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

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7
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1answer
70 views

Find all pair of cubic equations

Find all pair of cubic equations $x^3+ax^2+bx+c=0$ and $x^3+bx^2+ax+c=0$, where $a,b$ are positive integers and $c$ not equal to $0$ is an integer, such that both the equations have three integer ...
0
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0answers
24 views

Why doesn't Horner's method work with the following cubic equation?

I'm trying to factor $$2x^3 - 4x^2 + 2x$$ I use the Horner's method │ 2 4 2 │ 0 --------------- │ 0 0 │ 0 --------------- 0│ 2 4 2 │ 0 and I obtain ...
-1
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1answer
24 views

How to find stationary points of two-variable cubic [on hold]

I need to differentiate this cubic function to get the stationary points: $$f(x,y) = x^3 + ax^2 + bxy^2 + cxy + dx + e,$$ where $a$, $b$, $c$, $d$ and $e$ are constants. How do I do this?
2
votes
0answers
21 views

Find cubic equation roots: what choice to make for cube root to avoid circularly permuting the roots?

My work involves solving a cubic equation similar to this $a x^3+b x^2+c x +d$ which has three roots, $x_1, x_2, x_3$ (as given in https://en.wikipedia.org/wiki/Cubic_function.) ...
0
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0answers
33 views

Software that uses Cardano's method to solve a cubic?

I am writing an essay for my school. I am think of testing the speed of the Cardano's formula for different types of cubic polynomial. Is there any software or calculator that uses the Cardano's ...
1
vote
1answer
33 views

Finding all possible cubic equations from two/three points

I'm trying to find all possible cubic equations that can be found from two scenarios. The first scenario is a lot like the one I asked a couple of days ago on Stack Overflow, found here: ...
6
votes
1answer
174 views

The asymptotic of the number of integers that are sums of three nonnegative cubes

Let $c(n) $ be the number of distinct integers between $0 $ and $n $ of the form $ a^3 + b^3 + c^3$, meaning the sum of $3$ nonnegative cubes. $C(n) = O( n \space \ln(n)^x ) $ Find and prove the ...
1
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0answers
23 views

Any pros and cons of the Cardano's method to solve cubic?

Cardano's method can be used to solve all arbitrary cubic equations. However, I doubt this method would be flawless and I am curious what limitations does the Cardano's method have. So far I only ...
1
vote
2answers
17 views

Cubic function range garauntees in 0-1 interval?

Let's say i had a cubic function $y = Ax^3+Bx^2+Cx+D$ and I know that $A,B,C,D$ are all in the range $[0,1]$, but i don't know their value. Is there any way to know any minimum or maximum value of ...
1
vote
3answers
89 views

how did Cardano obtain three solutions for cubic?

So, if I am not mistaken Complex numbers were discovered after Cardano's method. But from Cardano's Method on Wikipedia, it says to get the three solutions, we should use the root of unity. In that ...
2
votes
1answer
34 views

Find all ordered triplets $(p,q,r)$

Let $A,G^2,H$ be the roots of the cubic equation $x^3+px^2+qx+r=0$ which are in G.P., where $p,q$ are integers and $A,G,H$ are respectively AM,GM,HM of two positive numbers. If $p,q \in (-100,100)$, ...
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3answers
119 views

Harder-Than-Seems Inverse of $f(x)=x^3-x-12$?

This may seem simple but I have had long days of frustration with finding the inverse of this: $$f(x)=x^3-x-12.$$ I got this on some homework and it did not ask for the inverse. However I wanted to ...
-1
votes
1answer
51 views

Cubic Depressed Form ! What can we deduce form it?

Cubic depressed form with equation $f(x) = x^3 + px + q$ The question is, when $p$ is positive, will the function have $3$ real roots ? or does it have to have $1$ real and $2$ complex roots? My ...
0
votes
1answer
55 views

Roots of cubic equation

If$\frac{1+\alpha}{1-\alpha},\frac{1+\beta}{1-\beta},\frac{1+\gamma}{1-\gamma}$ are the roots of the cubic equation $f(x)=0$ where $\alpha,\beta,\gamma$ are the real roots of the cubic equation ...
0
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0answers
35 views

Trigonometric Substitution Method to solve Cubic Equation.

Here are the questions. IN the wiki page, it says p has to be smaller than 0. But they didnt really explain why... Therefore, I assume it is impossible to have a complex number inside arcosine, is ...
0
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0answers
23 views

cubic polynomial cardano method.

When the discriminant is negative where the three roots are real, according to wiki, we have to use $u^3 $ and equation $(t = u - p/3u) $ to find the roots. However, cant we just use $t = u + v $ ...
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0answers
37 views

Confusions about Cardano's method to solve cubic

Here are some questions that I don't regarding to the Cardano's method. 1, Is $q^3/27 + p^2/4$ the discriminant for cubic? 2, what is casus irreducibilis? Does it mean using radicals to represent ...
0
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1answer
61 views

Cubic function: Cardano's method

(Wikipedia link) So I am writing an essay on different ways to solve cubics. But I get stuck in the Cardano's method... Mainly is the part with Cardano's method's condition $\frac{q^2}{4} + ...
0
votes
1answer
41 views

How to derive the general formula to determine the equation of a given cubic function

My question is: When determining the equation of a cubic function, we can separate the general cubic equation into it's solutions and we end up with the equation $y = a(x-r_1)(x-r_2)(x-r_3)$ We ...
4
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0answers
205 views

Integer solutions of a cubic equation

With $\mathrm {gcd}(x,y)=1$ I have the following equation: $$x^3-xy^2+1=N$$ I want to find the integer solutions, given an N, of the variables $x$ and $y$. I have tried factoring the equation into ...
1
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1answer
41 views

Is the cubic formula numerically unstable?

Are there numerical rounding issues in using the cubic formula to find roots of cubic equations? Similarly with the quartic formula? I do know for the quadratic formula to solve $ax^2+bx+c = 0$ that ...
0
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3answers
64 views
5
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2answers
63 views

Find a polynomial with cubic values for consecutive integers.

Can we find a cubic polynomial (except the obvious ones, i.e. cubes of linear polynomials), say, $f(x)\in \Bbb{Q[x]}$ whose values are cubes for four consecutive integers? What about five consecutive ...
0
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0answers
53 views

Matlab Coding finding zeros without using fzero or roots function

So i am a completely new at Matlab. I'm basically suppose to develop a function in Matlab that finds the zeros of a cubic polynomial. real and complex. I'm pasting below what I have so far. I started ...
3
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1answer
56 views

Finding roots of a cubic equation

Question: If $6(8a + c) = 16b+ 3d$ then $f(x) = ax^3 + bx^2 + cx + d$ has at least one root in: $(-3,0)$ $(-4,0)$ $(-4,-3)$ $(0,2)$ Attempt: Having solved several such ...
0
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0answers
36 views

How to find a perfect regression fit in R?

I have a set of points, which I know can be described with some equation. How can I find this equation? The scatter plot for this set looks like this: I look at the plot and assume that I can use a ...
0
votes
1answer
28 views

Calculate Volume m3?

I'm attempting to calculate the value that is m3 but i don't know how the person got these values. The first row first column is 22mm(width) x 100mm(height) .. then the bold in the second column is ...
0
votes
5answers
51 views

Find the cubic equation of roots $α, β, γ$.

Taken from Fitzpatrick $4$ unit course textbook. The question says: If the cubic equation $\ ax^3+bx^2+cx+d$ has roots $α, β, γ$. Find the cubic equation who's roots are $α^2, β^2, γ^2$ I keep ...
6
votes
4answers
657 views

Guessing one root of a cubic equation for Hit and Trial

Suppose I have a cubic equation as $$15x^3-4x^2-25x+14=0$$ By Hit and Trial method I know that one of the roots is $x=1$. And hence I can solve the cubic equation wit ease as it will take the form ...
11
votes
5answers
1k views

Finding cubic with golden ratio as root

I want to find a cubic such that it meets the following criteria: Has the golden ratio as its only real root Has integral coefficients Has a leading coefficient of $1$ and a final coefficient of ...
0
votes
1answer
52 views

Roots of a real cubic equation

I have a cubic equation of the form $$x^3-a^2x-b^2=0.$$ It is given that all roots are real, moreover, only one root is positive and the other two are negative. Let the positive root be ...
0
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0answers
34 views

What is known about the Heronian primes?

A Diophantine equation $$x^3 - Dy^3 = 1$$ always has a trivial solution $x = y^3 + 1$. It appears that a non-trivial (that is those with $x$ smaller than trivial) solution exists iff $y$ is a Heronian ...
2
votes
2answers
40 views

Concerning Roots of the cubic equation $f(x)=x^3+x^2-5x-1$ and the Greatest Integer (or Floor) function

The Question I got into a rather tight corner with this question. It says: Let $\alpha, \beta, \gamma$ be the roots of $f(x)=0$, where $f(x)=x^3+x^2-5x-1$. Then, the value of ...
0
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0answers
30 views

How to prove that the dimension of the twisted cubic is 12?

The twisted cubic $C=AX$, $A$ is $4*4$ non-singular matrix. $X=[1\quad t \quad t^2 \quad t^3]^T$. How to prove that the dimension of $C$ is $12$? As I have known that the dimension of $A$ is $15$ ...
1
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1answer
59 views

Finding the real root of the polynomial $2x^3-3x^2+2 $

I want to get exactly roots of this equation... $2x^3-3x^2+2 = 0$ I try to solve it but can not find the solution. wolframealpha just give me aproximation.. I know the real root is $-1< root ...
1
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1answer
38 views

Finding cubic bezier curve endpoints based on relationship between endpoints and a point on the curve.

I have the following information about a bezier curve: The curve begins at $x=0$ and ends at $x=1$. The curve has two control points each at the same height as their closest endpoints, one at ...
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votes
2answers
94 views

Advanced secondary school mathematics [closed]

Solving the equation: ${x^3 + 2x^2 + 2\sqrt2x + 2\sqrt2 = 0}$ Please help me, I have no idea to this problem.
2
votes
2answers
39 views

Find the possible values of a in the cubic equation.

Given that $(x-a)$ is a factor of $x^3-ax^2+2x^2-5x-3$, find the possible values of the constant $a$. I believe you first have to find the $a$ in the cubic equation then the other $a$ in $(x-a)$, but ...
0
votes
1answer
82 views

finding roots when polynomial does not equal zero

I was trying to solve this polynomial $$x(3-x^2)=1$$ I worked for the term $(3-x^2)$, I thought that this term cannot be $0$, thus $$3-x^2 >0$$ $x< \sqrt{3}$, $x<-\sqrt{3}$ is rejected ...
3
votes
2answers
82 views

Find a second root of $x^3+px+q$ given the first root

This is a problem from Artin where given one root $a$, you have to find an equation for a second root in terms of $a$, $p$, $q$, and the square root of the discriminant $\delta$. Here's what I have ...
1
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2answers
428 views

Solving a cubic equation involving trigonometric functions

Question: If $$\sin(x) - \cos(x) = \frac{\sqrt{3}}2$$ Then $$\sin^3(x) - \cos^3(x) = ?$$ I have turned first equation into a quadratic so I got $$\sin(x) = \frac{\sqrt{3}\mp\sqrt{5}}4$$ and $$\cos(x) ...
0
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0answers
25 views

Minimum of a cubic fitted to two points and their derivatives

I'm trying to understand a line search method used to find a step length in a minimsation algorithm. There is an interval $[a, b]$ containing desirable step lengths and there are two previous ...
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1answer
44 views

Solving $x^3 = -1$ for complex numbers [duplicate]

How can I solve for the complex solutions of $$ x^3 = -1 $$
0
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1answer
62 views

Show that S is a cubic spline (natural or clamped)

Please see question. I believe the answer should be: $S_0(2)=\frac12(x^3-3x+2)=2$ $S'_0(2)=\frac12(3x^2-3)=\frac{9}{2}$ $S''_0(2)=\frac12(6x)=6$ $S_1(2)=\frac12(x^3-12x^2+45x-46)=2$ ...
1
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2answers
94 views

Solving a cubic equation $x^3+15x^2+24x-40=0$

I have : $x^3+15x^2+24x-40=0$ When I use $x=u-a/3$ where $a=15$ and I replace it gives : $u^3-51u+90=0$ Now, my discriminant is inferior to $0$... How do I find out atleast one solution of this ...
1
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2answers
26 views

The sum of the abscissae of the intersections of a cubic and a line

I remember being told in passing in a talk once the following theorem: Let $y=x^3$, and let $x_1,x_2,x_3$ be the abscissae ($x$ co-ordinates) of three distinct points on this cubic. Then ...
0
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3answers
62 views

Find the real root $\alpha$ of the cubic equation $z^3-2z^2-3z+10=0$

Find the real root $\alpha$ of the cubic equation, $$z^3-2z^2-3z+10=0$$ The exam paper is giving just 2 marks for this and the mark scheme isn't very helpful. My idea is that you can use some of this ...
0
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1answer
49 views

The roots of the cubic equation $z^3-2z^2+pz+10=0$ are $\alpha$, $\beta$ and $\gamma$. Show that $\alpha^2+\beta^2+\gamma^2=p+13$

$$z^3-2z^2+pz+10=0$$ $$ax^3+bx^2+cx+d=0$$ $$\Rightarrow\,\,\,\,\,\,\,\,\,a=1,\,\,\,\,\,\,\,\, b=-2,\,\,\,\,\,\,\,\, c=p,\,\,\,\,\,\,\,\, d=10$$ ...
1
vote
2answers
96 views

Derivation for the general cubic formula

It's a long equation, and Wikipedia writes it to be $$x_k = -\frac{1}{3a}(b + u_kC + \frac{\Delta_0}{u_kC}), \quad k \in \{1,2,3\}$$ But there is no derivation of it. The sources I've read so far ...
0
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1answer
53 views

Find if sphere is inside parallelepiped

I have many spheres in a 3D space, with their center's position and their radius to be known. I also have 1 parallelepiped ( wiki link) with its 8 vertices' positions to be also known. How can I ...