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

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most general form of $X - A = X^{-1}B (X^{-1}BX^{-1}+ C)^{-1}$ that has a real solution $X = f(A,B,C)$?

What is the most general form of the cubic matrix equation $X - A = X^{-1}B (X^{-1}BX^{-1}+ C)^{-1}$ that has a real solution of the form $X = f(A,B,C)$, where $A,B$ and $C$ are positive definite ...
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2answers
62 views

Motivational example for complex numbers

Years ago I was introduced to complex numbers. In class we had been talking about the cubic polynomial and its solutions. At one point we saw an example where, when using the formula, one had to stop ...
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1answer
15 views

How to calculate the length of a cubic hermite spline between two points

I am using the following equation to create a cubic hermite spline: $$p_n(t) = a_nt^3+b_nt^2+c_nt+d_n$$ $$1\geq t\geq 0$$ $p_n(t)$ is the unit interval interpolation equation for dimension n. $t$ is ...
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3answers
58 views

The real cubic root expression

$2x^3-2x^2-3x+2=0$ has 3 real root, but they are all express in such way: $x=\dfrac{1}{3}\left(1+\dfrac{\sqrt[3]{-23+3i\sqrt{237}}}{\sqrt[3]{2^2}}+\dfrac{11}{\sqrt[3]{2(-23+3i\sqrt{237}})}\right)$ ...
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0answers
31 views

Issue with modelling word problems as cubic equations

I am having problems with a specific set of word problems, which are meant to be modeled as cubic equations in order to be solved. I will give some examples to specify where I can't solve it. The ...
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1answer
25 views

The cubic interpolation

I try to understand the cubic interpolation for my studies. The following website says " (1) The four equations above can be rewritten to this (2):" but how? Can anyone explain me the the necessary ...
2
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1answer
55 views

Solving a cubic polynomial equation.

Overview I have tried finding a solution to this problem myself and I have flailed. Its just a challenge for me. could you please tell me how far am I in solving this question? My approach for ...
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2answers
43 views

Cubic roots of determinant.

If x=a+2b satisfies the cubic (a,b element of R) f(x)= $$\left|\begin{matrix} a-x & b & b \\ b & a-x & b \\ b & b & a-x\end{matrix}\right|$$ =0, then it's other 2 roots are?
6
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2answers
184 views

Find the maximum possible value.

For all ordered triples $(p,q,r)$ define the polynomial $$f_{p,q,r}(x)=x^3-px^2+qx-r$$ Let $a_{1},a_{2},a_{3},b_{1},b_{2},b_{3},c_{1},c_{2},c_{3}$ be (not necessarily distinct) positive reals such ...
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What is an example of real application of cubic equations?

I didn't yet encounter to a case that need to be solved by cubic equations (degree three) ! May you give me some information about the branches of science or criterion deal with such nature ?
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1answer
28 views

Positive integer solutions of $0=10x³-(2y+5)x²+(y-4)x+76$

To practise my mathematical skills, I often solve some problems in my free time. In this case, one should find every positive integer solution $(x,y)\inℤ^+\timesℤ^+$ of $0=10x³-(2y+5)x²+(y-4)x+76$. ...
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2answers
130 views

Best way to solve $X^3-X^2-X-1=0$

can anyone help me for this cubic equation ? can be solved without delta method? $X^3-X^2-X-1=0$ (answer is $\sim 1.8393$)
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0answers
49 views

Link between a cubic polynomial and a trig identity

Alright, so I am told to prove that: $$\tan (3A) = \frac{3\tan(A)-\tan^3(A)}{1-3\tan^2(A)}$$ This can be pretty easily done by applying the $\tan$ addition formula, taking the angles $2A$ and $A$, ...
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2answers
46 views

Prove that system of equation implies statement

How to prove that $$ \begin{cases} x_1 + x_2 + x_3 = 0 \\ x_1x_2 + x_2x_3 + x_3x_1 = p \\ x_1x_2x_3 = -q \\ x_1 = 1/x_2 + 1/x_3 \end{cases} $$ implies $$ q^3 + pq + q = 0 $$ ?
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5answers
62 views

Cubic trig equation

I'm trying to solve the following trig equation: $\cos^3(x)-\sin^3(x)=1$ I set up the substitutions $a=\cos(x)$ and $b=\sin(x)$ and, playing with trig identities, got as far as $a^3+a^2b-b-1=0$, but ...
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1answer
66 views

Tangent at average of two roots of cubic with one real and two complex roots

I was able to easily prove that the tangent at the average of two roots of a real cubic polynomial passed through the third root of the function. But I have only done this for functions with three ...
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2answers
19 views

Express one root of depressed cubic equation via another and square root of discriminant

Given a cubic polynomial $f(x)=x^3+px+q, p,q\in \mathbb{Q}$ and one of its roots $x_1$, how to express another root $x_2$ in terms of $x_1$, square root of the discriminant $d=\sqrt{-4p^3-27q^2}$, and ...
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3answers
78 views

Find all values of $a$ for which there are two real solutions of $x^3-2ax^2+a^2x-3=0$

Find all values of $a$ for which there are two real solutions of the equation. $$x^3-2ax^2+a^2x-3=0$$ Ans = $1.5\sqrt[3]{6}$ I tried to research the function by dint of derivative, but it didn't ...
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1answer
50 views

Estimate on a positive root of a cubic equation

Suppose that the cubic equation \begin{equation} a\,x^3+b\,x^2+c\,x+d=0, \end{equation} where $a,d>0$ and the discriminant $\Delta>0$. (refer to http://en.wikipedia.org/wiki/Cubic_function) ) ...
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1answer
86 views

Solving Cubic Equations Using Origami

I have to write a research paper on a mathematical topic for my class; I chose the above topic. I understand that a parabola can be formed using a focus and directrix, both created by origami folds, ...
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3answers
71 views

Roots of a cubic expression and arithmetic progression

If the roots of the equation $x^3+ax^2+bx+c=0$ are in an arithmetic, then what is the value of $2a^3-9ab$? Please explain!
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3answers
51 views

How to find the values of a and b?

If the polynomial 6x4 + 8x3 - 5x2 + ax + b is exactly divisible by the polynomial 2x2 - 5, then find the values of a and b.
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2answers
194 views

Why doesn't the Rational Root Theorem work on this cubic?

I tried using the Rational Root Theorem on $x^3-6x^2+4x-5=0$. However, I could not find a rational root of the cubic. When I plugged the cubic into Wolfram alpha, it yielded a very messy real ...
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2answers
86 views

Find the zero of the polynomial

Find the zero of the polynomial $f(x) = x^3 - 5x^2- 2x + 24,$ if it is given that the product of the zeroes is 12. Thanks in advance.
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2answers
60 views

Depressed cubic roots

According to Wikipedia, a depressed cubic has only one root and 2 imaginary roots. Is this true? Can a depressed cubic of the form $x^3+px+q=0$ have 2 or 3 real roots? Edit: Here is a screenshot of ...
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2answers
54 views

Topological space underlying this curve

I have to solve this exercise but I have really no clue even how to start with it: Identify the topological space underlying the cubic $Y^2Z=X^2(X-Z)$ in $\mathbb{PR}^2$. How does it fit with the ...
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2answers
51 views

How to find the second derivative?

I use this article from Wikipedia to build it in my program. How to find the second derivative in $(x_i, y_i)$ point of this cubic interpolation, if I know other $(x_j, y_j)$ points?
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36 views

Prove that $x^3+px+q$ has multiple root $\Longleftrightarrow$ the discriminant is $0$. Is my solution correct?

$f'=x^2+p$ $f'=0,p=-3x^2$ Using $p=-3x^2,f=x^3-3x^3+q$ $f=0,q=2x^3$ Then the discriminant is: $$D=-108\cdot\big((2x^3/2)^2+(-3x^2/3)^3\big)=-108\cdot(x^6-x^6)=0$$ Even if it's correct, can you ...
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1answer
93 views

What cubic problems did Tartaglia and Fior pose to each other?

I have been researching the history of finding roots to general polynomials and the story of solving for the roots of cubic polynomials ($ax^3+bx^2+cx+d=0$) lead me to find several sources describing ...
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2answers
87 views

Bombelli's wild thought of cubic equations

In many books, like Visual Complex Analysis. talk about the real original of complex number. the author begin with this equation: $$x^3=15x+4$$ Then the author use the formula ...
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0answers
56 views

Möbius transformation that permutes roots of a cubic polynomial

The roots of the polynomial $x^3-3x-1$ can be permuted by the function $z\mapsto \dfrac{-1}{1+z}$ which is easily checked by a direct calculation. Is there a simple formula for a Möbius ...
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0answers
22 views

A Question about Cubic and Galois Fields

Let $f \in \mathbb{Q}[x]: f(x) = x^3 + p x + q, p \geq 0, f$ irreducible therein; $\mathbb{k}$ be the splitting field of $f$ over $\mathbb{Q}$; $G = Gal(\mathbb{k} / \mathbb{Q})$. I have to show that ...
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71 views

How was this solution found?

Consider an empty spherical bowl of radius $r$. I was trying to find the height to which I would need to fill the bowl with water so that it would be one quarter full (in terms of volume). The total ...
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1answer
40 views

Equations for Cubic Regression

So, I'm making a simple program for drawing graphs, and I'm looking at making some simple best-fit curves using some basic regression analysis. I've happily got linear and quadratic regression working ...
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1answer
67 views

A Cubic Equation

$2x^3+ax^2+bx+4=0$, $(a,b \in R^+)$ has three real roots. Then : A. $a\geqslant 4.2^{\frac 1 3}$ B. $a\geqslant 1.2^{\frac 1 3}$ C. $a\geqslant 6.2^{\frac 1 3}$ D. $a\geqslant 2.2^{\frac 1 3}$ ...
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2answers
74 views

Solution for cubic algebra

For a cubic equation: $$x^3-xb+a=0 \\$$ EDIT: the above equation has three real solutions for x. one of the solutions is: $$a=2\cdot \left(\frac b3\right)^{3/2}$$ EDIT: "one of the solutions is:" ...
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2answers
193 views

How to solve for a non-factorable cubic equation?

I want to know how one would go about solving an unfactorable cubic. I know how to factor cubics to solve them, but I do not know what to do if I cannot factor it. For example, if I have to solve for ...
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1answer
33 views

Approximations to the Roots of a Function

I want to find approximations to the root of a function in two variables using the Newton-Raphson method. I can use the method on a function in a single variable but I'm lost as to how you can use it ...
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2answers
87 views

Cubic function that has no y intercept

Is there a cubic function that is only in quadrants 1 and 2 of the coordinate plane and so never crosses the y axis? If so can you give me a cubic function that does that?
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2answers
53 views

Solving an equation of order $x$

Two questions 1) Looking at Vieta's formula for cubics and quadratics I think I noticed something. Is it just my imagination or are the following statements true? a) There will be a number of ...
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109 views

How do i solve the cubic equation?

$$x^3 - 3x^2 - 3x +2 = 0$$ The rational root test does not work; there are no rational roots.
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1answer
114 views

Galois group of an irreducible cubic polynomial without using the discriminant

Let $f\in\mathbb Q[x]$ be irreducible of degree $3$. Since the Galois group $G$ of $f$ is a transitive subgroup of $S_3$, it is either $S_3$ or $A_3$. Those two possibilities are easily ...
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3answers
71 views

Finding X intercept of a cubic equation?

What is the $x$ intercept of $y=(x-2)(x^2+25) $? To find $x$ intercept:$ 0=(x-2)(x^2+25) $ I tried $ 0=(x-2)(x+5)(x+5)$ in which the $X$ intercepts are $2,-5$ and $-5$. Is this correct?
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131 views

Finding X intercepts, Domain and Range of a cubic? [closed]

y=x(x^2-9) or y=x^3-9x I know that the Y intercept is 0. What is the X intercept, Domain and Range?
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1answer
41 views

Minimal Discriminant of An Elliptic Curve

I want to determine the minimal discriminant of $$ y^2 + xy = x^3-x^2-50x+111 $$ as an elliptic curve over the rationals. I managed to reduce it to the form $y^2=x^3+Ax+B$ where $A,B$ are rational, ...
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1answer
30 views

Cubic function growth

Hopefully this question isn't too simple for this site. I'm doing a CompSci algorithms course, and trying to understand various growth rates. A function with cubic complexity such as the 3Sum ...
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1answer
17 views

Sketching these two cubic polynomials

I've went with through my text book and finished the 2 page answer section out of all the 23 questions these two were the hardest. Could I please have some help or advice on working it out. How ...
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4answers
227 views

Real and distinct roots of a cubic equation

The real values of $a$ for which the equation $x^3-3x+a=0$ has three real and distinct roots is
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3answers
343 views

Cubic equation with two complex roots and one real root?

Find the real root of the equation $z^3 + z + 10 = 0$ given that one complex root is $1 – 2i$. I've realized that the roots are $(1-2i), (1+2i)$, and a real number we'll call $a$. So using the ...
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1answer
58 views

Exception on the Cubic Formula

I have searched for the cubic formula, which is: $$ \sqrt[3]{\frac{-B^3}{27A^3} + \frac{BC}{6A^2} - \frac{D}{2A} + \sqrt{\left(\frac{-B^3}{27A^3} + \frac{BC}{6A^2} - \frac{D}{2A}\right) ^ 2 + ...