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

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4
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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 ...
3
votes
2answers
38 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 ...
1
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2answers
26 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: $$ ...
2
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1answer
60 views

Roots of $x^3-x+1$

I am trying to find nice explicit formulas for the roots of the polynomial $x^3-x+1$. Is there some clever way to write down the roots in a reasonably easy way? I found the roots, but my expressions ...
0
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1answer
20 views

Trisecting angle equivalence of constructing a segment

After reading Wikipedia and some previous questions asked in this site, I still don't understand this. Following the Pierre Wantzel. Triple angle formula cos(3theta ) and getting a polynomial p(x). ...
2
votes
2answers
81 views

When does a cubic equation has two roots with same absolute values

Let $$z^3+bz^2+cz+d=0$$ be a cubic equation with complex coefficients. Suppose $z_1, z_2$ and $z_3$ are its roots. I need to find a condition on $b,c,d$ so that $$|z_1|=|z_2|.$$ How can I find such a ...
0
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2answers
53 views

How to ind the complex roots of $y^3-\frac{1}{3}y+\frac{25}{27}$

I've been trying to solve this for hours and all found was the real solution by Cardano"s formula. I vaguely remember that if $\alpha$ is a root of a complex number, the other roots are $\omega ...
2
votes
1answer
41 views

How to solve $t^3+pt+q=0$

I thought this shouldn't be too hard, but evidently not. I am asked to solve $t^3+pt+q=0$ given $27q^2+4p^3<0$, using $\cos{3 \theta}=4 \cos ^3{\theta} - 3 \cos {\theta}$. At first I thought, ...
1
vote
1answer
20 views

Geomteric progression of cubic equation roots

I am kind of stuck on the following... There is a cubic equation like this: $x^3-px^2+qx-r=0$ We are given that the three roots are: $ak^{-1},a,ak$ for some constants $k$ and $a$. Also given ...
0
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1answer
35 views

STEP III 1999 Q1 - Cubic polynomial exercise

STEP is a Cambridge University devised examination paper for mathematics. I have been solving Q1 from the STEP III paper And I have stumbled upon a 'weird', not even sure whether it is correct, way ...
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0answers
13 views

Cubic curve as approximation of Euler spirals?

I was reading the wiki article about Euler spirals and I reached this passage: Unaware of the solution of the geometry by Leonhard Euler, Rankine cited the cubic curve (a polynomial curve of ...
6
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2answers
365 views

I know there are three real roots for cubic however cubic formula is giving me non-real answer. What am I doing wrong?

I want to solve the equation $x^3-x=0$ using this cubic equation. For there to be real roots for the cubic (I know the roots are $x=-1$, $x=0$, $x=1$), I assume there must be a positive inside the ...
1
vote
1answer
29 views

cubic spline interpolation - derivative known -

I at the moment trying to understand how to apply the interpolation method stated above. I have been given a start and end position, and for both position i know what their slope is. $\dot{X_a} = ...
1
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2answers
62 views

How to solve $y=2x^3+x$ for $x$

I'm doing something that has throw out: $$y=nx^{2n-1}+x$$ and do make any progress with this problem I needed to make x the subject. Correct me if I'm wrong but this seems impossible to work with for ...
0
votes
1answer
37 views

Solving cubic equation with general expression $ax^3+bx^2+(c-j)x+(d-m)=0$

I am trying to solve the equation $ax^3+bx^2+(c-j)x+(d-m)=0$ to find an expression for all three roots. I found this 1 but whenever I try and solve it by substituting values in I always get imaginary ...
7
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0answers
84 views

Use the identity $\cos 3\theta = 4 \cos^3\theta- 3 \cos \theta$ to solve the cubic equation $t^3 + pt + q = 0$ when $p, q \in \mathbb{R}$.

I'm self studying Ian Stewart's Galois Theory and this is Exercise 1.8 from his Third Edition: Use the identity $\cos 3\theta = 4 \cos^3\theta- 3 \cos \theta$ to solve the cubic equation $t^3 + ...
1
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2answers
28 views

If the zeros of a cubic form an arithmetic sequence, its point of inflection lies on the $x$-axis [closed]

If the zeros of the cubic $$y=x^3+ax^2+bx+c $$ form an arithmetic sequence, then show that the point of inflexion lies on the $x$-axis.
2
votes
2answers
39 views

Show that the difference between the greatest and the least of them is not less than $\sqrt{a^2-3b}$ nor greater than $2\sqrt{a^2-3b}$

Let the roots of the cubic equation $x^3+ax^2+bx+c=0$ be real.Show that the difference between the greatest and the least of them is not less than $\sqrt{a^2-3b}$ nor greater than $2\sqrt{a^2-3b}$. ...
2
votes
1answer
37 views

How to merge two (or more) continuous functions into one?

I have two functions, $g(x) = 4x^3$, domain $0<x<0.5$ $h(x) = \frac{1}{2}((2x-1)^3+1)$, domain $0.5\le x<1$ which I derived from: $g(x) = ((2x)^3)/2$, domain $0<x<0.5$ ...
2
votes
1answer
97 views

Integer solutions to $\frac{d^3}{r}+r=a^2$

What are the positive integer solutions $(a,d,r)$ to $\frac{d^3}{r}+r=a^2$? This is a revised version of my deleted question. Alternate forms are $d^3 = r(a^2-r)$ and from the quadratic formula ...
2
votes
2answers
91 views

How to solve the cubic equation $56z^3-70z^2-21z-4=0$?

$56z^3-70z^2-21z-4=0$ how to solve for $z$. I formed three equations but not getting the answer. If I get a start or suggestion it would be a great help.
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3answers
39 views

Coefficient of a cubic expression $x^3-2x^2+ax+10=0$ such that sum of two roots is zero

So the given cubic is $x^3-2x^2+ax+10=0$. The condition is one of the root is additive inverse of another. I need to find the coefficient $a$. I did some algebraic calculations and cancelling of the ...
0
votes
3answers
55 views

How can I solve for $x$ values of a cubic function where $x$ intersects a given $y$?

Given the cubic function: y = -3 + 10x - 7x^2 + 1x^3, how can I find a value of x when ...
1
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2answers
65 views

Having trouble finding roots/factorizing a cubic equation

I've been trying to find a method that can work for any cubic equation, but I can't seem to find one. Right now, I'm trying to find the roots/ factorize the following equation: $x^3-5x^2+3x+9 = 0$. ...
1
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1answer
28 views

Finding the condition for point of intersection of three normals to a given parabola

Question: Suppose that the normals at three different points on the parabola $y^2=4x$ pass through the point (h,0). Show that h>2. My attempt: Equation of normal to parabola $y^2=4x$: ...
0
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1answer
36 views

Let S be a cubic spline that has knots t0 < t1 < · · · < tn.

Let S be a cubic spline that has knots $t_0 < t_1 < · · · < t_n$. Suppose that on the two intervals $[t_0, t_1]$ and $[t_2, t_3]$, S reduces to linear polynomials. What does the polynomial S ...
2
votes
2answers
30 views

$c$ is the value of $x^3+3x-14$ where $x=\sqrt[3]{7+5\sqrt2}-\frac{1}{\sqrt[3]{7+5\sqrt2}}$.Find the value of $a+b+c$

$a=\sqrt{57+40\sqrt2}-\sqrt{57-40\sqrt2}$ and $b=\sqrt{25^{\frac{1}{\log_85}}+49^{\frac{1}{\log_67}}}$ and $c$ is the value of $x^3+3x-14$ where ...
2
votes
2answers
58 views

Show that $x^3+2x=\cos x$ has exactly one solution in the interval $[0,\pi/2]$

Show that $x^3+2x=\cos x$ have exactly one solution in the interval $[0,\pi/2]$. Don't even know where to start. Thanks in advance
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0answers
16 views

Understanding Ferrari's Solution

I'm trying to understand how Ferrari's Solution works. Thanks to this post I understand that we are solving this for $y$ to find the perfect square: $$y^3 + \frac{5\alpha y^2}{2} + (2\alpha^2 - ...
3
votes
3answers
310 views

Solving Cubic when There are Known to be 3 Real Roots

When solving for roots to a cubic equation, the sign of the $\Delta$ tells us when there will be 3 distinct real roots (as long as the first terms coefficient, $a$, is non-zero.) Namely when $\Delta$ ...
0
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1answer
65 views

Finding All 3 Roots of a Cubic

I'm trying to find all real roots of a cubic. I wanted to use Cardando's Method but I'm not sure I'm correctly understanding how to obtain all 3 roots given the depressed cubic: $$t^3 + pt + q = 0$$ ...
0
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1answer
31 views

Cubic spline. What is symmetrical form and why?

I'm trying to understand the algorithm for cubic spline from Wikipedia. It says the polynomial can be written in symmetrical form: A third order polynomial $q(x)$ for which $q(x_1)=y_1$, $(x_2)=y_2$, ...
1
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2answers
52 views

Find the natural cubic spline function whose knots are $-1$, $0$, and $1$ and that takes the values $S(-1)=13$, $S(0)=7$, and $S(1)=9$.

Find the natural cubic spline function whose knots are $-1$, $0$, and $1$ and that takes the values $S(-1)=13$, $S(0)=7$, and $S(1)=9$. I'm not sure how to go about this. Any solutions/hints are ...
2
votes
1answer
34 views

Solving cubics with complex numbers, before complex numbers.

An aside in another website reads: Complex numbers were used to solve cubic polynomials, before complex numbers were invented. I tried Googling this technique but didn't get anywhere. What is ...
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3answers
54 views

Finding the roots or interval in which the (roots) lie of the given equation.

If an equation $$\frac{1}{x-3} + \frac{1}{x-4} + \frac{1}{x-5}+ \frac{1}{x-7}=0 $$ has been given to us then how can we find out-- 1) the number of roots of the equation ??...if it was one of those ...
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1answer
56 views

The sum of three cubes

I'm reading an article about numbers that can be expressed as the sum of three cubes: http://www.ams.org/journals/mcom/2007-76-259/S0025-5718-07-01947-3/S0025-5718-07-01947-3.pdf It states in the ...
0
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3answers
42 views

Generalised formula for fitting a cubic between two points with specified slopes at each point

As the question says, I'd like to fit a cubic between two points, $(x_1,y_1)$ and $(x_2,y_2)$. The reason I need a cubic is that I want to specify the first differential at each point. I can see ...
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1answer
41 views

Finding the zero of a weird cubic function

Volume of Silo: A silo grain consists of a cylindrical main section and a hemispherical roof. If the total volume of the silo (including the part inside the roof section) is $\mathbf{15,000\ \ ft^3}$ ...
3
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0answers
67 views

Motivation of Vieta's transformation

The depressed cubic equation $y^3 +py + q = 0$ can be solved with Vieta's transformation (or Vieta's substitution) $y = z - \frac{p}{3 \cdot z}.$ This reduces the cubic equation to a quadratic ...
3
votes
4answers
55 views

Show that if $a\neq b$ then $a^3+a\neq b^3+b$

Show that if $a\neq b$ then $a^3+a\neq b^3+b$ We assume that $a^3+a=b^3+b$ to show that $a=b$ $$\begin{align} a^3+a=b^3+b &\iff a^3-b^3=b-a\\ &\iff(a-b)(a^2+ab+b^2)=b-a\\ &\iff ...
1
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1answer
51 views

System of three equations with lots of symmetry and 6 unexpected (?) solutions.

I'm interested in the system of equations: $a(b^2+c)=c(c+ab)$ $b(c^2+a)=a(a+bc)$ $c(a^2+b)=b(b+ac)$ It is easy to see that $a=b=c=t$ are solutions for all $t$, in fact these are the only real ...
8
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2answers
211 views

Applications of the formula expressing roots of a general cubic polynomial

I know that the mathematics related to finding the general formula by expressing the roots of a third (and fourth) degree polynomial by means of radicals has had an impressive impact on mathematics ...
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1answer
38 views

Finding irrational and complex roots of a cubic polynomial

I've got a question which shows short answers and no method so I'm trying to find a hand performed method of solving the cubic polynomial for the roots: ...
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2answers
46 views

cardano's method - I'm unable to find my mistake

I'm currently trying to calculate zeros of a cubic function using the Cardano formula I somehow miscalculated really bad and I suspect that I've done a really cheap beginners mistake. I searched but ...
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0answers
13 views

Approximating a value based on a cubic function (using a tangent of a graph )

Click here to see the question So for question iii, the answer is $-1/(sqr3)$, saying it the negative value will be towards infinity when the denominator is close to zero. I understand that. ...
0
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0answers
24 views

Find roots of a cubic function thank to roots of derivative function

Let $f(x) = x^3 + px + q$ I have to find how many roots $f$ has. If $p\geq 0$, the answer is one. But now I have to find the answer for $p<0$. We know that $f'(x) = 3x^2 + p$ The roots of $f'$ ...
2
votes
3answers
61 views

Find all the solutions of diophantine eq: $x^3-2xy^2+y^3-s^2=0$

Given $x,y,s$ are natural numbers: $$x^3-2xy^2+y^3-s^2=0$$ I found the solutions using wolfram alpha $$(x,y,s) = (1,2,1), (6,10,4), (4,8,8)$$ But how do I prove these are the only solutions? Any ...
3
votes
3answers
76 views

Solve $2n^3 + 3n^2 + n = d$

how to solve the following Diophantine equation? $$ 2n^3+3n^2+n=d $$ Here, $n$ and $d$ are non-negative integers. It would also be great if it were possible to formulate an algorithm for $$ an^3 +bn^2 ...
1
vote
2answers
30 views

How to solve a system of non linear equations of the form $2ax-2x(ax^2+by^2) = 0$ [closed]

Solve this system for $x,y$: $$2ax-2x(ax^2+by^2) = 0\tag{1}$$ $$2by-2y(ax^2+by^2) = 0\tag{2}$$ where $a$ and $b$ are constants such that $a>b>0$. I re-arranged (1) to $x = ...
1
vote
1answer
44 views

If a cubic has three real roots, must its factor quadratic equation have real roots too?

The question I'm trying to answer is for what values of $p$ does $f(x)=x^3 + px^2 + qx - 4$ have three real roots. It is given that $(x-1)$ is a factor of $f(x)$ and that $p + q = 3$. Using algebraic ...