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

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Solving for a cubic polynomial's roots using Viete's Theorem

I am asked to find the roots $f(x)=x^3-24x^2-24x-25$. However, the only thing I am aware of in regards to finding the roots of cubic polynomials is Viete's Theorem. However, this theorem requires that ...
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1answer
48 views

Expressing the roots of a cubic as polynomials in one root

All roots of $8x^3-6x+1$ are real. (*) The discriminant of $8x^3-6x+1$ is $5184=72^2$ and so the splitting field of $8x^3-6x+1$ has degree $3$. Therefore, all three roots can be expressed as ...
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1answer
21 views

Random Walk with overshoot, step sizes $+1, -2$. Solve the polynomial in $e^λ$ [on hold]

If the moment generating function is $$mS(θ) = E(e^{θS}) = pe^θ + qe^{−2θ} = 1$$ Show that setting $$mS(\lambda) = 1$$ yields the unique positive solution: $$ \lambda = \log { \frac{q + \sqrt{4pq + ...
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1answer
66 views

A simple cubic equation problem:

Consider the cubic equation $$az^3-bz^2+\bar{b}z-\bar{a}=0$$ where $a$ and $b$ are non-zero complex numbers. Suppose $z_1, z_2$ and $z_3$ are the roots. Question: Which $a$ and $b$ gives ...
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1answer
28 views

Solving a cubic function with P and Q

I have been struggling a little bit over solving cubic functions. I have been trying to use the P and Q method. So the question is What is the approximate value of the greatest zero of $f(x) = x^3 - ...
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1answer
47 views

Symmetric system of equations problem

Solve the following simultaneous eqations on the set of real numbers: $$a^2+b^3=a+1$$ $$b^2+a^3=b+1$$ I have found two trivial solutions: $$a=b=1$$ $$a=b=-1$$ but I can't prove that there are no ...
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11answers
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What equation produces this curve?

I'm working on an engineering project, and I'd like to be able to input an equation into my CAD software, rather than drawing a spline. The spline is pretty simple - a gentle curve which begins and ...
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0answers
59 views

The probability that a random (real) cubic has three real roots

We can formalize the notion of the probability that a randomly selected quadratic real polynomial has real roots as follows: Suppose $R > 0$, and suppose the random variables $a, b, c$ are ...
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1answer
25 views

How to draw cubic plane curve?

In Python, using MatPlotLib, given [vector] parameters $a$ and $b$ and [scalar] parameter $c$, I want to draw a general cubic plane curve in 2-dimensional space (regular plane with $x$ and $y$ axes): ...
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1answer
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I don't understand this process in solving the cubic

My previous question was about the cubic. Now, the following confuses me Q. Use $\cos{3 \theta}=4 \cos^3{\theta}-3 \cos{\theta}$ to solve the cubic $t^3+pt+q=0$ for $p,q$ real when ...
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1answer
35 views

Discriminant of the cube, quartic…

I was told the discriminant of the cubic is $$\Delta=-27q^2-4p^3$$ and that $\Delta>0$ means that there are three real roots. Simply put, why is this the discriminant? I ask this because, looking ...
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29 views

A cubic relationship for perfect numbers

Combining the identity (the composition of the polynomial $P_3(x)=\frac{x(x+1)(2x+1)}{6}$ with the arithmetical function $N(n)=n$ and the known proof by induction) ...
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2answers
55 views

'Elegant' ways on solving for roots for this cubic function?

I have this interesting cubic equation, $$ x^{3} - 80\alpha x^{2} + (1744\alpha^{2}-81)x + (3240\alpha-5760\alpha^{3}) = 0 $$ where $\alpha$ is some constant. I went about the method of Cardano, ...
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8answers
145 views

Why is $x^3-x^2+x$ injective?

The function $$ f(x) = x^3-x^2+x$$ is injective (as seen on the graph), but by doing f(a) = f(b) I can't get to the point where a = b. Can this be indicated in an analytical way (without graph) ?
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4answers
81 views

Cubic polynomial with three (distinct) irrational roots

I am looking for an equation $$x^3+ax^2+bx+c=0, \qquad a, b, c \in \Bbb Z,$$ of degree $3$ that has $3$ different roots. For an equation of degree $2$ it is easy---for example $x^2-2=0$---but I ...
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0answers
57 views

Without solving it, is there an elementary way to show that $X^3+Y^3=Z^3$ has a finite number of primitive [and non-trivial] integer solutions?

Considering the cubic case of Fermat’s Last Theorem, I make the following claim: Proposition: The Diophantine equation $$ X^3 + Y^3 = Z^3 \tag{$\star$} $$ has a finite number of primitive [and ...
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3answers
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Three questions about the form $X^2 \pm 3Y^2 = Z^3$ and a related lemma

In Ribenboim’s Fermat’s Last Theorem for Amateurs, he gives the following lemma [Lemma 4.7, pp. 30–31]. Lemma. Let $E$ be the set of all triples $(u, v, s)$ such that $s$ is odd, $\gcd(u,v) = 1$ and ...
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1answer
12 views

Substitution of factors of the free term to find factors of a cubic equation

I was taught that finding a factor (and hence a solution) of a cubic equation may be easier if I try if the factors of the free term are roots of the equation. For example, if one has an equation ...
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2answers
30 views

Suppose I have a cubic equation with unknown coefficient. How can I find the right coefficient to settle the roots condition?

Suppose I have this equation, $S^3 + 19S^2 + 25S - 75 + K=0$ How can I find $K$ that makes all the root values negative? So far, all I'm doing is substituting $K$ with random values until it gives ...
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24 views

draw a sketch of the derivative graph

The turning points of a graph of a cubic function $h(x)$ are $(2, -3)$ and $(5, 4)$ Draw a sketch of $h'(x)$ showing the $x$ intercepts only. My personal attempt is: $(x-2)(x-5)=0$ which is $ ...
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1answer
75 views

Trying to prove $c^3a^2+(9c^2-b^2)a+(27c-10b)=0$ has no positive integer solutions

I'm trying to prove (or, I suppose, disprove) the following claim, in either version. Conjecture (Strong Version): There are no positive integers $a,b,c$ such that $$c^3a^2+(9c^2-b^2)a+(27c-10b)=0.$$ ...
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0answers
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Solving for $t$ in a cubic function

I have a function $h(t)=5t^3 + 30t^2 + 45t + 4$ I would like to solve for t when $h(t)=50$ I have tried factorizing by inspection and by table but the function does not appear to have a real ...
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4answers
60 views

Finding the roots of a cubic equation

I trying to find the roots of the equation $$ax^3 + bx^2 + cx + d = 0$$ By using some changes of variable (which does not really matter now) I was able to rewrite this equation as $$z^3 - ...
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2answers
31 views

Finding the complex roots of a cubic equation [closed]

I'm trying to solve the equation $$ (w^{3})^{2} - qw^{3}-\frac{p^{3}}{27}=0$$ as part of the solution to the generic cubic equation (but I will not start from there. It's pointless). This is a ...
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2answers
175 views

Cubic Equation Related

How do I solve the following cubic equation? $(x+1)^2(x-2)-2nx-6n=0,n \in \mathbb{N}$ $ \therefore (x+1)^2(x-2)-2n(x+3)=0$ I don't know how to solve further.
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1answer
27 views

bad explanation in how to solve the cubic

The question asked me to solve $t^3+pt+q=0$ where $p,q$ real and $27q^2+4p^3<0$ using the identity $\cos{3\theta}=4\cos^3{\theta}-3\cos{\theta}$. Answer goes like this and I have stopped ...
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1answer
32 views

Vieta's Theorems Clarification

For applying Vieta's Theorems on cubic equations, I've seen two conflicting answers from two different books. Book 1 says that for a cubic equation $px^3 + qx^2 + rx + s = 0$ has three different ...
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2answers
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Is this polynomial irreducible? $f(x)=x^3-117x+53$

I am trying to find if $f(x)=x^3-117x+53$ is irreducible or reducible. I tried Eisenstein's criterion but this fails as $53$ does not divide $117$ After some thinking, if f(x) has any roots then ...
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1answer
91 views

How can $ (x+b)^{2/3} - x^{2/3} $ be simplified using difference of cubes formula?

How can $ (x+b)^{2/3} - x^{2/3} $ be simplified using difference of cubes formula? I'm solving a question on differentiation and it led to this expression. How can this be simplified?
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50 views

How is $ [(x+h)^{1/3} - x^{1/3}] [(x+h)^{2/3} +x^{1/3}(x+h)^{1/3}+ x^{2/3}] $ simplified to become $ (x+h-x) $?

How is $ [(x+h)^{1/3} - x^{1/3}] [(x+h)^{2/3} +x^{1/3}(x+h)^{1/3}+ x^{2/3}] $ simplified to become $ (x+h-x) $ ?? I'm currently reading a text and I've been trying to get the hang of this for a ...
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2answers
23 views

Find the Cubic equation given 1 Point and Slope

The question asks to find a cubic equation given point A (0/18) slope here is 0. Point B only given the x=20 and slope is -0.6. Please help, been trying to solve it for so long now. :( 1.) ...
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1answer
62 views

Resolving the tedious cubic

The equation given to me is $$4x^4 + 16x^3 - 17x^2 - 102x -45 = 0$$ I'm asked to find it's resolvent cubic which is not so difficult to find. But the problem is that the question further asks to find ...
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1answer
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System of equations and inequalities involving quartic and cubic eq's

Given the system of equations/inequalities: $$\left \{ \begin{array}{llll} 16x^4-40ax^3+(15a^2+24b)x^2-18abx+3b^2 = 0 \\ 5ax-4x^2-b>0 \\ 15ax-20x^2-3b<0, \end{array}\right.$$ where $x<0$, and ...
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3answers
43 views

alternate forms of $t^3 -1$

I'm looking to understand how $t^3-1$ factorises to $(t-1)(t^2+t+1)$. I know how to find the first factor $(t-1)$, but have trouble finding the second factor $(t^2+t+1)$. I've tried doing long ...
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0answers
67 views

Calculate the eigenvalues of the following 3x3 matrix and solving a cubic with complex roots?

I have a question regarding a 3x3 matrix and its eigenvalues. The matrix is $A= \begin{bmatrix}1 & 1 & 2\\-2 & -1 & 2 \\ -1 & -2 & 3\end{bmatrix}$ What I have attempted is to ...
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2answers
55 views

Determining whether there are solutions to the cubic polynomial equation $x^3 - x = k - k^3$ other than $x = -k$ for a given parameter $k$

Let $k$ be a real parameter, and consider the equation $$x^3 - x = k - k^3 .$$ Obviously, $x=-k$ is a solution. Is it the only one? How to prove it?
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1answer
72 views

How to solve cubic equations with given coefficients?

I have a large data set that requires a cubic equation to be solved for each point. There are too many points to use Goal Seek (numerical Excel method) on them all. For example: $$y=7\cdot ...
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3answers
65 views

Compare five ways of solving cubic equation by iterations (nested expressions)

Say we have a depressed cubic equation in the general form: $$x^3-bx-c=0$$ There are basically five ways of solving it by iterations. Let's consider them in no particular order (the names are my ...
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1answer
35 views

Using Vieta's formula to solve a Cubic Polynomial

I am using Vieta's formula on a cubic polynomial to solve for the sum of the roots $p_1+p_2+p_3$ and the product $p_1\times p_2\times p_3$. I've solved for both $p_1+p_2+p_3$ and $p_1\times p_2\times ...
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1answer
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Cubic equations

I have a solution to a differential equation $y^3/3+2y^2+4y = t + k$. I want it in the form $y = f(t)$. Wolfram alpha tells me the answer is $y = \sqrt[3]{k - 3t -8} - 2 $. Now how did it do that?
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Given the roots, how can I define the cubic polynomial function?

Let's say that I have A, B and C as the roots of a cubic polynomial function. How can I find this function?
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1answer
72 views

Coefficents of cubic polynomial and its least root

Let $x^3-(m+n+1)x^2+(m+n-3+mn)x-(m-1)(n-1)=0$, be a cubic polynomial with positive roots, where $m,n \ge2$ are natural nos. For fixed $m+n$, say $15$, it turns out that least root of the polynomial ...
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0answers
34 views

What is relation between a particular root of two polynomials?

We have $$x^3+(m+n+p-1)x^2-((m+n)(1-p)+2p-1-mn)x-(p-1)(m-1)(n-1)=0$$ in which $m,n\ge2, p\ge1$ are natural numbers. All the three roots of this cubic are positive. Let $\lambda$ be the least of them. ...
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1answer
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Is it possible to write $\cos \left( \frac{1}{3}\arccos \frac{37}{64}-\frac{\pi }{3} \right)$ as a radical expression of real number

$\cos \left( \frac{1}{3}\arccos \frac{37}{64}-\frac{\pi }{3} \right)=\frac{{{\left( -37-3\text{i}\sqrt{303} \right)}^{1/3}}+{{\left( -37+3\text{i}\sqrt{303} \right)}^{1/3}}}{2}$, the number inside the ...
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0answers
47 views

How do I derive the cubic formula? (without substitutions)

I've heard of a number of ways that people have derived a cubic formula (I've even heard of a number of different ways to show the formula itself too). What I want to know is how to derive it without ...
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2answers
43 views

How to solve $8x^3-6x+6xy^2=0=4y^3-4y+6xy^2$

How can I solve this system? $$ \left\{\begin{matrix} 8x^3-6x+6xy^2=0\\ 4y^3-4y+6xy^2=0 \end{matrix}\right. $$ I do $$ \left\{\begin{matrix} 8x^3-6x+6xy^2=0\\ 4y^3-4y+6xy^2=0 \end{matrix}\right. ...
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3answers
57 views

Find the two values of $k$ for which $2x^3-9x^2+12x-k$ has a double real root.

Find the two values of $k$ for which $2x^3-9x^2+12x-k$ has a double real root. I've found one method which is to equate $$2x^3-9x^2+12x-k=2(x-r)^2(x-c)$$ Expanding and equating coefficients I ...
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2answers
41 views

All zeroes of monic cubic $x^3+ax^2+bx+c$ are negative reals and $a\lt3$. Range of $b+c$?

$a,b,c$ are real numbers. I have to find the range of values of $b+c$. So, I started off by assuming $\alpha , \beta , \gamma$ as the roots. This gives us $\alpha \beta \gamma = -c$ and ...
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2answers
31 views

Finding roots of cubic (trig)

The question is By putting $x$ $=$ $\frac 23 cos (\theta)$ Find the exact roots of the equation in terms of $\pi$ $$ 27x^3 - 9x = 1 $$ What I have attempted: $$ ...