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Questions tagged [cubic-equations]

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

2
votes
1answer
51 views

Let : $P(x)=x^{3}+ax^{2}+bx+c$ where $(a,b,c)\in Z^3$

Question : If : $P(x)=x^3+ax^2+bx+c$ where $(a,b,c)\in Z^3$ And $m,n,k$ root of $P(x)$ such that : $m.n=k$ Then show that : $2P(-1)$ multiple of $P(1)+P(-1)-2[1+P(0)]$ My try : We known ...
0
votes
0answers
49 views

Equation : $3x^4+x^3-10x^2-x+3=0$ [duplicate]

Solve in $R$ the following equation : $3x^4+x^3-10x^2-x+3=0$ Im try use sub $y=x+\frac{1}{x}$ But I don't understand whene and where I use this sub Please give me ideas or hint to approach it
0
votes
2answers
54 views

Solve in $C$ : $p(x)=x^4+2x^3-x^2-2x+7=0$ [duplicate]

Find all root : $p(x)=x^4+2x^3-x^2-2x+7=0$ Where $p(\alpha)=0$ , $\alpha=\sqrt{2}+\omega$ $\omega=e^{\frac{2iπ}{3}}$ My try : Since : $\alpha$ root of equation then $\bar\alpha$ also root ...
-1
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2answers
48 views

Finding all prime $p$ for which there exists a positive integer $n$ such that $p^n+1$ is a perfect square?

$p=7~(n=1)$ is a solution. But how to prove that there are not any other solutions?
1
vote
1answer
38 views

How can we solve the expression explicitly for $X$ in terms of $Y$?

I am thinking how to write $X$ explicitly in terms of $Y,A,B,C$? I have $AX^3 + X^2(B-1) + X(-C) + \alpha = Y$ I thought of using symbolic maths but could not find any. Any help is nice!
2
votes
1answer
69 views

Roots of a cubic polynomial in $[-1,1]$ [closed]

I am preparing for an entrance and this question has taken too much of my time. Suppose function $f : \mathbb R \to \mathbb R$ is given by $$f(x) = x^3 - 3x + b$$ Find the number of points in the ...
0
votes
1answer
62 views

How to solve the cubic equation $-1.6x^3-2.1x^2-2.9x-0.3=0$

Is there any method similar to the quadratic formula to solve the equation; {$-1.6x^3-2.1x^2-2.9x-0.3=0$} If not, how would I go about finding the solutions for cubic equations like these?
6
votes
4answers
470 views

Further factorisation of a difference of cubes?

We know that a difference of cubes can be factored using $a^3-b^3=(a-b)(a^2+ab+b^2)$. How do we know that the quadratic can't be factored further. For example, $$ 27x^3-(x+3)^3=((3x-(x+3))(9x^2+3x(x+...
4
votes
2answers
70 views

Is $y=|x^3| $a parabola?

I'm just curious. It seems to have the same shape and a similar form as parabolas such as $ x^2$ and $x^4 $. The odd exponent would normally give negative outputs for negative inputs, but the absolute ...
0
votes
1answer
28 views

Finding general solution of a third-order ODE

I am given the equation: $$y''' - 7y'' + 16y' + 10y = 0$$ where a root of the characteristic equation is $\lambda= -1$. I gathered from this that $y=e^{-x}$ is a solution of the third order ODE. I ...
2
votes
1answer
35 views

How can I create a cubic spline to connect these two line segments?

I am trying to create an audio envelope smoothing curve to smooth a steady signal (y=1) as it goes into exponential decay (y=c^-x). I have expressed the graphs of the two line segments I am trying to ...
0
votes
2answers
43 views

Apply Newton-Raphson method to find the solutions to

Apply Newton-Raphson method to find the solutions to the equation $x^3-5x=0$ starting with an initial guess of $x_0 = 1$. While using Newton Raphson method, the value doesn't converge to a specific ...
0
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0answers
22 views

Cubic surfaces normal forms classification

Is there a classification for the the normal forms of cubics surfaces? I can find such a classification for quadrics on Wikipedia, but didn't find much on the subject for cubics apart from straight ...
0
votes
2answers
95 views

How to show $(2 + \frac{10}{3\sqrt{3}})^{1/3} + \frac{2}{3} (2 + \frac{10}{3\sqrt{3}})^{-1/3} = 2$

I used the method shown in the link (the second answer) to solve $0 = x^3 - 2x - 4$: Is there a systematic way of solving cubic equations? I got that one solution is $x = (2 + \frac{10}{3\sqrt{3}})^{...
3
votes
4answers
88 views

Solving a linear system of reciprocals.

Solve for $\begin{cases}\frac{1}{x} +\frac{1}{y}+\frac{1}{z}=0\\\frac{4}{x} +\frac{3}{y}+\frac{2}{z}=5\\\frac{3}{x} +\frac{2}{y}+\frac{4}{z}=-4\end{cases}$ I turn the equations into $\begin{cases}yz+...
2
votes
0answers
44 views

Given any cubic polynomial with real coefficients, find the argument of the complex roots

I have the cubic equation $ y = 2x^3 - 2x^2 - 6x - 3 = 0 $, which I am given has one real root and two complex conjugate roots. Is it possible, without explicitly finding these roots, to find the ...
1
vote
1answer
42 views

cubic Hermite interpolation

Professor gave us this little bastard of a question and I'm at a complete loss about what to do. Some help or hints would be immensely appreciated, translated to the best of my abilities. Let $x_0=0$,...
-4
votes
2answers
60 views

How to solve $x^3-19x^2+96x-144=0$ without using vieta's formula? [closed]

How to solve $$x^3 - 19x^2 + 96x - 144=0$$ without using Vieta's formula ?
0
votes
1answer
23 views

Median of a continuous r.v.

Let X be a continuous r.v., with pdf $f_X(x) = kx(1-x), 0 < x <1$ I evaluated k = 6 and found the cdf $F_x(x) = 3x^2 - 2x^3$, but then I am asked to find the median. The equation $F_x(x) = 1/2$ ...
1
vote
1answer
41 views

Inverse function of $ax + bx^3$

I am trying to find the inverse of the function $y = f(x) = ax + bx^3$, i.e. $x = f^{-1}(y)$. (The equation arises in the modeling of a certain type of transmission used in robots) Looking at the ...
0
votes
1answer
46 views

A particular cubic can be written as $y_1^2 = (x_1 - e_1)(x_1 - e_2)(x_1 - e_3)$. Show that $e_1, e_2,$ and $e_3$ are distinct.

I am working through Algebraic Geometry: A Problem Solving Approach and am stuck on exercise 2.4.22. The previous problem was to consider $y^2 = 4x^3 + b_2x^2 + 2b_4x + b_6$ and transform this with $...
0
votes
0answers
60 views

Inverse of $f(x) = x^{3}-x^{2}$

Can anybody find the inverse of $f:(-1,0) \to \mathbb{R}$ such that $f(x) = x^{3} - x^{2}$ ?
0
votes
0answers
32 views

An integer with cubic root that is not constructible

Prove that for an integer $n$ which is not of the form $a^3$, $\sqrt[3]{n}$ is not constructible using that fact that if a cubic equation with rational coefficients has a constructible root then the ...
0
votes
1answer
35 views

calculate cubic equation from two points and two slopes, variably [closed]

I'm trying to variably calculate the cubic equation between two points and the slopes at said points. (Ultimately to get an approximation of a bezier spline where i can calculate the y value from a ...
3
votes
3answers
91 views

What are the integer coeffcients of a cubic polynomial having two particular properties?

Let $f(x) = x^3 + a x^2 + b x + c$ and $g(x) = x^3 + b x^2 + c x + a\,$ where $a, b, c$ are integers and $c\neq 0\,$. Suppose that the following conditions hold: $f(1)=0$ The roots of $g(x)$ are ...
1
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1answer
24 views

Just Another question on Probability

Of three independent events, the chance that only the first occurs is a, the chance that only the second occurs is b and the chance of only third is c. Show that the chances of three events are, ...
0
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0answers
18 views

A system of bivariate cubics

It is asked to solve for rational numbers $x$ and $y$ the following pair of equations: $$ a=x^3+3xy^2\left(\frac {a-b^3}{3b}\right), $$ $$ \frac {9b^3+a}{3b}=3x^2y+y^3\left(\frac {a-b^3}{3b}\right) $$ ...
2
votes
1answer
57 views

sum of coefficients of $k(x)$

The polynomial $x^3-3x^2-4x+4=0$ has $3$ real roots $\alpha,\beta,\gamma$ and equation $k(x)=x^3+ax^2+bx+c=0$ has $3$ roots $\alpha',\beta',\gamma'$ and $\alpha'=\alpha+\beta\omega+\gamma \omega^2\;$ ...
0
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2answers
36 views

Complicated algebraic roots

I'm working on a problem: Suppose that the roots of $$3x^3 +18x^2 +9x -2 = 0$$ are $a$, $b$ and $c$; and the roots of $$x^3 +rx^2 +sx +t = 0$$ are $a+b$, $a+c$ and $b+c$. Determine the cubic ...
4
votes
1answer
61 views

Does $x^3 - \frac{m}{n}\sqrt{5}x - 1$ has rational root?

I am trying to show whether $p(x) = x^3 - \frac{m}{n}\sqrt{5}x - 1$ has a rational root or not, where $\frac{m}{n}$ is rational. My attempt so far is to turn $p(x)$ into another polynomial $q(x) = ( - ...
3
votes
3answers
96 views

Prove that the roots $\in \Bbb R$ of $x^3+x+1=0$ aren't rational without RRT

I need to prove that the roots $\in \Bbb R$ of $x^3+x+1=0$ aren't rational. Obviously, it's easy to use the rational root theorem to prove that there are not rational solutions to this equation, but i ...
1
vote
2answers
47 views

How to prove that there exist exactly two non-similar isosceles triangles ABC such that tanA+tanB+tanC=100?

I actually have a doubt about the solution of this question given in my book. It uses the equations tan 2A = - tan C (from A=B, A+B+C = 180 degrees) and 2 tan A + tan C = 100, thereby formulating the ...
2
votes
1answer
54 views

Common root of cubic and quadratic equation

If equations $ax^3+2bx^2+3cx+4d=0$ and $ax^2+bx+c=0$ have a non zero common root, prove that $(c^2-2bd)(b^2-ac) \geq 0$. I know the condition of common root of two quadratic equations but I have no ...
1
vote
3answers
57 views

Factor $2x^3 + 3x^2 - 2x$

I am having the most difficult time factoring this equation out. I should have paid more attention in precalculus in high school :( $2x^3 + 3x^2 - 2x = x(2x^2 + 3x - 2) = x(2x-1)(x+2)$ How do you ...
0
votes
3answers
38 views

Real and complex solutions of cubic implicit equation [closed]

I have faced this differential problem: $(y'(x))^3 = 1/x^4$. From the fundamental theorem of algebra i know there exist 3 solutions $y_1$, $y_2$, $y_3$, but formally how can I procede to deduce that?...
2
votes
2answers
64 views

Showing when $x^3 = px+q$ only has one real solution ($p,q \gt 0$)

My question is in determining when the equation $x^3 = px+q$ has only one real solution. For this question we are using the restriction $p,q \gt 0$ The trouble is that I am getting what at first ...
-3
votes
2answers
104 views

Roots of equation $x^3-x-1=0$

If α , β , γ are the roots of equation $x^3 -x -1 =0$ then $$ \frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma} $$ My ...
1
vote
1answer
42 views

Find out if cubic equation has real solution or complex ones?

Without knowing any solution to that equation, is there a way to quickly tell if it has 3 real solution or 1 real and 2 conjugated ones?
0
votes
2answers
41 views

Product of cubic root differences

Given a cubic equation $x^3 + ax^2 + bx + c$, with three real roots $r_1, r_2, r_3$, how can we express $(r_1 - r_2)(r_2 - r_3)(r_3 - r_1)$ in terms of $a,b,c$ and how do we prove it using Vieta's? ...
-1
votes
3answers
36 views

The real solution from a system of equation

I found this question from my friend's math competition, I don't know where I must start it There are 3 couples of real numbers $$(x_1,y_1) (x_2,y_2)$$ and $$(x_3, y_3)$$ that satisfies the system of ...
0
votes
1answer
38 views

Solution to quadratic and cubic equation with partial root

I am having trouble understanding how to resolve quadratic and cubic equations using the method described by my university lecturer (I am really interested to know if this method has a name). My ...
3
votes
1answer
81 views

Requirements for an integer root of cubic equation

If we have the quadratic equation $$a x^2 + b x + c = 0$$ with $a,b,c$ integers, then a requirement for $x$ to have an integer solution is for $b^2 - 4ac$ to be a square integer. This condition is ...
-1
votes
3answers
48 views

Solve cubic root equation [closed]

$ \sqrt[3]{x+5} - \sqrt[3]{x-5} = 1 $ How many values of x satisfy the equation?
1
vote
3answers
55 views

Roots of polynomials and their formulae relating to coefficients

Write down the cubic equation given that $\alpha + \beta + \gamma = 4$, $\alpha^2 + \beta^2 + \gamma^2 = 66$, and $\alpha^3 + \beta^3 + \gamma^3 = 280$ Ok so, the sum of roots is given and I'm able ...
5
votes
1answer
131 views

For $c\in\mathbb{F}_p^*$, the cubic $t^3-3ct^2-3t+c$ has exactly one root $r\in\mathbb{F}_p$. Express $r$ in terms of $c$ without cubic roots.

For some $c \in \mathbb{F}_p^*$ consider the polynomial $$ f(t) = t^3 - 3ct^2 - 3t + c $$ for $p \equiv 1$ (mod $3$) and $p \equiv 3$ (mod $4$). In this case $3$ is a quadratic non-residue modulo $p$ ...
0
votes
1answer
33 views

Find Equation of line to solve a cubic function

A cubic graph for $x^3 + 4x^2 + 2x - 4$ is given. Question is: Find an equation of the line you would draw on that graph to solve graphically: $x^3 + 4x^2 + 3x - 1 = 0.$ Give the answer in $y = mx + ...
3
votes
1answer
42 views

Proving there is only one set of positive integer solutions

I am looking to prove there is no perfect cube root to any thing of the form $3k^2 + 3k + 1$. As has been discussed previously, we know this must be true by invoking Fermat's last theorem, because $3k^...
0
votes
1answer
78 views

Expressing, in terms of real radicals, the trigonometric functions generated by cubic equations with integer coefficients

When solving a cubic equation, one might have to use trigonometric functions. In some cases, these trigonometric functions can be expressed in terms of real radicals. The goal is to find all the ...
0
votes
4answers
99 views

Solving a cubic diophantine equation $ax^3+bx^2=cy^3$

I want to know how to solve the equation $ax^3+bx^2=cy^3$ in positive integers $x$ and $y$, assuming $a, b, c$ are positive integers and $gcd(a, b, c) = 1$. If there is not a general algorithm, I ...
1
vote
0answers
177 views

Integer solutions to $x^3+y^3+z^3=29$

What are all the triples $(x,y,z) \in \Bbb Z^3$ with $$x^3+y^3+z^3=29, x \geq y \geq z$$ ? We find immediately $(3,1,1)$, but are there other? According to this question, it could be a difficult ...