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 (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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6
votes
1answer
72 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 ...
4
votes
3answers
114 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
50 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
56 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
3answers
79 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
81 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
42 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
114 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 ...
0
votes
0answers
38 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
54 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
42 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
23 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 ...
13
votes
1answer
288 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
26 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
175 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
33 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
38 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
55 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
86 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
33 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
24 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
38 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
166 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
33 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
27 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
58 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 ...
0
votes
2answers
49 views

Cubic roots of determinant.

If x=a+2b satisfies the cubic (a,b element of R) f(x)= $$\left|\begin{matrix} a-x & b & b \\ b & a-x & b \\ b & b & a-x\end{matrix}\right|$$ =0, then it's other 2 roots are?
1
vote
4answers
92 views

Roots of $x \cos{x}-1$ and $\cos{x}-x^{-1}$

To finding the roots of $x \cos{x}-1=0$ we can write the equation as $$ f(x)=x \cos{x}-1=0 \to x \cos{x}=1 \to \cos{x}=\frac{1}{x} \to \cos{x}-\frac{1}{x}=u(x)-v(x)=g(x)=0 $$ The roots of $f(x)$ are ...
1
vote
2answers
52 views

Fastest way to obtain the parametric value t of a bezier curve, for a given set x coordinates.

The problem is the following: Having a bezier curve B(t) we have coordinate x from the curve, and we need to obtain the y values from it, hence we need to compute the t values. What is the fastest ...
2
votes
0answers
71 views

Explicit expression for root of equation

Is it possible to find an explicit expression for the root(s) (except $x=0$) for the following function $$f(x)= x-2 + 2b^x$$ where $0\leq b \leq 1$. Numerically this is no problem at all. But what ...
3
votes
1answer
42 views

Points of intersection between circle and parabola

Find the points of intersection between circle and a parable: circle: $x^2 + y^2 - 2x + 4y - 11 = 0$ parable: $y = (-x^2+ 2x + 1 - 2\sqrt{3})$ I don't understand how to solve this, I really tried, ...
3
votes
0answers
55 views

Roots of derivative of q-expontial function

Let the q-deformation of the exponential function be defined by $$ e_q(z)=\sum_{n=0}^\infty{\frac{z^n}{[n]_q!}}. $$ Eq. (1.8) of this paper provides the product representation $$ ...
0
votes
1answer
26 views

Find roots of binomial expression by replacing some variables?

So we have the binomial expression * I might be not using the correct term,english isnt my first language* $$ \left[1- \frac 34e^{-j2\pi\cdot f} + \frac 18e^{-j4\pi \cdot f} \right]$$ How do I find ...
1
vote
1answer
49 views

roots of sum of exponential functions

Could anyone point me in the right direction of finding the roots of equations of the form $$ \sum_{i=1}^n a_ie^{f_i(x)}, $$ where $a_i \in \mathbb{R}$ and the $f_i$ are each first degree polynomials ...
1
vote
4answers
65 views

Finding all roots of $z^4-4z^3+9z^2-4z+8$

I need to know all the roots of $z^4-4z^3+9z^2-4z+8$. I know only one root: z=i. Is there an easy way to find the 3 roots that are unknown? thanks.
3
votes
1answer
38 views

Is the set of continuous function with Lebesgue zero set a Borel set in continuous space?

Let $D$ be a domain in $\mathbb{R^d}$ and denote the continuous function space on $D$ as $X := C(\overline{D})$ where we can define the $\sigma$-algebra $\mathscr{B}(X)$ of $X$, that is sets in $X$ ...
1
vote
0answers
105 views

Is there a relationship between a function's period and number of roots?

Let: $f(x,a,l)=\prod _{k=a}^l\sin \left(\frac{\pi x}{k}\right)$ and $f(x,k)=\sin \left(\frac{\pi x}{k}\right)$ I came up with this equation to find the period $T(f(x,a,l))=2\,{{\pi }^{l-a+1} ...
0
votes
1answer
48 views

Methods for solving equations with exponents?

In the following equation, capital letters represent arbitrary real numbers that are constant with respect to $x$: $$A\left(x+B\right)\left(1 + \frac{C}{x+D}\right)^E + Fx + G = 0$$ I'm trying to ...
3
votes
3answers
36 views

Roots of real polynomial

$f(x)$ is a real polynomial. Show that $z=a+bi$ and $\bar z=a-bi$ have the same algebric multiplicity. I know that if $z=a+bi$ is a root of $f$ then $\bar z=a-bi$ is too, but don't know how to use ...
14
votes
2answers
359 views

Does this polynomial have all its roots both distinct and real?

Recently, I wondered about the following problem: let $n\geq 5$ and let $$ P_n(x)=(x-1)(x-2)\ldots (x-n)-1 $$ Is it true that $P_n(x)$ has $n$ distinct real roots for any $n\geq 5$ ? I checked it ...
3
votes
2answers
71 views

Root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$

Is there an analytic way to obtain the highest root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$ in terms of $K$ and $C$? The integer $K \ll x$ and the constant $C$ are known. The other way to ask ...
1
vote
3answers
47 views

How do you solve two equal absolute value expressions?

I'm having trouble understanding how the following is solved. $$|x+1| = |x-2|$$
6
votes
2answers
192 views

Find the maximum possible value.

For all ordered triples $(p,q,r)$ define the polynomial $$f_{p,q,r}(x)=x^3-px^2+qx-r$$ Let $a_{1},a_{2},a_{3},b_{1},b_{2},b_{3},c_{1},c_{2},c_{3}$ be (not necessarily distinct) positive reals such ...
16
votes
3answers
360 views

The function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$.

Show that if $f\in \mathcal{C}^3$ and $2\cdot\pi$ periodic then the function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$. My attempt : f is $2\pi$ periodic and $\mathcal{C}^3$, we have : ...
2
votes
2answers
48 views

Information about the roots of a polynomial without their calculation

Suppose I have a polynomial (of any order) and I'm not able to calculate the roots. Is there a way to get at least some information about the roots such as how many of them are complex, negative or ...
6
votes
3answers
82 views

Solving Equation of Degree n, where n is any value between 1 and 2

How does one solve an equation of the form: $$ax^n + bx + c = 0$$ where n is a non integer value between 1 and 2. Is there a formula to provide an analytic solution?
30
votes
4answers
870 views

$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x)$, then all the roots of $p_k(x)$ are real

$p_0(x)=a_mx^m+a_{m-1}x^{m-1}+\dotsb+a_1x+a_0(a_m,\dotsc,a_1,a_0\in\Bbb R)$ is a polynomial, and $$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x),\qquad n=1,2,\dotsc$$ then, there exist $N\in\Bbb N$, such ...
1
vote
1answer
50 views

Using argument principle on $e^z + z$

I want to use the argument principle to estimate the number of zeros of $e^z + z$ inside the rectangle with sides $y=2\pi n i, 2 \pi (n+1)i$ and $x = R, -R$ for $R$ large. But $\int \frac {e^z + ...
0
votes
2answers
55 views

Ideas on how to proceed with a proof?

Sorry if this is a nonspecific question - I can provide more details but at this point I need general ideas on a proof strategy. So I recently reduced a rather difficult optimization problem to ...
6
votes
4answers
125 views

Solve $x^{3}-3x=\sqrt{x+2}$

Solve for real $x$ $$x^{3}-3x=\sqrt{x+2}$$ By inspection, $x=2$ is a root of this equation. So, I squared both sides and divided the six degree polynomial obtained by $x-2$. Then I got a ...