Questions tagged [cubics]
This tag is for questions relating to cubic equations, these are polynomials with $~3^{rd}~$ power terms as the highest order terms.
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Failing to solve cubic equation
I'm trying to solve a more complex cubic equation but to simplify things as a start I picked this one:
$$ 3\cdot 4^3+2\cdot 4-200=0 $$
Here $x$ is $4$.
I'm looking at wikipedia and trying to solve ...
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cubic equation edge cases
Working on general cubic equation solver in form ax^3+bx^2+cx+d=0 And have no clue for special cases:
In terms of cubic there should be one real root and two complex, or 3 real roots if coefficients ...
2
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3
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Prove that $a_3 \lambda^{3} + a_2 \lambda^{2} + a_1 \lambda + a_0 = 0$ has three real roots
I'm trying to prove that the cubic equation
$a_3 \lambda^{3} + a_2 \lambda^{2} + a_1 \lambda + a_0 = 0$
has three real roots. The coefficients are
$a_3 = - 1 - \sigma - \tau - \chi$
$a_2 = -2 (\sigma +...
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There is a compass-like tool that can draw $y=x^2$ on paper. Is there one for $y=x^3$?
Is there a tool that can draw $y=x^3$ on paper?
I'm referring to low-tech tools, e.g. not computers.
I only know of tools that can draw $y=x^2$. The YouTube video "Conic Sections Compass" ...
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Prove this potential cubic theorem/formula [closed]
Prove that $\dfrac{x³+y³+z³}{x+y+z}=x²+y+z$; if $x<y<z$; $y=x+1$; $z=y+1$; and $x$, $y$ and $z$ are positive whole numbers.
If you prove this, I technically discovered a new formula since I ...
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Prove $2(a+b+c)\left(1+\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\ge 3(a+b)(b+c)(c+a)$ for $abc=1.$
Let $a,b,c>0: abc=1.$ Prove that: $$2(a+b+c)\left(1+\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\ge 3(a+b)(b+c)(c+a). $$
I've tried to use a well-known lemma but the rest is quite complicated for me.
...
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Three real roots of a cubic
Question: If the equation $z^3-mz^2+lz-k=0$ has three real roots, then necessary condition must be _______
$l=1$
$ l \neq 1$
$ m = 1$
$ m \neq 1$
I know there is a question here on stack about ...
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Why are all real inflection points on a cubic projective algebraic curve on 1 line?
Say we have $C\subset \mathbb{CP}^2$, a smooth curve of degree 3. I am aware of the group structure on cubics, what I don't get, is why are all inflection points with only real coordinates lie on a ...
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More $a^3+b^3+c^3=(c+1)^3,$ and $\sqrt[3]{\cos\tfrac{2\pi}7}+\sqrt[3]{\cos\tfrac{4\pi}7}+\sqrt[3]{\cos\tfrac{8\pi}7}=\sqrt[3]{\tfrac{5-3\sqrt[3]7}2}$
I. Solutions
In a previous post, On sums of three cubes of form $a^3+b^3+c^3=(c+1)^3$, an example of which is the well-known,
$$3^3+4^3+5^3=6^3$$
we asked if there were polynomial parameterizations ...
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Disjoint exceptional lines on non-minimal cubic surface
A diagonal cubic surface $\sum_{i=0}^3a_iT_i^3=0$ is not minimal if, for example, $a_1a_2a_3^{-1}a_4^{-1}\in(k^*)^3$. This should be because there is an exceptional line $D$ such that no element in ...
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Prove $(1+a^3) (1+b^3)(1+c^3) \ge (\frac{ab+bc+ca+1}{2})^3$
This is a question from 4U maths (highest level of Y12 maths in Australia) from a generally difficult paper. The question itself does not define what a, b, and c are - based on the comments, we assume ...
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Descartes folium
The geometry of Descartes' folium, $x^3+y^3=3axy$ has been well studied. Can someone tell me which geometric property characterizes the following cubic curve:
$$bx^3+y^3=3axy$$
The previous curve is a ...
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Set of coefficients of degree three monic real polynomial with three real roots is connected.
Let $p(x)=x^3+ax^2+bx+c$ be a cubic polynomial with real coefficients $a, b, c,$ and define:
$$D=\{(a,b,c)\in \mathbb{R}^3\mid \text{the polynomial}\ p(x)\ \text{factors into linear factors over }\ \...
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Can you tell me about the (probably) well known relationship between the coefficients of a cubic and some features of a rectangular solid?
If we look at the expansion of this
$$(x+a)(x+b)(x+c)=x^3+(a+b+c)x^2+(ab+bc+ca)x+abc$$
And consider a rectangular solid with length, width and height of $a, b, c$ respectively.
Then
$$l_{edges}=4(a+b+...
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For translation of axes, is there a definite equation for any of "the 27 lines" on the Clebsch Diagonal Cubic?
For the Clebsch Diagonal Cubic (related to a Quanta mag article on Hilbert's 13th Problem), I want to generate a point at will on any of these lines along the surface. Wolfram calls these "...
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Prove a relation between the coefficients of a depressed cubic.
The equation $x^{3}+px^{2}+q=0$ where p and q are non-zero constants, has three real roots $\alpha$, $\beta$ and $\gamma$. Given that the interval between $\alpha$ and $\beta$ is p and that the ...
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find $a$ and $b$ where $x^3 - 4x^2 -3x + 18 = (x+a)(x-b)^2$
I have a problem solving this as when I match the coefficients to the expanded brackets I end up with $2$ unknowns $a \& b$. So cannot substitute any values to find the other. According to the ...
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Proof of conditions for polynomials
Find the conditions for the roots $\alpha, \beta, \gamma$ of the equation $x^3-ax^2+bx-c=0$ to be in: $(i)$A.P.; $(ii)$G.P.
If the roots are not in A.P. and if $\alpha+\lambda,\ \beta+\lambda,\ \...
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How to prove that the real solution to this equation is greater than 1 without solving it? $x^3 - x^2 +2x - 4 =0$ [closed]
I was attempting a problem online and reached a point where I needed to prove that the real solution to this equation is greater than 1 but couldn't seem to find a way to do it. Please advise!
If you ...
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2
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Prime numbers $p$ such that $7p+1$ is a cube [closed]
I am stuck on this assignment. I have to find every prime number $p$ such that $7p+1$ is a cube number. After exploring enough I must say there is no prime $p$ satisfying this condition. I have tried ...
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Why Can't Cubic Equation Have Fractional Solutions When Its Coefficients Are All Integers? [duplicate]
In Leonhard Euler's book, "The Elements of Algebra" he seems to say that if we convert any cubic equation into the form $x^3 + ax^2 + bx + c$, and make sure that $a$, $b$ and $c$ are integer ...
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Finding rational coefficients of a cubic polynomial that fits 4 data points that have been floored to an integer
I have 4 data points:
(204, 5422892)
(205, 5722486)
(207, 6343357)
(213, 8386502)
I have information that these data points were generated with a cubic polynomial
$y = ax ^ 3 + bx ^ 2 + cx + d$
with ...
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Prove that $\mathbb Q(\cos\tfrac\pi7)\neq\mathbb Q(\cos\tfrac\pi9)$
Let $\tau=2\pi$ be the full angle. (tau)
For any integer $k$ and any angle $\theta$, $\cos(k\theta)$ is a polynomial in $\cos\theta$. In particular, $\cos(2\theta)=2\cos^2\theta-1$, which shows that $\...
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How to solve $x^3−x+1=0$
I am interested in finding a solution for the equation:
$$ x^3 - x + 1 = 0 $$
I've noticed that there are numerous polynomial equations where one of the coefficients is zero. Could you provide ...
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Numerical Analysis - natural cubic spline and clamped cubin spline
a question from first exam period (A).
True or false ( it is false, but I want to understand ).
Given the following intersection points $x_0, x_1,...,x_n$ (interpolation nodes ) and the values of ...
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Who first deduced the formula for distance between the roots of a cubic equation and what does it look like?
$ax^3+bx^2+cx+d=0$
$p\equiv|x_i-x_j|=$ formula ?
3
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Simplifying Coefficients of a Cubic Polynomial with Complex Roots
I am currently encountering difficulties while trying to solve the following question, and I would greatly appreciate any assistance you can provide.
Let $a,b,c$ be complex numbers.
The roots of $z^{3}...
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If the complex roots of $x^3-x-2=0$ are $r\pm si$, and $As^6 +Bs^4 + Cs^2 =26$ for integers $A$, $B$, $C$, find $A+B+C$
The question:
In the cubic $x^3-x-2=0$, there is one real root and two complex roots of the form $r\pm si$, with $r$ and $s$ real. If there exists integers $A,B,$ and $C$ such that $As^6 +Bs^4 + Cs^2 ...
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Prove irreducible cubic polynomial over $\mathbb{Q}$ with a cyclic galois group has real roots
I want to prove the following:
Let $f\in \mathbb{Q}[x]$ be an irreducible cubic polynomial, whose Galois group is cyclic. Prove that all of the roots of $f$ are real.
I know that the Galois group $G$...
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Let a, b, c, d be complex numbers satisfying $a+b+c+d=a^3+b^3+c^3+d^3=0$. Prove that a pair of the a, b, c, d must add up to 0 [duplicate]
When doing this I tried using the identity
$x^3+y^3+z^3=3xyz$ if $x+y+z=0$
I take $x=a$, $y=b$, and $z=c+d$
So $a+b+(c+d)=0$
$a^3+b^3+(c+d)^3=3ab(c+d)$
$a^3+b^3+c^3+d^3+3cd(c+d)=3ab(c+d)$
$(a^3+b^3+c^...
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Can I use this algorithm for solving cubic equations?
I am trying to find the root solutions for a cubic equation including the eigenvalues of each root.
I tried to put the equation into my calcualtor but the calculator doesn't show solutions that has ...
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Convert an expression with radicals into simpler form
It was pointed out in a mathologer video on the cubic formula that $\sqrt[3]{20 + \sqrt{392}} + \sqrt[3]{20 - \sqrt{392}}$ is actually equal to $4$. Is there a series of transformations that can be ...
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Solving a cubic using triple angle for cos (i.e $\cos(3A)$)
a) Show that $x=2\sqrt{2}\cos(A)$ satisfies the cubic equation $x^3 - 6x = -2$ provided that $\cos(3A)$ = $\frac{-1}{2\sqrt{2}}$
I did not have a difficulty with this question, I have provided it for ...
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1
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How do I find a cubic equation given only one root?
Given the root of a cubic equation $Z = \sqrt[3]{Y + \sqrt{Y^2 - \frac{X^6}{27}}} + \sqrt[3]{Y - \sqrt{Y^2 - \frac{X^6}{27}}} - X$ and the assumption that both $X$ and $Y$ are greater than zero, is ...
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Best way to solve $\frac{x^3+3}{x^2+1}>\frac{x^3-3}{x^2-1}$
I was wondering what the best way to solve questions like these are?
$$\frac{x^3+3}{x^2+1}>\frac{x^3-3}{x^2-1}$$
I can get the answer, which is $(-\infty,-1)\cup(1,3)$. But I'm not sure if I have ...
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Is there any faster way to factor $x^3-3x+2$?
$$x^3-3x+2$$
$$x^3-3x+x^2+2-x^2$$
$$x^2-3x+2+x^3-x^2$$
$$(x-2)(x-1)+x^2(x-1)$$
$$(x-1)[x^2+x-2]$$
$$(x-1)(x+2)(x-1)$$
Is there a better, faster way to factor this cubic trinomial?
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Classification of curves passing through 7 points. (Hartshorne III ex 10.7)
This is the exercise III 10.7 in Hartshorne's Algebraic Geometry
I am not sure if I misunderstood the question.
The seven points of the projective plane over $\mathbb{F}_2$, I think, means
$\{[x_0,...
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Prove $\lim_{n\to\infty}\int_0^{a} \left(\sqrt{2n/\pi-x^2}-\sqrt{2n/\pi-a^2}\right)dx=1/6$ where $a$ is the largest real root of $4x^6+x^2=2n/\pi$.
I've never seen anything like this before: an unsolvable cubic, within a definite integral, within a limit (which applies to the cubic and the integral), resulting in a simple closed form.
Prove $\...
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2
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Condition for the existence of positive solution to cubic equation
In a physics textbook I have encountered a cubic equation of the form:
$$Ax^3-Bx+C=0$$
The book states that there exists a positive solution $x>0$ to this equation if and only if the following ...
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Find sum of all integral values of $r$ such that all roots of the equation $x^3-(r-1)x^2-11x+4r=0$ are also integers
Find sum of all integral values of $r$ such that all roots of the equation $$x^3-(r-1)x^2-11x+4r=0$$
are also integers.
What I could do was $$r=\frac{x^3+x^2-11x}{x^2-4}=x+1+\frac{4-7x}{x^2-4}$$
Since ...
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Find all real numbers $a$ for equation $x^3 + ax^2 + 51x + 2023=0$, has two equal roots.
Problem:
Find all real numbers $a$ for which the equation, $x^3 + ax^2 + 51x + 2023=0$, has two equal roots.
This problem is from an algebra round of a local high school math competition that has ...
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Why doesn't simultaneous equations work to find co-efficients of a cubic that passes through four points?
I'm trying to find the equation of a cubic that passes through three specific points (technically it's four but that point is y-intercept). The equation would look something like this:$f(x)=ax^3+bx^2+...
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Order $3$ linear transforms invariating a binary cubic form
Consider $P(x,y)$ a homogenous polynomial of degree $3$ in two variables (a binary cubic). To it we associate first the $2\times 2$ matrix
$$\frac{1}{2}\operatorname{Hess}(P) = \frac{1}{2}\cdot\left( ...
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Involution on monic cubic polynomials related to nesting/denesting of cubic radicals
Consider the involutive transformation
$$\mathbb{R}^3 \ni (a,b,c) \overset{\phi}{\mapsto} \left( \frac{a + 2 c}{\sqrt{3}}, \frac{a^2 + a c + c^2}{3} - b , \frac{a - c}{\sqrt{3}}\right)$$
Show that if $...
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Solving Cubic Equation using Cardano's method: $x^3-4x+1=0$
Substituting $x=u+v$, we get $$3uv=4$$ $$u^3+v^3=-1$$
and it follows that, $$u^3=-\frac{1}{2}+\iota \frac{\sqrt{687}}{18}$$
assume, $z=x+\iota y=-\frac{1}{2}+\iota \frac{\sqrt{687}}{18}$
$$r=\sqrt{x^2+...
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Rational solutions for $x^3+y^3=1$ where both x and y are non-negative
How can I find rational solutions for $x^3+y^3=1$ where both x and y are non-negative?
Edit: One of the answer in this post for general form of solutions
$$(a,b) \mapsto \left( \frac{a(a^3 + 2b^3)}{a^...
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Non-constant cubic spline interval
I have the following Numerical Methods homework question that I am working on.
Question: The velocity profile of a rocket against time is give as below
\begin{array}{c|ccccc}
\bf t & 0& 10&...
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0
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75
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Involution on $2\times 2$ matrices
Show that the map on $2\times 2$ matrices
\begin{eqnarray}
\left( \begin{matrix} a & b\\ c & d \end{matrix} \right)\overset{\Phi}{\mapsto} \left( \begin{matrix} a & b\\ c & d \end{...
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2
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Vieta's Formula and Polynomials
The following below is a question i was asked. The question left me stunned as i was unable to solve it. The question is as follows:
If $A$,$B$ and $C$ are roots of the cubic equation $ax^3+bx^2+cx+d=...
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Homographic relation between two roots of a cubic
Consider a cubic equation $ x^3 + 3a x^2 + 3 b x + c=0$ with distinct roots. Show that any two roots $x$, $y$ are connected by a homographic relation
$$(a^2-b) x y + \frac{1}{2}\ (\ (a b-c+\delta) x +...
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