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

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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
42 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
51 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
35 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
48 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
31 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
33 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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2answers
47 views

Find the value of this cubic polynomial [closed]

If $x=3+3^{1/3}+3^{2/3}$, then what is the value of $x^3-9x^2+18x-10$?
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3answers
45 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
30 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
34 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
36 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
73 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
61 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
86 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
41 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
44 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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1answer
33 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
36 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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0answers
44 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
50 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
54 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
57 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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0answers
68 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
18 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
28 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
27 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
38 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
48 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
24 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
58 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
69 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!
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3answers
61 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 ...
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1answer
52 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
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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. ...
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1answer
184 views

What's so special about primes $x^2+27y^2 = 31,43, 109, 157,\dots$ for cubics?

While trying to find a closed-form solution for particular cubics as sums of cosines (related to this question), I came across this family with all roots real, $$F(x) = x^3+x^2-mx+N = ...
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0answers
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A cubic equation: $u^3−2u^2−2v^3−20v^2+16v=0$

Update (Dec. 22): I have already solved this question with Magma. Recently, I read a paper [1] and saw the following equation: $$u^3−2u^2−2v^3−20v^2+16v=0.$$ The author then got a Weierstrass ...
4
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1answer
63 views

Solving cubic with a nice real solution

Solve the cubic for $x\in\mathbb{R}$ $$x^3-9 x^2-15x-6 =0$$ The only real solution is $x=3+2\sqrt[3]{7}+\sqrt[3]{7^2}$. Given the regularity of this solution, can we solve this constructively ...
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0answers
32 views

How to find the Coefficient of the Quadratic Term?

Given $4x^3 +bx^2+cx+d$ and two roots of this cubic function $(0,0)$ and $(2,0)$ Find the coefficient of the quadratic term? When I first read this I had no idea how to solve this and still ...
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0answers
74 views

How can I check these equations if they have a solution?

I have two equations which are: $p^3+k\equiv0 (mod \quad h) $ and $(3p^2+3mp+m^2)m\equiv 0(mod \quad h)$ where $k,h,m >0$ and $p\ge0$ and $h\nmid m$ I need to show for given k,m,h and for all ...
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2answers
31 views

Can all cubic/quartic polynomials be expressed in a form with only one x term?

Quadratic expressions $ax^2+bx+c$ can all be expressed in a form with only one x term: $$a(x+\frac{b}{2a})^2+c-\frac {b^2}{4a}$$ Is the same true for all cubic or quartic expressions? Is there a ...
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1answer
26 views

Determine parameter so that the absolute value of real solution of the equation is larger than the modulo of complex solution

Given the equation: $$x^3+x+\lambda=0$$ determine real parameter $\lambda$ so that the real solution is greater by absolute value than modulo of the complex solutions. My attempt: Let $x_1$ be the ...
3
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4answers
59 views

How do I find the sum of the cubes of the roots in a cubic polynomial?

I have an equation, $x^3-x^2+x-2$, with three distinct roots, $p$, $q$ and $r$. What is the value of $p^3+q^3+r^3$? I'm not sure how to do this. Using Vieta's formula, we know that: $pqr= 2$ ...
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1answer
25 views

cubic integral roots

I am trying to find the integral roots (if they exist) of the following polynomial. Additionally, it would be helpful if someone could explain an algorithmic approach to solving this. $$ f(x) = 2x^3 ...
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1answer
75 views

What is the condition for the first root of a cubic function to be positive?

Is there any way to determine if the "first root" of a cubic equation is positive, assuming that it's real, given coefficients $a,b,c,$ and $d$? I tried following along with Wikipedia's explaination ...
4
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0answers
113 views

$15a+6b+4c+8d=0$ implies $ax^3+bx^2+cx+d$ has a positive root

Let $a,b,c,d$ be real numbers such that $15a+6b+4c+8d=0$. Show that $f(x)=ax^3+bx^2+cx+d$ has a positive root. (Komal, Problem N. 170.) I want to try to use the intermediate value theorem, showing ...
0
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2answers
65 views

Cubic spline solving equation

$$S(x)=\begin{cases} x^3 +4x^2 -2x +7 & \text{ if } -1\leq x\leq 0, \\ x^3 - 2x^2 +4x +5& \text{ if } 1\leq x\leq 2, \end{cases}$$ is a cubic spline with knots $\{-1, 0, 1, 2\}$ ...