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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3
votes
1answer
51 views

A contest math problem

Let $P(x)$ be a polynomial with integer coefficients of degree $d>0$. If $\alpha $ and $\beta $ are two integers such that $P(\alpha)=1$ and $P(\beta)=-1$, then prove that $|\beta ...
2
votes
3answers
56 views

The number of distinct real roots of a polynomial of degree 4

Suppose I have a equation of a degree of 4 and I don't know a proper method of solving this type of equation (like completing the square is a proper method to solve the quadratic equation) so how or ...
1
vote
0answers
24 views

Roots of the Taylor approximation of the exponential

While answering another question, I looked at the roots of the $n^{th}$ degree Taylor approximation of the exponential. $$e^x\approx E_n(x):=\sum_{k=0}^n\frac{x^k}{k!}.$$ Apparently, these root are ...
3
votes
1answer
545 views

Working with casus irreducibilis

I read about casus irreducibilis here. As an example of casus irreducibilis, it says we can factor $x^3 - 15x - 4$ to find $4$ as a root and it also has two other real roots. Using Cardano's method we ...
0
votes
2answers
20 views

Roots of a Non-Monic Cubic Polynomial

Find all roots of $f(x)=231x^3+68x^2-9x-2$ I cannot use the cubic formula or Viete's theorem here because the polynomial is not monic. The only other way I can think of doing this is by the rational ...
11
votes
1answer
106 views

Finding all real roots of the equation $(x+1) \sqrt{x+2} + (x+6)\sqrt{x+7} = x^2+7x+12$

Find all real roots of the equation $$(x+1) \sqrt{x+2} + (x+6)\sqrt{x+7} = x^2+7x+12$$ I tried squaring the equation, but the degree of the equation became too high and unmanageable. I ...
3
votes
1answer
58 views

Find real roots of the equation

Find all real solutions to $$\dfrac{\sqrt{x+1}}{2+\sqrt{2-x}} - \dfrac{\sqrt{x^2-x+2}}{2+\sqrt{-x^2+x+1}} = x^3-x^2-x+1$$ This question is very similar to one of my previous problem, ...
2
votes
1answer
37 views

Solving a mixed radical and quadratic equation

Solve for $x \in \mathbb{R}$ $$4x^2(x+2) +3(2x^2-4x-3)\sqrt{4x+3} +6x = 0$$ I tried taking square by isolating the radical, but the resultant equation couldn't be solved. Any help ...
5
votes
1answer
75 views

Solve for real $x$

Solve for $x \in \mathbb{R}$ $$ 1 + \dfrac{\sqrt{x+3}}{1+\sqrt{1-x}} = x + \dfrac{\sqrt{2x+2}}{1+\sqrt{2-2x}} $$ I tried some substitutions and squaring but that didn't help. I also ...
5
votes
1answer
82 views

Solving a radical equation for real $x$

Solve for $x \in \mathbb{R}$ $$\dfrac{\sqrt{x^2-x+2}}{1+\sqrt{-x^2+x+2}} - \dfrac{\sqrt{x^2+x}}{1+\sqrt{-x^2-x+4}} = x^2-1$$ I tried squaring the equation but it became a sixteen degree ...
2
votes
0answers
19 views

Finding all complex roots of an equation with exponentials.

I know that $$ (-1)^x + 2^x - 2 x - 1 = 0 $$ has a single real root $(x =3)$ and an infinite number of complex roots whose real part appears often negative. Don't the complex roots also have their ...
6
votes
1answer
85 views

Is there any nice explanation of why the complex exponential function has no roots in the complex plane? [duplicate]

Here I am not looking for an explanation that uses basic properties that complex exponential function has, such as $e^{z+w}=e^ze^w$ or $e^0=1$ or any other, if this fact can be explained by using ...
0
votes
1answer
30 views

Cant find roots for this eq'n

Can someone explain how to factor/find roots to this polynomial: $$ s^4 + 14s^3 +45s^2 +650s + 1800 = 0 $$ its such a nightmare ive been stuck for hours, any help would be appreciated :)
0
votes
0answers
14 views

Basic Linear Algebra/Root finding question

What is the general method for solving this problem? $\theta_n.1_T'.z_T=0_n$ where $\theta_n$ is a n x 1 vector of parameters that are free to vary, $1_T'$ is a 1 x T vector of ones, $z_T$ is a T x ...
-2
votes
0answers
14 views

combine the two solutions $2n\pi \pm i\ln(\sqrt2 +1)$ [on hold]

the equation with variable $n$ has two solutions that can be combined together properly and i for complex numbers $$ 2n\pi \pm i\ln(\sqrt2 +1) $$ how this can be solved and how the solutions can be ...
1
vote
0answers
22 views

Conjugate roots of a polynomial

If $\sqrt 2 - i$ is a root of $x^5-x^4-2x^3+mx^2+9x+m-11=0$, $m \in \Bbb Q$ find m and the other roots. My question is what other roots can i deduce from what is given? Is $\sqrt 2 + i$ the only one ...
1
vote
0answers
22 views

Computing $n^{\text{th}}$ root of a positive integer to arbitrary precision using integer arithmetic

There are various questions on this forum that appear similar, but my question pertains to writing code that can compute the $n^{\text{th}}$ root of a number $a$ correct to $p$ decimal places, where ...
1
vote
1answer
38 views

Let $α$ and $β$ be the roots of equation $px^2+qx+r=0,p≠0$

Let $α$ and $β$ be the roots of equation $px^2+qx+r=0,p≠0$, If $p,q,r$ are in A.P and $\dfrac{1}{α}+\dfrac{1}{β}=4$, then the value of $|α−β|$ is $:$ $\dfrac{\sqrt{61}}{9} $ $\dfrac{2\sqrt{17}}{9}$ ...
-2
votes
0answers
22 views

What is the order of the zeros of a complex differentiable function? [on hold]

If $f(x) = x^2 + 4x + 4$ is complex differentiable, meaning with cauchy equations fulfilled, what is the order of the zeros of the function? How to compute it?
0
votes
1answer
34 views

13th root of 2 in field $\mathbb{F}_{13}$

Is there an easy to find $13^{th}$ root of $2$ in the field $\mathbb{F}_{13}$? I'm having trouble finding one here. Thanks!
0
votes
2answers
397 views

Numerical root finding of function with unknown parameters

I have a multivariate function of which I want to find one of (or all) its roots. However, besides the variables, it also depends on a bunch of parameters. Now I only want to find roots which are ...
0
votes
0answers
18 views

solving equation with variable on both sides

Is there a way to solve the following equation for $v_n^2$? I am working through a problem and I feel like there should be a way to solve it, but I am not sure how to do it: ...
-1
votes
1answer
22 views

Show that any polynomial of odd degree 2n+1: $f(x)=\sum_{k=0}^{2n+1} a_kx^k $, $a_{2n+1}\neq0$ has at least one real root.

Show that any polynomial of odd degree 2n+1: $$f(x)=\sum_{k=0}^{2n+1} a_kx^k $$ $a_{2n+1}\neq0$ has at least one real root. I would like to prove this using IVT, how would I go about starting ...
0
votes
1answer
27 views

solving a pair of simultaneous equations

I have a rather messy pair of simultaneous equations, which I need to solve for x: ...
5
votes
2answers
671 views

Solving a 6th degree polynomial equation

I have a polynomial equation that arose from a problem I was solving. The equation is as follows: $$-x^6+x^5+2x^4-2x^3+x^2+2x-1=0 .$$ I need to find $x$, and specifically there should be a real ...
1
vote
0answers
28 views

Quadrant in which the zeros of a polynomial lies

Consider a polynomial $$p(z) = z^6 + 9z^4 + z^3 + 2z + 4 $$ I need to find which quadrant of the complex plane contains how many zeros that lie in unit circle. Also, I need to find which quadrant ...
0
votes
0answers
20 views

Newton Method Variant with convergence of order 3

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be twice continuously differentiable for all $x$ in the neighborhood $U(\xi)=\{x\in\mathbb{R}:|x-\xi|<r\}$ of a simple zero $\xi$ of $f$ such that ...
1
vote
2answers
24 views

Roots of a perturbed equation

I'm looking to show that the equation $$\displaystyle \psi(\delta) := e^{\alpha g(\delta)} - \delta$$ has a real root for $\alpha$ sufficiently small that converges to $\delta = 1$ as $\alpha ...
-1
votes
1answer
40 views

Algebra Roots (Cubic/Complex)

Show that the equation $3z^3+(2-3ai)z^2+(6+2bi)z+4=0$ (where both $a$ and $b$ are real numbers) has exactly one real root, and find this root. I've dealt with quadratics in this form but never with ...
2
votes
1answer
66 views

A simple cubic equation problem:

Consider the cubic equation $$az^3-bz^2+\bar{b}z-\bar{a}=0$$ where $a$ and $b$ are non-zero complex numbers. Suppose $z_1, z_2$ and $z_3$ are the roots. Question: Which $a$ and $b$ gives ...
0
votes
1answer
20 views

Zero functions on open interval

Are there non-constant differentiable functions that are zero on an open interval of real line? I've tried using the product integral: $$ f(x) = \exp(\int_0^1 \log(x-u) \mathrm{d}u ) = \frac{x^x ...
1
vote
0answers
9 views

Finding the roots with the largest magnitude

Given a non-constant polynomial $p\in\mathbb{Z}[x]=\alpha\prod_{k=1}^nx-\alpha_k$ how can I find the roots $R=\{\beta_1,\ldots,\beta_t\}\subseteq\{\alpha_1,\ldots,\alpha_n\}\subseteq\mathbb{C}$ with ...
0
votes
0answers
13 views

Is there an algebraic solution for this rootfinding problem?

I would like to solve for the roots of $f(x)=a_0 + a_1x^\gamma + a_2x^{\gamma+1}$, where $a_0,a_1 \in \mathbb{R}$ and $\gamma \in \mathbb{R}_{\geq 0}$ are arbitrary coefficients. This is possible ...
6
votes
3answers
185 views

Why four roots to this equation: $(7x+1)^{1 \over 3}+(8+x-x^2)^{1 \over 3}+(x^2-8x-1)^{1 \over 3}=2$ [closed]

$$(7x+1)^{1 \over 3}+(8+x-x^2)^{1 \over 3}+(x^2-8x-1)^{1 \over 3}=2$$ I figured the roots are $0$, $1$, $-1$, and $9$. But why?
1
vote
2answers
40 views

$e^z=3z^5$ - Rouche's theorem

Question : Show that the equation $e^z=3z^5$ possesses five distinct real roots. In using the Rouche's theorem with the function $f(z)=-e^z+3z^5$ and $g(z)=-3z^5$, I succeeded to prove the ...
2
votes
1answer
66 views

zeros of two functions are alternate

Let $a,b,c,d$ be real numbers. Show that the zeros of the functions $f(x)=a\cos x+b\sin x$ and $g(x)=c\cos x+d\sin x$ are distinct and alternate whenever $ad-bc\neq 0$. Suppose $x_0\in \mathbb{R}$ ...
0
votes
0answers
20 views

Computationally check for roots/positiveness of a big polynomial in a given interval

For a proof, I need to check that given a little interval $(0, 0.28)$ some concrete polynomials $\in \mathbb{Q}[w]$ (polynomials in one variable ranging over the real numbers, with degrees around 50) ...
3
votes
3answers
150 views

Is my proof for the existence of roots of an odd-degree polynomial correct?

$\color{crimson}{\text{Problem}}$ Show that if $f:\mathbb{R}\to\mathbb{R}$ be a polynomial of odd degree with real coefficients then it has at least one real root. $\color{green}{\text{Proof}}$ Let ...
0
votes
1answer
27 views

Construct a degree $n$ polynomial with roots $a_1, a_2, a_3, \ldots, a_n$

We have the numbers: $a_1, a_2, a_3, \ldots, a_n$ Show that there is a polynomial $P(x)$ of degree $n$ such that $a_1, a_2, a_3, \ldots, a_n$ are roots of $P(x)$
0
votes
0answers
12 views

Equation involving Bessel and Struve functions

I need to solve the equation $Z(\gamma) = r$ of the function $$Z(\gamma) = 1 - \frac{2}{\gamma} \left(J_1(\gamma) - i H_1(\gamma)\right),$$ where $J_1$ is the Bessel function and $H_1$ the Struve ...
0
votes
2answers
381 views

Relation between real roots of a polynomial and real roots of its derivative

I have this question which popped in my mind while solving questions of maxima and minima. First Case:Let $f(x)$ be an $n$ degree polynomial which has $r$ real roots. Using this can we say anything ...
0
votes
1answer
30 views

Roots of a fourth order polynomial [duplicate]

I am looking for the roots of $x^4=-1$, I have written $ -1 $ using Euler as $e^{j180}$. Therefore, $x=\pm e^{j45}$. But the fourth order equation should have two other roots, how can I get them?
3
votes
2answers
35 views

Number of real roots of $f ' ( x )$

Let $$f(x)=(x-a)(x-b)^3(x-c)^5(x-d)^7 $$ where $a,b,c,d$ are real numbers with $a < b < c < d$ . Thus $ f ( x )$ has $16$ real roots counting multiplicities and among them $4$ are ...
1
vote
1answer
62 views

How many roots does the polynomial $p(z) = z^8 + 3z^7 + 6z^2 + 1$ have inside the annulus $1 < |z| < 2$?

How many roots does the polynomial $p(z) = z^8 + 3z^7 + 6z^2 + 1$ have inside the annulus $1 < |z| < 2$? I know I can use Rouche's Theorem. I'm just not sure how. It states that $|f(z) − g(z)| ...
0
votes
1answer
23 views

Estimation for points in a neighbourhood of a root of a polynomial

Let $p(x)$ be a polynomial with complex coefficients and $p(\tilde x)=0$. Choose $\delta>0$ small enough, such that $\tilde x$ is the only root of $p$ in $B_\delta(\tilde x)$. I want to show that ...
3
votes
2answers
83 views

Solve the equation $x^3-6x-6=0$

Evaluate the roots of $$x^3-6x-6=0$$ I solved it using Cardano's method, but I'm looking for other elementary approaches through substitutions and properties of polynomials. ...
0
votes
0answers
25 views

Finding root from implicit definition

I have the following implicit equation which defines $x$, $x$ is such that: $ (x-y) - \frac{x-(2y - f(y))}{2} \Phi( \frac{(2y - f(y)) - x}{\sigma}) - \frac{x-f(y)}{2} \Phi(\frac{f(y) - x}{\sigma}) + ...
3
votes
2answers
53 views

Finding Roots of tenth degree polynomial

I know that there are no explicit formulas to find roots for polynomials of degree higher than $4$. I have to find all the roots of the polynomial $ f(z) = 1+z^2+z^4+z^6+z^8+z^{10}$ I found two ...
0
votes
1answer
47 views

Prove that the roots are equal

Suppose that all roots of the polynomial equation $x^4 - 4x^3 + ax^2 +bx + 1 = 0$ are positive real numbers. Show that all roots of the polynomial are equal. I am not getting any idea as to how to ...
0
votes
0answers
29 views

If x is a rational number expression

Can the expression $\dfrac{\sqrt{x+1}}{\sqrt{x-1}}$ be expressed with a rational denominator as $\dfrac{\sqrt{x^2-1}}{x-1}$ provided that $x$ is a rational number?