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

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How to Solve a “Cubic Equation”? [on hold]

-Hello, Guys! How to solve a "Cubic Equation"?
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1answer
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Formula for solutions of a cubic equation [duplicate]

I am looking for a simplified formula for a cubic equation in the form: $$Ax^3+Bx^2+Cx+D=0$$ That solves for roots $r_1, r_2 $, and $r_3$ when the discriminant is positive and $r$ when it is ...
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4answers
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The cubic equation $x^3-5x^2+6x-3 = 0$ has solutions $\alpha$, $\beta$ and $\gamma$. [on hold]

The cubic equation $x^3-5x^2+6x-3 = 0$ has solutions $\alpha$, $\beta$ and $\gamma$. Find the value of $$\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}$$
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1answer
48 views

If $a>b>0$ and $a^3 +b^3 +27ab=729$ then the quadratic equation $ax^2+bx-9=0$ has roots $P,Q(P<Q)$. Find the value of $4Q-aP$?

If $a>b>0$ and $a^3 +b^3 +27ab=729$ then the quadratic equation $ax^2+bx-9=0$ has roots $P,Q(P<Q)$. Find the value of $4Q-aP$? How will I begin with the solution just a hint would be enough....
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Find the limit when X--->2

Please HELP. I've tried doing it with L'Hopital but I can't. The answer is supposed to be 1/3 .Please help. $$ \lim_{x\to 2/3} \frac{\sqrt[3]{9x^7+ 4}- 2 }{3x-2} $$
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1answer
117 views

Points on the elliptic curve for Ramanujan-type cubic identities

Given the rational Diophantine equation, $$t^3 - t^2 - \tfrac{1}{3}(n^2 + n)t - \tfrac{1}{27}n^3=w^3\tag1$$ Two points are, $$t_0 = 0\tag2$$ $$t_2 = \frac{-(1 + 2 n) (1 + 11 n + 42 n^2 + 14 n^3 + 13 ...
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1answer
30 views

Cubic Polynomial fitting with defined ranges for coefficients

Is there a way, given a set of values $(x,y)$, to find a cubic polynomial $f(x)$ that fits the values? My cubic polynomial is defined as $c_0 + c_1x +\frac {1}{2} c_2 x^2 +\frac {1}{6}c_3 x^3$ ...
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119 views

Generalizing Ramanujan's cube roots of cubic roots identities

(This extends this post.) Define the function, $$\sqrt[3]{G(t)} = \sqrt[3]{t+x_1}+\sqrt[3]{t+x_2}+\sqrt[3]{t+x_3}\tag1$$ where the $x_i$ are roots of the cubic, $$x^3+ax^2+bx+c=0\tag2$$ While $G(t)...
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If $f(x)=x^3-3x^2-x+2$ ,Find an expression for the function $y$, which is obtained by rotating the graph of $f (x)$ through $180°$.

Question: $f(x)=x^3-3x^2-x+2$ where $x\leq 1$ Find an expression for the function $y = g(x)$, where $x\geq1$ , which is obtained by rotating the graph of $y = f (x)$ through $180°$ about ...
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What's the explanation for these (infinitely many?) Ramanujan-type identities?

Define the function, $$F(\beta) := \sqrt[3]{\beta+x_1}+\sqrt[3]{\beta+x_2}+\sqrt[3]{\beta+x_3}\tag1$$ where, $$x_1 =2\cos\big(\tfrac{2\pi }{7}\big),\;x_2 =2\cos\big(\tfrac{4\pi }{7}\big),\; x_3 = ...
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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 ...
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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
52 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
45 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
44 views

Integrating Inverse square root of a polynomial

I'm Looking for integrals of the following kind $\int\limits_{0}^{M} \left( -x^3 +bx^2 -\omega \right)^{-\frac{1}{2}} \,dx \, ,$ for positive constants $b, \omega$, and $M$ is on of the roots of ...
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1answer
58 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) $ ...
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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 ...
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1answer
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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 ...
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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
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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
47 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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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
41 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 $$x_1^3 + x_2^3 + x_3^3$$ ...
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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....
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1answer
27 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
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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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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
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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
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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
41 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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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=(...
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1answer
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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 ...
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3answers
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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 ...
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1answer
45 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 ...
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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
81 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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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
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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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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 ...
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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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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): $...
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1answer
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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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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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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
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'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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157 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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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
63 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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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
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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-...