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

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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
49 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
26 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
74 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
36 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
47 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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2answers
61 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 ...
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2answers
395 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
12 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
39 views

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
35 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
77 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 ...
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2answers
22 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
54 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
46 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
82 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
39 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 ...
3
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1answer
53 views

Show that $(Y^2-X^3)|f$ if $f$ vanishes on the curve $C: (t^2,t^3)$, and determine what property of a field $k$ will ensure that the result holds.

Let $\phi: \mathbb{R^1}\rightarrow \mathbb{R^2}$ be the map given by $t \mapsto (t^2,t^3)$; prove directly that any polynomial $f\in \mathbb{R[X,Y]}$ vanishing on the image $C=\phi(\mathbb{R^1})$ is ...
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1answer
33 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
35 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
50 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
29 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
31 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 ...
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1answer
35 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
39 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
87 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 ...
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1answer
65 views

Taking the cube root of a sum of radicals

I am wondering how to derive the following simplification without knowing it beforehand: $$^3\sqrt{10 + 6\sqrt{3}} = 1 + \sqrt{3}$$ After the fact, it is easy to verify algebraically. The problem ...
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5answers
89 views

How to find the cubic with roots: $k$, $k^{-1}$ and $1-k$?

This is the second part of a question that asks the same thing but for a quadratic, that part seemed to be fine. The next part asks you to show that: $$x^3-\frac{3}{2}x^2-\frac{3}{2}x+1=0 $$ is the ...
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4answers
48 views

Complex roots of irreducible cubic in $\mathbb{Q}[x]$

Let $$f(x) = x^3 +ax^2 + cx + d \in \mathbb{Q}[x] $$ with one real root, and two complex roots: α and β α and β are conjugates. My task is to show that: $$β \notin \mathbb{Q}(α)$$ I'm confused as I ...
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1answer
50 views

Solving a cubic equation

Solve $y=ax^3+bx^2+cx+d$ I need $x$ in terms of $y$ . I do not need the roots of the cubic equation . I need to express $x$ in terms of $y, x>0$
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34 views

Solving the cubic $2t^{3} - 3t_{1}t^{2} + t_{1}^{3} = 0$

The question is: "4. Find the equation of the tangent and the equation of the normal to the curve $x = 3t^{2}$, $y = t^{3}$ at the point whose parameter is $t_{1}$. Find the parameter of the point at ...
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1answer
37 views

I don't understand this proof about graphs of cubic functions

This proof on proofs.wiki shows that: All graphs of cubic functions are transformations of an odd cubic function. I have no problems with the steps. I just do not understand what is the idea behing ...
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47 views

“Factorizations” of $a^3+b^3+c^3+mabc$?

It is easy to see that $$a^2+b^2+c^2+ab+bc+ca=\frac{1}{2}((a+b)^2+(b+c)^2+(c+a)^2),$$ $$a^2+b^2+c^2+2(ab+bc+ca)=(a+b+c)^2,$$ $$a^2+b^2+c^2-ab-bc-ca=\frac{1}{2}((a-b)^2+(b-c)^2+(c-a)^2),$$ and ...
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1answer
51 views

Factoring of $x^3-3x^2+30x-1$

I need help factoring \begin{align} x^3-3x^2+30x-1=0.\tag{1} \end{align} Any thoughts? I've tried the old guess and check method with long division and ...
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1answer
57 views

what is the inverse of this function

I'm weak at math and I need the inverse of this function if it's computable: $f(t) = A + (-2t^3 + 3t^2)(B-A)$ Note that $A$ and $B$ are constants. thanks for your help.
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1answer
64 views

How to solve cubic vector equation?

A cubic equation $$x^3+ax^2+bx+c=0$$ has three solutions, which can be found analytically. Likewise, a vector equation like $$\underline{\underline{A}} \underline{x} + ...
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83 views

Solving cubic equation modulo prime

I'm trying to an algorithm that can solve an elliptic curve equation for constant y: $y^2 = x^3 + ax + b \text{ mod } p$ p is 57 digits long I've tried to solve it using like a regular cubic ...
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0answers
20 views

How many cubic inches of lead are in a one-pond sample of lead?

The average weight for cast lead is 708 pounds per cubic foot. Consider a one-pound sample of cast lead. How many cubic inches of lead are in the sample? Choose the closest answer. One cubic foot is ...
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1answer
39 views

Cubic: Finding turning point when given x and y intercepts

I have tried substituting in the two points (-4,0) and (0,28) and solving simultaneously for b and c with no success, and the book gives two separate but equally correct solutions for b and c that ...
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1answer
37 views

Finding the real irrational root of a cubic polynomial?

I just wanted to check if anyone can see a simpler way to solve this. Because I am not looking forward to using the cubic formula to solve it! $$ det(\lambda-AI) = \left| \begin{array}{ccc} \lambda + ...
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1answer
42 views

CHKMO 2015 and cubic equations

Let $a,b,c$ be distinct real numbers. If the equations $E_1: ax^3+bx+c=0, E_2: bx^3+cx+a=0$ and $E_3: cx^3+ax+b=0$ have a common root, prove that at least one of these equations has three real ...
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1answer
61 views

stuck with a cubic equation

I picked an equation $0= x^3 +x^2 -2x -1$ I plotted it with geogebra, to see if it had more than $1$ real root. It definitely cuts the $x$-axis $3$ times. But when I checked wolfram alpha, to see ...
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1answer
27 views

Tschirnhausen cubic - expressing in terms of x

Is there a way to express the following function $$y=x\sqrt{x+3}$$ in the form $x$ as a function of $y$? Thanks
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1answer
60 views

Solve $z^3 + 5z^2 + (9 - 5i)z + 10 - 10i = 0$ [duplicate]

Solve $$z^3 + 5z^2 + (9 - 5i)z + 10 - 10i = 0$$ I have never dealt with equations with complex numbers in them so this is interesting; first Ill expand. $$ \implies z^3 + 5z^2- 5iz + 9z + 10 - 10i = ...
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1answer
81 views

Super conic sections?

I know graphs of the form $A x^2 + B xy + C y^2 + D x + E y + F = 0$ are conic sections. But what would happen if I changed the highest power to 3? Would this be a new 3D shape, a 4D version of it, or ...
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2answers
27 views

Equation of a cubic function with inflection point on (0.5,0.5) and contains (0,0), (1,1)

The title basically summarizes my question, but the reason I'm asking this is for use as a timing function for a translation in my game. Thanks in advance!
5
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3answers
63 views

Find a and b of $x^3+ax^2+bx−26=0$

I am doing practise papers and one of the questions is: The cubic equation $x^3+ax^2+bx−26=0$ has $3$ positive, distinct, integer roots. Find the values of $a$ and $b$ The mark scheme ...
0
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1answer
68 views

Determining how many roots a cubic equation has.

I am working through some of the quizes on brilliant.org I came across this question. Suppose that the following cubic polynomial has one rational root and two non-real complex roots: $$ x^3 - ...
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0answers
39 views

What are some elementary books which discuss projective lines on surfaces with examples?

I have the books: W. H. Blythe, On models of cubic surfaces (1905) and A. Henderson, The twenty-seven lines upon the cubic surface, and a couple more modern algebraic geometry books including I. R. ...