These are polynomials with 3rd power terms as the highest order terms. Usually used with polynomial tag.

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What's the explanation for these (infinitely many?) Ramanujan-type identities?

Define the function, $$F(\beta)=\sqrt[3]{\beta+\sum_{k=0}^1 \cos\Big(\frac{2\pi }{7}\cdot3^{3k}\Big) }+\sqrt[3]{\beta+\sum_{k=0}^1 \cos\Big(\frac{2\pi}{7}\cdot3^{3k+1}\Big) }+\sqrt[3]{\beta+\sum_{k=0}...
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0answers
30 views

Finding the unique Nash equilibrium

$$ m^2(1-m)[(1+m)^2 - R] x^3 + [6m^2R + 12mR - 2m(1-m^2)(6+2m) - 4tm^2(1+m)^2] x^2 + [(1-m)(6+2m)^2 + 8tm(1+m)(6+2m)] x - 4t(6+2m)^2=0 $$ where: m ∈ (0, 0.5), R ∈ [0, 0.25], t ∈ [0, 1], x ∈ [0, 2]...
0
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1answer
31 views

Solving a Linear Recurrence with a Cubic Characteristic Equation

I've been learning Linear Recurrences in my Discrete Math course and I've learned how to solve them when the characteristic equation is a quadratic. Is solving a linear recurrence with a cubic ...
0
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0answers
17 views

Cubic's Irreducible case.

I know how to solve a cubic equation using Cardano's method, but some equation when solved through Cardano's method give complex numbers under a cube root sign. For Example: $x^3-15x=4$. I want to ...
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1answer
43 views

Why does the “printing neatly” algorithm use cubes rather than squares?

In Introduction to Algorithms, 2nd ed. (Cormen, Leiserson, Rivest, and Stein), ch. 15, Dynamic Programming, problem 15-2 Printing neatly (a copy of which is here), the official solution given in ...
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1answer
41 views

non-complex cubic roots formula?

Suppose we have a cubic equation $$ ax^3 +bx^2 +cx +d =0 $$ for which we know that all three distinct roots are real. Do we have a formula for them that does not involve complex roots of unity? The ...
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1answer
40 views

Definite eliptic integral with trigonometric method

I'm Looking for integrals of the following kind $\int\limits_{0}^{M} \left( a +bx^2 + cx^3 \right)^{-\frac{1}{2}} \,dx $ Where $M$ is on of the roots of the polynomial. I used the trigonometric ...
0
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1answer
51 views

Precision in Cubic spline interpolation

I am working on cubic spline interpolation with set of data points from CAD with following steps: Form piecewise spline equations between points. cubic equation : $ ax^3 + bx^2 +cx + d = P(x) $ ...
0
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0answers
18 views

Solving a cubic equation by factorization

I have the cubic equation $G^3 - (C - C_0)G - \frac{F}{\eta}$, with the constants $C, C_0, F, \eta \in \Re$. I want to find the solutions, so I tried factorising the equation, but I have after ...
2
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1answer
56 views

Deriving Formulae for the roots of the quartic and cubic polynomials

I have seen derivations of the general solution for the roots of fourth and third degree polynomials of 1 variable in Dummit & Foote's Abstract Algebra; however, it was by no means simple to me. I ...
13
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3answers
642 views

How would you find the exact roots of this equation?

My friend asked me what the roots of $y=x^3+x^2-2x-1$ was. I didn't really know and when I graphed it, it had no integer solutions. So I asked him what the answer was, and he said that the $3$ roots ...
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5answers
111 views

Difference between real and complex solutions of cubic equations [closed]

Take for an example, this equation. $$x^3+15x+4=0$$ This equation has two complex solutions and a real one. $$x≈0.1327-3.8798i$$ $$x≈0.1327+3.8798i$$ $$x≈-0.26542$$ What's extra in the complex ...
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1answer
45 views

When are cubic solutions “simple”

Is there any way to tell from the coefficients of a cubic equation (with integer coefficients) whether or not the solutions will all have a "simple" form. By "simple" I mean of the form $a + b\sqrt{...
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2answers
67 views

Solve this equation $z^3-(2+4i)z^2-3(1-3i)z+14-2i=0,z\in C$

Solve following equation $$z^3-(2+4i)z^2-3(1-3i)z+14-2i=0,z\in C$$ Try $z=a+bi$,then It's ugly can you more simple ?
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3answers
40 views

Cubic problem regarding solutions to a cubic.

Let $x_1$, $x_2$, $x_3$ be the solutions of the equation $x^3 - 3x^2 + x - 1 = 0$. Determine the values of: $$\frac{1}{x_1x_2}+\frac{1}{x_2x_3}+\frac{1}{x_3x_1}$$ and $$x_1^3 + x_2^3 + x_3^3$$ ...
0
votes
1answer
60 views

root of $g$ is smaller than that of $f$.

For a fixed natural no. $n\ge4$, consider $$f(x)=x^3-(n+2)x^2+2nx-2,$$ $$g(x)=x^3-(n+3)x^2+2(n+1)x-2,$$ It seems that smallest root of $g$ is smaller than that of $f$. Can someone show how to prove it....
0
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1answer
23 views

Robust to solve a given cubic equation: Given roots are complex

Devise the algorithm to find the roots of cubic equation: $7x^3 + 11x^2 + 5x-2=0$. This question was asked in one of the quant finance interviews.
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1answer
38 views

Cubic polynomial with 1 real root and 2 complex conjugated roots (real coefficients)

I am stuck on this problem about cubic polynomials. I rely on the Wikipedia page on the topic. Using wikipedia notations (chapter "General formula for roots") : For the case where $\Delta > 0$, ...
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0answers
44 views

Solving for y in $4+x^3+y^3-3xy = 0$

I was trying to solve for $y$ in this equation: $4+x^3+y^3-3xy = 0$. I put it into WolframAlpha and it gave me an complicated answer. It appears like it used the cubic formula, except when I tried to ...
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1answer
47 views

Cardano's method returns incorrect answer for $x = u + v$

I'm trying to use Cardano's method to solve this equation: $$x^3+6x=20 \tag{1}$$ As described on Wikipedia, I let $x = u + v$ and expand in $(1)$: $$(u+v)^3+6(u+v)=20$$ $$u^3 + v^3 + (3uv+6)(u+v)-20=...
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1answer
18 views

Range of root of cubic equation.

Suppose $a$ and $b$ are two positive real numbers such that the roots of cubic equation $x^3-ax+b=0$ are all real. Let $p$ be a root of this equation with minimal absolute value. What is the range of ...
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1answer
37 views

Cubic spline interpolation results

I have a set of data points on which i am trying to do cubic spline interpolation. Below is the snapshot of the curve with the input data points marked in green color. And the red color marked point ...
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2answers
79 views

If the cubic equation with rational coefficients $x^3+ax^2+bx+c=0$ has a double root, the root is rational.

This question comes from a problem in Rational Points on Elliptic Curves by Silverman. The problem asks to show that For a cubic curve $C: y^2=x^3+ax^2+bx+c$ with $a,b,c\in \mathbb{Q}$, if $S=(...
2
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1answer
60 views

Cubic Spline Interpolation

My problem is to find a interpolating cubic spline to the points $$\left\{(0,0), \left(\frac{\pi}{2}, 1\right), \left(\pi,0\right), \left(\frac{3\pi}{2}, -1\right),(2\pi,0)\right\}$$ I did as ...
0
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3answers
57 views

Find the roots of cubic equation.

If the function $$f(x)=x^3-9x^2+24x+c$$ has three real and distinct roots $l,m,n$, the find value of $[l]+[m]+[n]$ where $[..]$ represents greatest integer function In my book there is no ...
0
votes
1answer
43 views

Solving for a cubic polynomial's roots using Viete's Theorem

I am asked to find the roots $f(x)=x^3-24x^2-24x-25$. However, the only thing I am aware of in regards to finding the roots of cubic polynomials is Viete's Theorem. However, this theorem requires that ...
5
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1answer
59 views

Expressing the roots of a cubic as polynomials in one root

All roots of $8x^3-6x+1$ are real. (*) The discriminant of $8x^3-6x+1$ is $5184=72^2$ and so the splitting field of $8x^3-6x+1$ has degree $3$. Therefore, all three roots can be expressed as ...
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1answer
79 views

A simple cubic equation problem:

Consider the cubic equation $$az^3-bz^2+\bar{b}z-\bar{a}=0$$ where $a$ and $b$ are non-zero complex numbers. Suppose $z_1, z_2$ and $z_3$ are the roots. Question: Which $a$ and $b$ gives $|z_1|=|...
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1answer
30 views

Solving a cubic function with P and Q

I have been struggling a little bit over solving cubic functions. I have been trying to use the P and Q method. So the question is What is the approximate value of the greatest zero of $f(x) = x^3 - ...
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1answer
57 views

Symmetric system of equations problem

Solve the following simultaneous eqations on the set of real numbers: $$a^2+b^3=a+1$$ $$b^2+a^3=b+1$$ I have found two trivial solutions: $$a=b=1$$ $$a=b=-1$$ but I can't prove that there are no ...
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11answers
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What equation produces this curve?

I'm working on an engineering project, and I'd like to be able to input an equation into my CAD software, rather than drawing a spline. The spline is pretty simple - a gentle curve which begins and ...
5
votes
0answers
76 views

The probability that a random (real) cubic has three real roots

We can formalize the notion of the probability that a randomly selected quadratic real polynomial has real roots as follows: Suppose $R > 0$, and suppose the random variables $a, b, c$ are (...
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1answer
29 views

How to draw cubic plane curve?

In Python, using MatPlotLib, given [vector] parameters $a$ and $b$ and [scalar] parameter $c$, I want to draw a general cubic plane curve in 2-dimensional space (regular plane with $x$ and $y$ axes): $...
2
votes
1answer
37 views

I don't understand this process in solving the cubic

My previous question was about the cubic. Now, the following confuses me Q. Use $\cos{3 \theta}=4 \cos^3{\theta}-3 \cos{\theta}$ to solve the cubic $t^3+pt+q=0$ for $p,q$ real when $27q^2+4p^3<...
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1answer
40 views

Discriminant of the cube, quartic…

I was told the discriminant of the cubic is $$\Delta=-27q^2-4p^3$$ and that $\Delta>0$ means that there are three real roots. Simply put, why is this the discriminant? I ask this because, looking ...
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0answers
30 views

A cubic relationship for perfect numbers

Combining the identity (the composition of the polynomial $P_3(x)=\frac{x(x+1)(2x+1)}{6}$ with the arithmetical function $N(n)=n$ and the known proof by induction) $$1^2+2^2+\ldots+n^2=\frac{n(n+1)(...
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2answers
58 views

'Elegant' ways on solving for roots for this cubic function?

I have this interesting cubic equation, $$ x^{3} - 80\alpha x^{2} + (1744\alpha^{2}-81)x + (3240\alpha-5760\alpha^{3}) = 0 $$ where $\alpha$ is some constant. I went about the method of Cardano, ...
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8answers
156 views

Why is $x^3-x^2+x$ injective?

The function $$ f(x) = x^3-x^2+x$$ is injective (as seen on the graph), but by doing f(a) = f(b) I can't get to the point where a = b. Can this be indicated in an analytical way (without graph) ?
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4answers
90 views

Cubic polynomial with three (distinct) irrational roots

I am looking for an equation $$x^3+ax^2+bx+c=0, \qquad a, b, c \in \Bbb Z,$$ of degree $3$ that has $3$ different roots. For an equation of degree $2$ it is easy---for example $x^2-2=0$---but I ...
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0answers
62 views

Without solving it, is there an elementary way to show that $X^3+Y^3=Z^3$ has a finite number of primitive [and non-trivial] integer solutions?

Considering the cubic case of Fermat’s Last Theorem, I make the following claim: Proposition: The Diophantine equation $$ X^3 + Y^3 = Z^3 \tag{$\star$} $$ has a finite number of primitive [and non-...
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3answers
138 views

Three questions about the form $X^2 \pm 3Y^2 = Z^3$ and a related lemma

In Ribenboim’s Fermat’s Last Theorem for Amateurs, he gives the following lemma [Lemma 4.7, pp. 30–31]. Lemma. Let $E$ be the set of all triples $(u, v, s)$ such that $s$ is odd, $\gcd(u,v) = 1$ and $...
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1answer
13 views

Substitution of factors of the free term to find factors of a cubic equation

I was taught that finding a factor (and hence a solution) of a cubic equation may be easier if I try if the factors of the free term are roots of the equation. For example, if one has an equation $x^3-...
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2answers
33 views

Suppose I have a cubic equation with unknown coefficient. How can I find the right coefficient to settle the roots condition?

Suppose I have this equation, $S^3 + 19S^2 + 25S - 75 + K=0$ How can I find $K$ that makes all the root values negative? So far, all I'm doing is substituting $K$ with random values until it gives ...
3
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1answer
75 views

Trying to prove $c^3a^2+(9c^2-b^2)a+(27c-10b)=0$ has no positive integer solutions

I'm trying to prove (or, I suppose, disprove) the following claim, in either version. Conjecture (Strong Version): There are no positive integers $a,b,c$ such that $$c^3a^2+(9c^2-b^2)a+(27c-10b)=0.$$ ...
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0answers
37 views

Solving for $t$ in a cubic function

I have a function $h(t)=5t^3 + 30t^2 + 45t + 4$ I would like to solve for t when $h(t)=50$ I have tried factorizing by inspection and by table but the function does not appear to have a real ...
0
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4answers
62 views

Finding the roots of a cubic equation

I trying to find the roots of the equation $$ax^3 + bx^2 + cx + d = 0$$ By using some changes of variable (which does not really matter now) I was able to rewrite this equation as $$z^3 - \...
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2answers
33 views

Finding the complex roots of a cubic equation [closed]

I'm trying to solve the equation $$ (w^{3})^{2} - qw^{3}-\frac{p^{3}}{27}=0$$ as part of the solution to the generic cubic equation (but I will not start from there. It's pointless). This is a ...
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2answers
183 views

Cubic Equation Related

How do I solve the following cubic equation? $(x+1)^2(x-2)-2nx-6n=0,n \in \mathbb{N}$ $ \therefore (x+1)^2(x-2)-2n(x+3)=0$ I don't know how to solve further.
2
votes
1answer
27 views

bad explanation in how to solve the cubic

The question asked me to solve $t^3+pt+q=0$ where $p,q$ real and $27q^2+4p^3<0$ using the identity $\cos{3\theta}=4\cos^3{\theta}-3\cos{\theta}$. Answer goes like this and I have stopped ...
2
votes
1answer
32 views

Vieta's Theorems Clarification

For applying Vieta's Theorems on cubic equations, I've seen two conflicting answers from two different books. Book 1 says that for a cubic equation $px^3 + qx^2 + rx + s = 0$ has three different ...