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

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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 ...
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1answer
40 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 ...
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0answers
26 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
21 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
28 views

Evaluating Cardano's method [on hold]

Consider the following evaluation of the cardano's method. When the discriminant of the depressed cubic is bigger than 0, there will be 3 distinct roots. -Pro for using cardano's method is the ...
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29 views

Why is cubic polynomial important? What are their applications in real life? [duplicate]

I need some examples of how solving cubic polynomials can be useful in real life. Someone suggested me that "in oil and gas industry, a single simulation of the process or of the reservoir involves ...
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0answers
34 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 ...
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2answers
73 views

Maths essay on solving cubic polynomial [closed]

I have to do a 4000 words essay on cubic polynomial. I will focus on evaluating and comparing cardano's and trigonometric substitution method (in detail).(including the discriminant part ) I will ...
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1answer
53 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} + ...
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1answer
28 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 ...
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132 views
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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 ...
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1answer
39 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 ...
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3answers
60 views
5
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57 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 ...
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0answers
44 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
54 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 ...
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0answers
31 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
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1answer
22 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 ...
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5answers
50 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 ...
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4answers
631 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
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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 ...
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1answer
51 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 ...
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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 ...
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2answers
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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 ...
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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 $15$ ...
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1answer
58 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 ...
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1answer
33 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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2answers
91 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.
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2answers
38 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 ...
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1answer
64 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 ...
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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 ...
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2answers
418 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) ...
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0answers
19 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
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Solving $x^3 = -1$ for complex numbers [duplicate]

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

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

Please see question. I believe the answer should be: $S_0(2)=\frac12(x^3-3x+2)=2$ $S'_0(2)=\frac12(3x^2-3)=\frac{9}{2}$ $S''_0(2)=\frac12(6x)=6$ $S_1(2)=\frac12(x^3-12x^2+45x-46)=2$ ...
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2answers
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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 ...
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25 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 ...
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3answers
61 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 ...
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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 $\alpha^2+\beta^2+\gamma^2=p+13$

$$z^3-2z^2+pz+10=0$$ $$ax^3+bx^2+cx+d=0$$ $$\Rightarrow\,\,\,\,\,\,\,\,\,a=1,\,\,\,\,\,\,\,\, b=-2,\,\,\,\,\,\,\,\, c=p,\,\,\,\,\,\,\,\, d=10$$ ...
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2answers
92 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 ...
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1answer
45 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 ...
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1answer
41 views

Solving Cubic Equation

$$ f(x) = ax^3 + (b - ad)x^2 + (c - bd)x - cd $$ where $a = 18, b = 4, c = 20$ and $d = 12$. What value of x satisfies the equation $f(x) = 0$? $$ f(x)=18x^3- 212x^2-28x-240. $$ i was told to slowly ...
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1answer
49 views

How to solve for $x^3$ relative to $(x-1)^3$

There is probably a better way of asking this question. There is a pretty simple formula to figure out $x^2$ from $(x-1)^2$. $$x^2 = (x-1)^2 + x + (x-1)$$ You can see how easy this formula is here: ...
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1answer
36 views

Does this sine satisfies this equation? [closed]

Is $\sin \frac{\pi}{9} \in \{ x | 8x^{3} -6x + 3^{\frac{1}{2}} = 0 \}$? It does seem so numerically but could you prove it why?
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3answers
60 views

If $x^3 +px -q =0$ then value of $(\alpha + \beta)(\beta + \gamma)(\alpha+\gamma)(1/\alpha^2 + 1/\beta^2+1/\gamma^2)$

I am given a cubic equation $E_1 : x^3 +px -q =0$ where $p,q \in R$ so what would be value of the expression $$(\alpha + \beta)(\beta + \gamma)(\alpha+\gamma)(\frac{1}{\alpha^2} + ...
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1answer
31 views

Does the equation $x^{3} - x - p = 0$ always has a solution in $\mathbb{R}$

Does the equation $x^{3} - x - p = 0$ always has a solution in $\mathbb{R}$ for all $p \in \mathbb{R}$. Is there a real solution for $x$ for each real number $p$? I am new to the theory of cubic ...
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1answer
38 views

What points help identify a cubic and an exponential graph?

What are the different points on the graph we need to know which would help determine the equation of the curve. For example in a quadratic graph, we can determine the equation if we know either any ...
1
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1answer
36 views

Solve for $x$: $6(x-3)^3 + 4(x+9)^2 +8x -5x^2 (x-10) = 0$

Solve for $x$ in: $$6(x-3)^3 + 4(x+9)^2 +8x -5x^2 (x-10) = 0$$ So far I've made it to: $x^3 +242x +162 = 0,$ but now I'm stuck.
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1answer
43 views

Solving Cubic root equation

This is the equation I have $$ [\frac{k^3\alpha^2x_{N-1}^2}{1-\alpha m}\lambda_{N-1}^3 + (k^2\alpha x_{N-1}^2 - \frac{k}{2}x_{N-1}^2)\lambda_{N-1}^2] - \frac{m}{2}(x_N - x_{N-1})^2 = 0 $$ I am ...
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1answer
94 views

Cubic Equation Finding Roots

How to prove that a particular cubic equation has three real and distinct roots without finding its discriminant via calculus method? Please do not use mathematical concepts beyond high ...