)x^{2}+ax-ab=0 $$ I know ...
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1 vote
2 answers
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Missing solutions from complex cubic root

From Cardano's work in the 1600s, we have this famous example of a cubic polynomial equation: $$x^3-15x-4=0.$$ Plugging the coefficients into the Cardano-Tartaglia formula, we get an expression for ...
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Meaning of $\left| (f\circ f)(x) \right|$ is not differentiable only on single point, when $f(x)$ is a cubic function?

$f(x)$ is a cubic function, and its leading coefficient is $1$. $f(x)$ intersects with $x$-axis only on $(1,0)$. $f'(1) \ne 0$. $\left| (f\circ f)(x) \right|$ is not differentiable only on $x=2$. ...
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2 votes
2 answers
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How to map cubic pixels to a 2d grid maintaining distance relationships

I have only basic math understanding, so please excuse any lack of clarity. I want to crochet an afghan using all the possible 3-color combinations of 6 yarns. Or maybe that's ridiculously too many, I ...
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2 votes
3 answers
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How does one show this complex expression equals a natural number?

We have: $$\left(\frac{10 }{3^{3/2}}i-3\right)^{1/3}+ \frac{7}{3 \left(\frac{10}{3^{3/2}}i-3\right)^{1/3}}=2$$ This comes from solving the cubic equation of $x^3-7x+6=0$ which factors as $(x-2)(x-1)(x+...
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2 votes
1 answer
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Getting the area between $y$-axis, $f(x)$, $f(2-x)$ when both function is given by their subtraction?

For a polynomial $f(x)$, let function $g(x) = f(x) - f(2-x)$. $g'(x) = 24x^2 - 48x + 50$ What is the area between $y=f(x)$, $y=f(2-x)$, and $y$-axis? My approach: $g(1) = f(1) - f(1) = 0$. From $g'(...
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3 answers
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Where does the root of this equation lie$?$

If $0\le p\le16$, then the equation $x^3-12x-p=0$ has one root in $(2,3)$ $(3,4]$ $(4,5)$ None of these My work: I know that the function $f(x)=x^3-12x-p=0$ is strictly increasing in the ...
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2 votes
1 answer
47 views

Stuck on problems about differential / derivatives

For the real number $t$ that $t \ge 6 $, let $f(x) = \frac{1}{t} \left( \frac{1}{8}x^3 + \frac{t^2}{8}x + 2 \right)$. Let the sum of all real numbers $k$ satisfying the following condition be $g(t)$ : ...
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  • 341
1 vote
0 answers
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Solving a Type of System of Cubic Equations

Is there a closed-form solution to the following type of system of cubic equations? $$ x_j=\sum_{i=1}^na_{ij}x_i^3,\quad\forall\,j=1,...,n $$ Here the $a_{ij}$'s are constants, and the $x_i$'s are the ...
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6 votes
2 answers
212 views

Integration formula for cubic polynomial $\int_a^bq(x)dx=\frac{b-a}{2}(q(b)+q(a))-\frac{(b-a)^2}{12}(q'(b)-q'(a))$

Show that $\forall a,b\in \mathbb{R}$, with $a<b$, we have$$\int \limits _a^bq(x)\,dx=\frac{b-a}{2}(q(b)+q(a))-\frac{(b-a)^2}{12}(q'(b)-q'(a)),$$ where $q\in \mathcal{P}_3$ is a cubic polynomial. I'...
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  • 75
-3 votes
1 answer
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Can any cubic polynomial be transformed into canonical form? [closed]

Can any cubic polynomial be transformed from $Ax^3+Bx^2+Cx+D$ to $a(b(x-h))^3 + k$? For example, how could $x^3+\frac{3x^2}{2}+\frac{x}{2}$ be transformed?
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2 votes
2 answers
83 views

Finding the roots of $x^5+x^2-9x+3$

I have to find all the roots of the polynomial $x^5+x^2-9x+3$ over the complex. To start, I used Wolfram to look for a factorization and it is $$x^5+x^2-9x+3=(x^2 + 3) (x^3 - 3 x + 1)$$ I can take it ...
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Confirming whether in the special case of roots nature cubic equation can be easily solved

Suppose we are given that $a+b+c = A $, $ab+bc+ca = B$ , $abc = C$ , We can observe that if we solve by eliminating two variables out of three(a,b,c) from three equations we would end up getting $x^3 -...
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-2 votes
1 answer
48 views

Condition for 2 real and distinct roots of a cubic equation

Determine the exact value of $k$ if the cubic equation $x^3 + kx +4 = 0$ has 2 distinct real roots
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2 votes
3 answers
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Is $a+5^{1/3}b+5^{2/3}c$ a root of any cubic polynomial in $\mathbb{Q}$?

For arbitrary $a, b, c \in \mathbb{Q}$, let $w := a + 5^{1/3} b + 5^{2/3} c$, is $w$ a root of any cubic polynomial in $\mathbb{Q}$? I guess the cubic polynomial always exists. But I am confused about ...
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1 vote
0 answers
46 views

How to find $x$ by $y$ in bezier curve.

I draw the lines connecting the different nodes on the screen with the cubic bezier curve. These lines must be suitable for interaction. So I need to know how close the user is to the line based on ...
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0 votes
0 answers
49 views

Meaning of $ g(x) = f(x-1) \times \left|\lim_{ h\to 0} \frac{f(x+h)-f(x)}{h}\right| $ has only one non-differentiable point?

$f(x)$ is a cubic function with leading coefficient of 1, $$ g(x) = f(x-1) \times \left|\lim_{ h\to 0} \frac{f(x+h)-f(x)}{h}\right| $$ $g(x)$ is not differentiable only at $x= -1$. $g(1)=0$, $f'(0)&...
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  • 341
0 votes
0 answers
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How to deal with functions like $|f(x) + x| = px$ when $f(x)$ is a cubic?

The problem is : There is a positive real number $p$, and a cubic function $f(x)$ with its leading coefficient $1$. Every real roots of an equation $f(x) = px$ is $\alpha - \beta$, $\alpha$, $2\beta$. ...
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  • 341
-1 votes
2 answers
30 views

finding the equation of a cubic function with 2 points [closed]

A cubic function is in the form $y=kx^3+c$. Find the equation if it passes through $(0,5)$ and $(2,-3)$. I’m not sure how i would work this out algebraically or if i even can without graphing.
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1 vote
1 answer
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What can be derived from the fact that $\lim_{x\to 0}\frac{f(x-3)f(x+3)}{x^3}$ is convergence?

Problem is : There is a cubic function $f(x)$ , with its positive coefficient. And, $\lim_{x\to 0} {\frac{f(x-3)f(x+3)}{x^3}}$ is convergence. there is only one natural number $k = k_1$ which makes ...
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  • 341
0 votes
1 answer
69 views

Polynomial and Complex Roots Problem

Suppose that there exist nonzero complex numbers $a,$ $b,$ $c,$ and $d$ such that $k$ is a root of both the equations $ax^3 + bx^2 + cx + d = 0$ and $bx^3 + cx^2 + dx + a = 0.$ find all possible ...
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2 votes
3 answers
118 views

Solving $a^3 - 33 ab^2 = -217$ and $3a^2 b - 11b^3 = 18$?

I am looking for ways of solving systems like that: $$\left\{\begin{array}{lcl} a^3 - 33 ab^2 = -217 \\ 3a^2 b - 11b^3 = 18 \end{array} \right.$$ I've tried turning it into a system of 2 equations ...
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4 votes
2 answers
103 views

Systems of cubic equations [closed]

I am looking for a general yet tailor-made methods to solve system of equations like that $a^3 + 6ab^2 = 7$ and $3a^2b +2b^3 = 5$ that involve terms of $(a +b)^3$. Does this have a name?
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4 votes
0 answers
84 views

A cute way to solve the quadratic. How to extend it to the cubic?

Playing around with complex numbers, I found a cute way of solving the quadratic equation. Let's start with the (monic) equation \begin{equation} z^2+pz+q = 0 \end{equation} where $z$ and the ...
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0 votes
1 answer
56 views

Why doesn't the Cardano Formula yield the result expected by the inverse equation?

My application is about indoor bicycles. Sometimes I need to estimate the virtual speed for certain amount of Watts on the pedal or the other way around (how much watts would a certain speed require ...
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0 votes
3 answers
105 views

Constructing a tangent line to a point on the curve $y = x^3$ using a compass and straight edge?

There are number of ways of constructing a tangent line to the curve $y = x^2$ using a compass and straight edge. Does anyone know of a way of constructing a tangent line to the curve $y = x^3$ using ...
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0 votes
0 answers
43 views

Determining the roots of the cubic polynomial

Let's say we have a third-order polynomial given as follows: $P(u)=-u^{3}+a_{1}u^{2}+a_{2}u-a_{3}^{2}$, where $a_{1},a_{2},a_{3}$ are all constants. In the specific problem that I am dealing with it ...
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15 votes
1 answer
204 views

Does the limit of the cubic formula approach the quadratic one as the cubic coefficient goes to $0$?

The formula for solving a cubic equation of the form $ax^3+bx^2+cx+d=0$ does not seem to yield the quadratic formula for the limit $\lim _{a \rightarrow 0} \text{(cubic formula)}$. But, if one tries ...
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  • 197
1 vote
2 answers
79 views

Solving the system $a^3 + 15ab^2 = 9$, $\;\frac 35 a^2b + b^3 = \frac 45$

I have problem solving the following system of two cubic equations. $$\begin{cases} a^3 + 15ab^2 = 9 \\ \frac 35 a^2b + b^3 = \frac 45 \end{cases} $$ I don't have any idea how to approach this kind of ...
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2 votes
1 answer
81 views

Discriminant of depressed cubic [duplicate]

If the cubic equation $x^3+px+q$ has roots $\alpha , \beta , \gamma $ then we know that $\alpha + \beta + \gamma =0 $, $\alpha \beta + \alpha \gamma + \beta \gamma =p $ and $\alpha \beta \gamma =-q $. ...
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  • 1,568
3 votes
1 answer
106 views

Solving a depressed cubic polynomial in modulus. [closed]

Is there a general technique for solving depressed cubic modulus polynomial? For instance, how would you solve the equation $a^3 + a + 21 = 0 \pmod{43}$?. My attempts eventually ended up with solving $...
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0 votes
2 answers
102 views

Factoring $x^3-6x^2+11x-6$ without using Rational Roots Theorem

To factor $P(x)=x^3-6x^2+11x-6$ one way is using Rational Roots Theorem and recognizing that $x=1$ makes $P(x)$ zero. But I want to factor without using it. I tried, $$x^3-6x^2+11x-6=x^2(x-6)+11(x-6)+...
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  • 1,113
0 votes
0 answers
74 views

Is this identity $ (1+6x^3+9x^4)^3+(1-6x^3+3x-9x^4)^3+(1-9x^3-6x^2)^3=9x^2+9x+3$ known with $x$ is an integer?

It is known that, let $x$ arbitrary integer $$(9x^4)^3+(3x-9x^4)^3+(1-9x^3)^3=1\tag{1}$$ discovred by Kurt Mahler in 1936, and $$(1+6x^3)^3+(1-6x^3)^3+(-6x^2)^3=2\tag{2}$$ discovred by A. S. ...
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3 votes
2 answers
61 views

Minimum value of M such that the cubic modulus value is always less than M for x in between -1 and 1 both included

Minimum value of $M$ such that $\exists a, b, c \in \mathbb{R}$ and $$ \left|4 x^{3}+a x^{2}+b x+c\right| \leq M ,\quad \forall|x| \leq 1 $$ What i considered was that putting x= 0, 1 and -1 we get ...
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  • 391
0 votes
1 answer
37 views

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
0 answers
16 views

What the extra 3 roots are of the 6th degree equation ? From which only the intial 3 are what we required in the problem

Question was to find the roots whose roots are square of the given roots of this polynomial $x^{3}+a x^{2}+b x+c=0$ I did used tranformation method to obtain $y^{3 / 2}+a y+b y^{\frac{1}{2}}+c=0$ but ...
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0 votes
2 answers
31 views

Finding equation whose roots are square of the original but intially some of them were negative .

Consider a cubic given as $x^3 + ax^2 + bx + c = 0$ which has roots $\alpha,-\beta,-\gamma$ , we need to find cubic equation whose roots are $\alpha^2 ,\beta^2 , \gamma^2$ is it possible to get it by ...
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1 vote
1 answer
34 views

Main Motivation behind the use of identity of $(a+b)^3$ ,so as to solve cubic equations using Cardons Method [closed]

Whats so special about the identity $(a+b)^3 = a^3 + b^3 + 3ab(a+b)$ which gave Cardano the clever idea to invent a method to solve cubic depressed equations?
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0 votes
1 answer
37 views

How to get the coefficients in a parametric cubic function

Let's say I have 4 points with x and y coordinates. And I want to determine the parametric cubic function: $x(t) = a_x t^3 + b_x t^2 + c_x t^3 + d_x$ and $y(t) = a_y t^3 + b_y t^2 + c_y t^3 + d_y$ So, ...
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0 votes
0 answers
27 views

2 Other Inverse Function of $f(x) = x^3+ax, \; a < 0$ for $x^2 \leq -\dfrac{4a^3}{27}$

For $a > 0$ the unique inverse of $f(x) = x^3+ax$ is $f^{-1}(x) = \dfrac{\sqrt[3]{9x+\sqrt{81x^2+12a^3}}+\sqrt[3]{9x-\sqrt{81x^2+12a^3}}}{\sqrt[3]{18}}$ But for $a < 0$ i know that this isn't ...
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0 votes
3 answers
114 views

Find the roots of $z^3 +3z^2 +3z +3=0$

Hello I have this problem: Find all $z \in C$ $z^3 +3z^2 +3z +3=0.$ With Mathematica I get 3 roots $z_1 = -1 -\sqrt[3]{2}$ $z_2 = -1 + (1 + i \sqrt{3})/2^{(2/3)}$ $z_3 = -1 + (1 - i \sqrt{3})/2^{(2/3)}...
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