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

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47 views

How to reduce a cubic function to 2nd degree function? [on hold]

I'm just stuck how to convert this equation to second degree equation and find all of its roots too. Equation is $x^3+2x^2-3x+6=0$
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1answer
22 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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35 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. ...
7
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0answers
110 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^2-2mx+N = ...
4
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0answers
90 views

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
54 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 ...
0
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0answers
27 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 ...
0
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0answers
70 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 ...
3
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2answers
25 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 ...
1
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1answer
24 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
57 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: $pqr= 2$ ...
0
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1answer
22 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 ...
0
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0answers
45 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
101 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. I want to try to use the intermediate value theorem, showing that $f(s)<0$ and ...
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2answers
50 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 $\{-1, 0, 1, 2\}$ ...
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2answers
44 views

Can we find the volume of a cube using a cubic equation. [closed]

As we know that cubic equations have a maximum power of $3$. Can we find the volume of a cube using it? Or it is something else? $$ax^3 + bx^2 + cx + d = 0$$ or $$x^3 = 0$$ If we can find the ...
2
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3answers
66 views

Why do we choose cubic polynomials when we make a spline?

Good morning, I want to learn more about cubic splines but unfortunately my class goes pretty quickly and we really only get the high level overview of why they're important and why they work. To me ...
2
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1answer
31 views

Modular Cubic Formula

What would be the process of solving a modular cubic equation? Eg. $$ax^3+bx^2+cx+d=0\pmod n$$ In the case that I was given, $d$ is a (very) large number, so rational root theorem isn't a viable ...
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3answers
34 views

Roots and Cubic equations

Let $\alpha$, $\beta$ and $\gamma$ be the roots of the equation $2x^3 + 4x^2 + 3x - 1 = 0$. Calculate $\frac{1}{\alpha^2 \beta^2} + \frac{1}{\beta^2 \gamma^2} + \frac{1}{\alpha^2 \gamma^2}$ GIVEN ...
5
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3answers
110 views

Solve $x^3 - x + 1 = 0$

Solve $x^3 - x + 1 = 0$, this cannot be done through elementary methods. Although, this is way out of my capabilities, I would love to see a solution (closed form only). Thanks!
2
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3answers
67 views

Without actually calculating the value of cubes find the value of $(1)^3+(2)^3+2(4)^3+(-5)^3+(-6)^3$. Also write the identity used

Without actually calculating the value of cubes find the value of $(1)^3+(2)^3+2(4)^3+(-5)^3+(-6)^3$. Also write the identity used
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1answer
64 views

Calculating originally arc approximated by cubic bezier curve

I have an cubic bezier curve, which is representing an arc by an approximation. The approximation was calculated with the kappa constant: $$ \\k = \frac43*(\sqrt{2}-1) $$ This means, that the ...
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2answers
69 views

$ay^3 + xy = ab^3$, can I isolate $y$?

I was wondering how much force it would take to compress a sphere of air (assuming Boyle's Law instead of the Real Gas laws, ignoring the engineering method of applying said force), so I started with ...
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2answers
44 views

Fully factorise $x^3-x^2-14x+24$ into linear factors

$$f(x)=x^3-x^2-14x+24$$ I've tried grouping the terms, but it just doesn't work out for me. Any help is appreciated.
2
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3answers
79 views

Roots of a cubic equation with coefficients based on unknown values $a$, $b$ and $c$.

I want to find the eigenvalues of the following matrix: $$ \left( \begin{array}{ccc} 0 & a & b \\ a & 0 & c \\ b & c & 0 \end{array} \right) $$ So, I found the characteristic ...
0
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0answers
60 views

Find factors of $0.08x^3 - 3.84x^2 + 42.66x - 137.7625$ using the Cubic Formula.

I have been going over this page as of late learning how to solve cubic formulas through depressing the equation, and solving for 'X'. Though, so far through numerous attempts, every single root I ...
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3answers
45 views

Number of cubes to fit into two bigger cube

I have following problem How many cubes, each with surface area of 36 square centi. are needed to form 2 cubes , each with surface area of 144 square centimeter To me, i would just divide 144*2 / ...
4
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0answers
88 views

Can -9 to 9 be placed in 41 lines of zero?

The cubic curve $2x^3-4x^2y+2xy^2-8x+y^3-y$ can be used to get lattice points allowing the placement of the numbers $-8$ to $8$ so that all 32 triplets that sum to 0 will be a straight line of three. ...
0
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1answer
33 views

Diophantus problem

I was given following problem as an example of early mathematics with the solutions. But it seems i can't understand from where they are getting the 35z^2 = 5 from in the solutions. Could someone ...
1
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1answer
61 views

Confused by the solution of $x^3+bx^2+cx+d=0$

From $x^3 + bx^2 + cx + d = 0$, we have $(x-x_1)(x-x_2)(x-x_3)=0$ for some roots $x_1$, $x_2$ and $x_3$. Expanding this second expression gives us $$x^3 + \left(x_1+x_2+x_3\right)x^2 + \left(x_1x_2 + ...
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0answers
87 views

Deriving General Solution for Depressed Cubics

I was reading An Imaginary Tale: The Story of the Square Root of Negative One and it begins with a derivation of the general solution for depressed cubics. It begins with the format for depressed ...
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2answers
40 views

Building a cubic polynomial with certain parameters

I am trying to build a cubic polynomial $y = ax^3 + bx^2 + cx + d$ with the following conditions: $y(0) = 3$ $y'(0) = 2$ $y'(1) = 0$ $y(1) = 4$ Unfortunately, solving the required system yields ...
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1answer
20 views

Application of the Rational Roots Theorem

Let f(x)=3x$^3$ - 40x$^2$ + 97x + 10 a. Find a rational number r such that f(r) = 0. (Hint: Use the rational roots theorem to narrow down possibilities for r.) So, I figured this part out. write r ...
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1answer
57 views

Factorizing Cubic Equations.

Factorization of Cubic Equations has always obstructed my way to the solution to a problem. Is there any simple technique to factorize them?
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2answers
68 views

Find the sum of the $9$th powers of the roots of $x^3+3x+9=0$

Let $$x^3+3x+9=0$$ and $x_1,x_2,x_3$ be the roots of this equation. Given that $\displaystyle S=x_1^9+x_2^9+x_3^9$. What is the exact value of $S$? I think that $S=0$, but I am not sure. I tried to ...
0
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1answer
36 views

Finding $N$ when sum is given

$1^2 + 2^2 + 3^2 + 4^2 + \cdots + N^2 = S$ Given $S$ How to find $N$. The Formula to Find $S$ from $N$ is: $S = \frac{N(N+1)(2N+1)}6$ so this gives me a cubic equation: $2N^3 + 3N^2 + N = 6S$ ...
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0answers
14 views

Interpolating With Hermite Cubics in 2D

I want to estimate the value of the function f(x,y) at a particular point. Suppose I am given two points, (x1, y1) and (x2, y2), along with the value of f and its partial derivatives fx and fy at the ...
2
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2answers
83 views

Find the value of $f(x)$ for $x = 2 + 2^{2/3} + 2^{1/3}$

If $x = 2 + 2^{2/3} + 2^{1/3}$, then find the value of $f(x)=x^3 - 6x^2 + 6x$. I am unable to get to the answer - end up with more than one term. Please help me solve this!
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30 views

Deriving the solution to an algebraic cubic equation

I want to derive the general formula to solve the general cubic equation $ax^3+bx^2+cx+d=0$ I will give my derivation of the simple quadratic equation to show what I'm looking for So we want to ...
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1answer
28 views

Find cubic equation using known points

I'm writing a javascript game, and are working on some interpolation functions, but I don't have enough mathematical knowledge to work out this problem. I want to use a cubic function for ...
3
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2answers
67 views

Simplify $x^3 - 4x^2 + 10x - 125$

I've been trying to simplify $x^3 - 4x^2 + 10x - 125$ for a while now, and I don't seem to progress. I know that the factors of $125$ are $1$, $5$, $25$ and $125$, but none of these seem to help here. ...
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1answer
42 views

Using sum/product of roots of a cubic equation to solve given expression $(a+b)^3+ (b+c)^3 + (c+a)^3$

If $a, b, c$ are the roots of the equation $7x^3- 25x +42 =0$, then the value of the expression $(a+b)^3+ (b+c)^3 + (c+a)^3$ is? I tried to solve this but wasn't able to simplify the term to be able ...
10
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2answers
218 views

Cubic polynomial equal to a cube

I've been researching cubes and I'm trying to solve this Diophantine equation over the integers. $$ax^3 + bx^2 + cx + d = y^3$$where a, b, c, d are parameters for a given $n$. For example, for $n = ...
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1answer
45 views

how to show that $\mathbb{Q}[\sqrt[3]{2}]$ is a field? (by elementary means)

To be very concrete, I want to show that every element of the form $1/(p+qx+rx^2)$ where $x=\sqrt[3]{2}$ where $p,q,r$ are rationals can be written in the form $a+bx+cx^2$ where again $a,b,c$ are ...
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3answers
92 views

Solving a Perturbed Cubic Equation

Consider a cubic equation $(1 + \epsilon)x^3 - 2ax^2 + (a - 3\epsilon)x + 2\epsilon = 0$ where $\epsilon > 0$ and $a \gg 1$. In the limit of $\epsilon \rightarrow 0$, $x(x^2 - 2ax + a) = 0$ so ...
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1answer
42 views

most general form of $X - A = X^{-1}B (X^{-1}BX^{-1}+ C)^{-1}$ that has a real solution $X = f(A,B,C)$?

What is the most general form of the cubic matrix equation $X - A = X^{-1}B (X^{-1}BX^{-1}+ C)^{-1}$ that has a real solution of the form $X = f(A,B,C)$, where $A,B$ and $C$ are positive definite ...
2
votes
2answers
69 views

Motivational example for complex numbers

Years ago I was introduced to complex numbers. In class we had been talking about the cubic polynomial and its solutions. At one point we saw an example where, when using the formula, one had to stop ...
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1answer
82 views

How to calculate the length of a cubic hermite spline between two points

I am using the following equation to create a cubic hermite spline: $$p_n(t) = a_nt^3+b_nt^2+c_nt+d_n$$ $$1\geq t\geq 0$$ $p_n(t)$ is the unit interval interpolation equation for dimension n. $t$ is ...
0
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3answers
69 views

The real cubic root expression

$2x^3-2x^2-3x+2=0$ has 3 real root, but they are all express in such way: $x=\dfrac{1}{3}\left(1+\dfrac{\sqrt[3]{-23+3i\sqrt{237}}}{\sqrt[3]{2^2}}+\dfrac{11}{\sqrt[3]{2(-23+3i\sqrt{237}})}\right)$ ...
0
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0answers
279 views

Issue with modelling word problems as cubic equations

I am having problems with a specific set of word problems, which are meant to be modeled as cubic equations in order to be solved. I will give some examples to specify where I can't solve it. The ...