Questions about the set of values at which a given function evaluates to zero. For questions about "square roots", "cube roots", and such, consider using the (radicals) and (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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2
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1answer
87 views

Limit solution to a transcendental equation

Let $n\ge 1$ be a positive integer. The question is to solve the following transcendental equation: \begin{equation} \left(1+q\right)^{2 n} = \frac{\sqrt{\pi}}{2} \frac{1-q}{\sqrt{q}} \sqrt{n} ...
1
vote
0answers
37 views

Roots of this trigonometric polynomial

Let $f:[0,2\pi) \rightarrow \mathbb{R}$ with $f(x):=\sum_{n=0}^{k}a_n \left(1+\cos(x)\right)^n$ for arbitrary $a_n$ with $a_k \neq 0$. My question is: What is the maximum number of zeros that this ...
3
votes
0answers
45 views

Gaps between roots of trigonometric polynomials?

Given a polynomial in $e^{\mathrm{i}k t}$ of the form $$ p(t) = \sum_{-n\leq k\leq n} c_k e^{\mathrm{i}k t} $$ with $\bar c_{-k} = c_k$, is there a good way of characterising how close its roots can ...
1
vote
3answers
166 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 ...
3
votes
3answers
232 views

Roots of polynomial of degree 6

I'm struggling to find the complex roots of $x^6-9x^3+8 = 0$. I've managed to find the real roots (1 and 2) by letting a variable, say $α = x^3$ and substituting where relevant, leading to a ...
3
votes
1answer
452 views

Checking tolerance of Newton-Raphson method to calculate square root

Finding the square root of $c$ is finding the solution to: $$x^2 - c = 0.0$$ We can use Newton's method to successively approximate the solution. My question is how to check whether we are within ...
1
vote
1answer
76 views

Roots of $\tan x - x$

The function $\tan x - x$ has exactly one root $x_n$ in the interval $(n\pi, (n + \frac{1}{2})\pi)$. Show that $$x_n = n\pi + \frac{\pi}{2} - \frac{1}{n\pi} + r_n$$ where $\lim_{n\rightarrow \infty} n ...
2
votes
4answers
56 views

Number of distinct real roots with $e^{-x}$ in the equation

How to find the number of distinct real roots of the equation $$13x^{13}-e^{-x}-1=0$$ I know that we generally find number of real roots by observing number of sign changes in $f(x)$ and $f(-x)$ but ...
1
vote
1answer
47 views

Sum of Non Real Roots of Bi Quadratic

Consider $$f(x)=8x^4-16x^3+16x^2-8x+k=0$$ where $k \in \mathbb{R}$,then find sum of non Real roots of f(x). My approach: we have $$f'(x)=32x^3-48x^2+32x-8=0$$ Also ...
2
votes
1answer
131 views

Solving equation $a^{-x} + \log x/\log a = 0$

Please can you instruct me how should I start writing an algorithm (pseudo-code, to be implemented) for finding all solutions for the following equation: $a^{-x} + \log x/\log a = 0$ where $a$ ($a$ ...
0
votes
1answer
88 views

Projective roots of a homogeneous polynomial

Suppose that $f(X,Y)\in\mathbb C[X,Y]$ is a homogeneous polynomial of degree $n$, then we can consider it as a function on $\mathbb P^1_\mathbb C$. It has at most $n+1$ projective roots (points of ...
1
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0answers
72 views

Quartic Polynomial Manipulation

I have a quartic polynomial in $x$ (too long to write here) $f(x,c_1, c_2, c_2)$ where $c_1, c_2, c_3$ are constants which I have complete freedom over how to fix their values, as long as they are ...
0
votes
2answers
32 views

Understanding Multiplicities

I am having troubles understanding what 'multiplicities' mean. In example what does $-1/3(multiplicity 2)$ translate into?? To clarify this is for finding zero's in a polynomial function Any help ...
5
votes
1answer
326 views

To prove this complex polynomial has all zeros on unit circle

I'm trying to prove a self-inversive polynomial $P(z) = \sum\limits_{n=0}^{N-1}a_nz^n$ has all its roots on the unit circle. The coefficients are such that $ a_n = e^{j(n-\frac{N-1}{2})\pi u_0} - ...
3
votes
1answer
106 views

Find roots of $3z^{100} - e^z$ in the unit disc.

This question was given in an exam in complex analysis: Let $f \left( z \right) = 3z^{100} - e^z$. Find all of $f$'s roots in $D \left ( 0,1 \right)$ and show that they are simple roots. I've seen ...
1
vote
1answer
188 views

The number of zeros in a polynomial

Does anyone know of a relationship between the number of zeros (complex and real) and the degree of a polynomial? Specifically, if a polynomial has a double root, is it compensated with complex ...
1
vote
1answer
94 views

Rouches Theorem Applied to a family of Polynomials

I would like to prove that the family of polynomials $z^{2j+2} + \alpha z^{2j+1} - \alpha z - 1$ has only one root inside the open unit circle when $|\alpha|$ is greater than 1. This seems like an ...
0
votes
1answer
62 views

If one root is unsolvable, can there be a solvable root?

Suppose you have some polynomial $p(x)$ with rational coefficients in which at least one root is unsolvable by radicals, does this imply that all other roots of $p(x)$ are unsolvable by radicals? ...
1
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0answers
67 views

What is the condition for roots of conjugate reciprocal polynomials to be on the unit circle?

Given an Nth order complex polynomial $P(z) = \sum\limits_{n=0}^N a_nz^n$ such that $a_n = a^*_{N-n}$ i.e. conjugate reciprocal, Lakatos and Losonczi mention that a necessary and sufficient condition ...
1
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0answers
34 views

Version of Chevalley's Theorem proving $\mathbb{F}_q$ is a $C_1$-field for $\mathbb{Z}/p^n\mathbb{Z}$?

Let $k=\mathbb{F}_q$ Chevalley's theorem states that if $f(x_i)\in k[x_1,...,x_n]$ is such that $f(0,...,0)=0$ and $deg(f)<n$, then there is a non-trivial $a_i\in k^n$ such that $f(a_i)=0$. Is ...
1
vote
1answer
295 views

Relation between the roots of a cubic equation and the coefficients

$ax^3 +bx^2 + cx + d= 0$ If the roots are $\alpha$ $\beta$ and $\gamma$, Is there any relationship between the sum of the squares of the roots and the coefficients of the quadratic equation.. In ...
0
votes
2answers
421 views

Find the cube roots of $ -8 i $ and plot them on a plane.

I can’t figure out the angle of this equation. I set it up like this: $$ z^{3} = 0 - 8 i. $$ I find that the $ r $-value is $ 2 $, but when I try to find the angle, I’m stuck. I can’t divide by $ 0 ...
7
votes
1answer
144 views

Find the maximum value of $ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $

If $x\in\mathbb{R}$ find the maximum value of $$ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $$ I tried this: Let $$y= \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1}$$ For maxima ...
1
vote
1answer
89 views

Proving that $\lim\limits_{z\to z_0}\frac{f(z)}{g(z)}=\lim\limits_{z\to z_0}\frac{f^\prime(z)}{g^\prime(z)}$

Let $f,g$ both analythic in neighbourhood of $z_0$ and they both have zero of multiplicity $n$ in $z_0$. Prove that $\lim\limits_{z\to z_0}\frac{f(z)}{g(z)}=\lim\limits_{z\to ...
6
votes
1answer
109 views

Solution of $\exp(z)=z$ in $\Bbb{C}$.

I have posted a related question here. I thinkg this one is more interesting: What about the solution of $\exp(z)=z$ in $\Bbb{C}$? My try : $z \mapsto e^z - z$ is entire non-constant. Perhaps ...
5
votes
3answers
160 views

all complex solutions of $z\sin(z)=1$?

A possibly easy question, Can we find all complex solutions of $z\sin(z)=1$ ? My try: Let $$\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}$$ so we have $$ z\frac{e^{iz} - e^{-iz}}{2i}=1 $$ Not sure how ...
1
vote
1answer
191 views

Kantorovich Theorem example

I need to write in C a program that finds roots of a 6th order polynomial. I was thinking of using Kantorovich Theorem convergence of Newton's method to find when can I use Newton-Rhapson method. I'm ...
1
vote
1answer
72 views

I need help proving a statement about rational roots

I have no idea where to start...this is the statement: If a polynomial of degree not greater than 5 with rational coefficients has multiple roots, it has also a rational root, except in the case ...
0
votes
4answers
624 views

Using sum/product of quadratic roots to solve cubic equation

Given $\alpha$ and $\beta$ are the roots of the quadratic equation $6x^2 + 2x - 3 = 0$, how do I find the value of: $$ \alpha^3 + \beta^3 $$ and: $$ \frac{1}{\alpha^3} + \frac{1}{\beta^3} $$ ...
1
vote
2answers
411 views

Most Efficient Method to Find Roots of Polynomial [duplicate]

I am designing a software that has to find the roots of polynomials. I have to write this software from scratch as opposed to using an already existing library due to company instructions. I currently ...
2
votes
1answer
56 views

Determine the coefficients of a polynomial knowing its roots

My prof. gave this problem as a bonus in an exam, and I couldn't figure out a solution. Some hints or a general method of solving it would be very nice. Given the following polynomial: ...
7
votes
4answers
134 views

Finding double root of $x^5-x+\alpha$

Given the polynomial $$x^5-x+\alpha$$ Find a value of $\alpha>0$ for which the above polynomial has a double root. Here's an animated plot of the roots as you change $\alpha$ from $0$ to $1$ I'm ...
1
vote
0answers
47 views

Extension of quadratic forms

A homogen multivariate polynomial with degree 2 is a quadratic form. It can be checked if the polynomial is positive for any non-zero vector by checking if the corresponding matrix A is positive ...
3
votes
1answer
135 views

What is the Most Efficient Way to Calculate the Internal Rate of Return?

I have built a program that prices financial assets and it does this in part by calculating the IRR. The problem is that it does not run as quickly as I would like it to. I currently use the ...
3
votes
0answers
70 views

How can I find my eccentricity (k, of the incomplete elliptic integral of the second kind) using a binomial series or root-finding algorithm

My main objective is to rearrange the following to find $A=$. I start with $C=\int^{B}_{0}\sqrt {1+\dfrac {A^{2}}{B^{2}}\cos^{2}\left(\dfrac {x}{B}\right) }dx$ By substituting $y=x/B$ and using ...
0
votes
1answer
31 views

complex conjugate pairs of a quartic

I tried my hand at this question, which included finding the partial fractions of $\frac{x^2}{1-x^5}$. I found a factor of $1-x$ for the denominator, but I do not know how to work out the complex ...
15
votes
1answer
326 views

Can we prove that the solutions of $\int_0^y \sin(\sin(x)) dx =1$ are irrational?

Can we prove that the solutions of $$\int_0^y \sin(\sin(x)) dx =1$$ are irrational? Wolfram Alpha gives two approximate sets of solutions as $\{4.58+2\pi k|k\in\mathbb{Z}\}$ and $\{1.69+2\pi ...
0
votes
1answer
44 views

Equation with binomial coefficients

Problem: Find the roots of $6z^5+15z^4+20z^3+15z^2+6z+1 = 0$. What I found: I realized that the coefficients were the binomial coefficients of $6$. Putting these values in, you would get ...
6
votes
1answer
192 views

Numerical inverse of logarithmic integral

What is the best way to numerically calculate the inverse of the logarithmic integral, defined by $$ \operatorname{li}(x)=\int_{0}^{x}\dfrac{1}{\log(t)}\operatorname{d}t $$ eg ...
3
votes
2answers
44 views

Build field extension and solve equation

Build quadratic extension of field that contains $5$ elements. And solve $x^2+x+2=0$ in this field. As I understand we need to build $\mathbb{F}_{5^{2}}$. Field $\mathbb{F}_5$ contains ...
0
votes
1answer
46 views

How to solve: $0 = -\sin \space 3x \cdot3, \left({\pi\over 12}, {7\pi \over12}\right)$

While working on some Rolle's Theorem problems I came to: $$f(x) = \cos 3x$$ This is both continuous on the given interval (and everywhere really) $[{\pi\over 12}, {7\pi \over12}]$ and ...
2
votes
1answer
84 views

Given roots (real and complex), find the polynomial

This is not a duplicate of theory of equations finding roots from given polynomial. Given that the roots (both real and complex) of a polynomial are $\frac{2}{3}$, $-1$, $3+\sqrt2i$, and $3+\sqrt2i$, ...
12
votes
2answers
99 views

Function such that zeros$=$order of the derivative

Does there exist a function $f\in C^n(\mathbb{R},\mathbb{R})$ for $n\ge2$ such that $f^{(n)}$ has exactly $n$ zeros, $f^{(n-1)}$ has exactly $n-1$ zeros and so on ? Where $f^{(n)}$ is the nth ...
0
votes
1answer
35 views

Common root of quadratic.

If the quadratic equations, $x^2+bx+c=0$ & $bx^2+cx+1=0$ have a common root. Prove that either $b+c+1=0$ or $b^2+c^2+1=bc+b+c$. Please also explain What should be the logic / approach we should ...
1
vote
1answer
53 views

Perturbation of complex polynomials

Let $f(z)=\sum\limits_{k=0}^N a_kz^k$ be a (monic) complex polynomial and $\{\xi_{k}\}_{k=1}^{N}$ be the roots of $f$ (with multiplicities). Let $\{\tilde{\xi_{k}}\}_{k=1}^{N}$ be the perturbed ...
1
vote
2answers
75 views

The domain of cubic root

The domain of cubic root and in general $(2n-1)$ th root is $\mathbb{R}$. But Wolframalpha says the domain of cubic root is all non-negative real numbers. Also Matlab return ...
4
votes
3answers
200 views

Square roots modulo powers of 2

Experimentally, it seems like every $a\equiv1\left(\bmod\,8\right)$ has 4 square roots mod $2^n$ for all $n \ge 3$ (ie solutions to $x^2\equiv a\left(\bmod\,2^n\right)$) Is this true? If so, how can ...
1
vote
2answers
79 views

Intermediate value theorem problem(proving only one root)

This problem is likely very trivial. Let $f(x) = x^5 - 5x + p$. Show that $f$ can have at most one root in $[0,1]$,regardless of the value of p. This seems to be an IVT problem, so I will go ...
1
vote
1answer
39 views

On the existence of polynomial roots

Assume $F$ is a field, and $f\in F[x]$ is polynomial. To see that $f$ has a root in some extension of $F$, without loss of generality we can assume $f$ is irreducible. Indeed any polynomial $f$ is ...
2
votes
1answer
228 views

Solving a cubic polynomial equation.

Overview I have tried finding a solution to this problem myself and I have flailed. Its just a challenge for me. could you please tell me how far am I in solving this question? My approach for ...