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

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3
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2answers
58 views

Simplify $x^3 - 4x^2 + 10x - 125$

I've been trying to simplify $x^3 - 4x^2 + 10x - 125$ for a while now, and I don't seem to progress. I know that the factors of $125$ are $1$, $5$, $25$ and $125$, but none of these seem to help here. ...
1
vote
1answer
36 views

Using sum/product of roots of a cubic equation to solve given expression $(a+b)^3+ (b+c)^3 + (c+a)^3$

If $a, b, c$ are the roots of the equation $7x^3- 25x +42 =0$, then the value of the expression $(a+b)^3+ (b+c)^3 + (c+a)^3$ is? I tried to solve this but wasn't able to simplify the term to be able ...
10
votes
2answers
203 views

Cubic polynomial equal to a cube

I've been researching cubes and I'm trying to solve this Diophantine equation over the integers. $$ax^3 + bx^2 + cx + d = y^3$$where a, b, c, d are parameters for a given $n$. For example, for $n = ...
1
vote
1answer
40 views

how to show that $\mathbb{Q}[\sqrt[3]{2}]$ is a field? (by elementary means)

To be very concrete, I want to show that every element of the form $1/(p+qx+rx^2)$ where $x=\sqrt[3]{2}$ where $p,q,r$ are rationals can be written in the form $a+bx+cx^2$ where again $a,b,c$ are ...
1
vote
3answers
51 views

Solving a Perturbed Cubic Equation

Consider a cubic equation $(1 + \epsilon)x^3 - 2ax^2 + (a - 3\epsilon)x + 2\epsilon = 0$ where $\epsilon > 0$ and $a \gg 1$. In the limit of $\epsilon \rightarrow 0$, $x(x^2 - 2ax + a) = 0$ so ...
0
votes
1answer
39 views

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 ...
2
votes
2answers
63 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 ...
0
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1answer
26 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 ...
0
votes
3answers
60 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: $x=\dfrac{1}{3}\left(1+\dfrac{\sqrt[3]{-23+3i\sqrt{237}}}{\sqrt[3]{2^2}}+\dfrac{11}{\sqrt[3]{2(-23+3i\sqrt{237}})}\right)$ ...
0
votes
0answers
58 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 ...
0
votes
1answer
26 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
votes
1answer
61 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 ...
0
votes
2answers
54 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
votes
2answers
192 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 ...
26
votes
13answers
3k views

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 ?
0
votes
1answer
29 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 $0=10x³-(2y+5)x²+(y-4)x+76$. ...
1
vote
2answers
146 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 $\sim 1.8393$)
3
votes
0answers
52 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 $A$, ...
3
votes
2answers
48 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 $$ q^3 + pq + q = 0 $$ ?
3
votes
5answers
67 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 ...
1
vote
1answer
97 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 ...
1
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2answers
24 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 ...
0
votes
3answers
81 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 ...
2
votes
1answer
53 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) ) ...
5
votes
1answer
92 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, ...
1
vote
3answers
107 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!
1
vote
3answers
65 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.
3
votes
2answers
205 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 ...
2
votes
2answers
99 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.
0
votes
2answers
66 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 ...
0
votes
2answers
57 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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vote
2answers
52 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?
0
votes
0answers
38 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 ...
6
votes
1answer
97 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 ...
2
votes
2answers
102 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 ...
3
votes
0answers
57 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 ...
7
votes
2answers
72 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 ...
1
vote
1answer
44 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 ...
1
vote
1answer
69 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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vote
2answers
76 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:" ...
0
votes
2answers
259 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 ...
0
votes
1answer
34 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 ...
0
votes
2answers
95 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?
0
votes
2answers
99 views

How to solve the cubic equation $ x^3+3x -2 = 0$ without using matrices?

I am trying to solve $ x^3+3x -2 = 0$ Using the remainder theroem but none of the factors of the constant make the equation equal to $0$. Is there any way I can get the answers without using matrices? ...
0
votes
2answers
54 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 ...
1
vote
2answers
114 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.
1
vote
1answer
116 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 ...
0
votes
3answers
91 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?
0
votes
2answers
216 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?