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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3
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1answer
145 views

Proof Verification: The polynomial $f(x) = (x+1)^n-x^n-1$ has a root of multiplicity 2 if and only if $n \equiv 1 \pmod 6$

Proof Verification: The polynomial $f(x) = (x+1)^n-x^n-1$ has a root of multiplicity 2 if and only if $n \equiv 1 \mod 6$ Let $r$ be a root, real or complex, of multiplicity 2 of $f(x)$. Then, by the ...
0
votes
2answers
64 views

Find roots of polynomial $f(X) = X^7 - 6 X^6 + 10 X^5 - 13 X^3 + 18 X^2 -22 X + 12 \in \mathbb Q[X]$

Find the roots of the polynomial $$ f(X) = X^7 - 6 X^6 + 10X^5 - 13 X^3 + 18 X^2 -22 X + 12 \in \mathbb Q[X] $$ in $\mathbb Q$, $\mathbb R$ and $\mathbb C$. We covered the factor-theorem in ...
-2
votes
3answers
60 views

A question on quadratic equations.. Given below in the picture.

PLease also tell how u got to the answer as I want to know the way to solve further questions
0
votes
0answers
63 views

Application of Hensel's lemma

Show that the polynomial $\Phi(x)=x^2 -2 \in O(\widehat{\Bbb Q_2})[x] $ has no root in $\widehat{\Bbb Q_2}$, even though $\bar\Phi(x)\in E(\widehat{\Bbb Q_2})[x]$ has a root in $E(\widehat{\Bbb Q_2}) ...
3
votes
1answer
120 views

Finding real cubic root of the equation

The cubic equation has one real root.Find it. $\displaystyle 8x^3-3x^2-3x-1=0$
2
votes
3answers
88 views

If $a$ and $b$ are the roots of $z^2 - 2z + 4 = 0$ then what is $a^n + b^n + ab$ ($n$ is a natural number)?

I don't know how to solve this question, any help would be appreciate it. If $z^2 - 2z + 4 = 0$, then what is the result of this $a^n + b^n + ab$ ($n$ is a natural number, $a$ and $b$ are the roots ...
2
votes
2answers
131 views

Find all solutions of $z^5+a^5=0$

The task is as follows: Find all solutions of $z^5+a^5=0$, where $a$ is a positive real number. My initial attempt (which leads nowhere) My guess is that i'll have to find the 5 5th roots of ...
3
votes
1answer
123 views

Newton-Raphson's method

Hello MathExchange community ! I am working on some "simple" numerical methods to solve 4th degrees and below equations. To make it easier I am working on the $[0, 1]$ interval and I know for sure ...
0
votes
4answers
183 views

How can I solve $y^{3}-3y^{2}+2=0$?

I am stuck at this equation $y^{3}-3y^{2}+2=0$. How do I solve it without calculator? It might be very trivial so I think I just need a hint. It is actually a substitution $y=\log x$,but I think it ...
1
vote
1answer
82 views

Convergence of order 3 of a Newton's method variant

Let $f\in C^2$ and $x^*$ be a simple root of $f$, i.e. $f(x^*)=0\wedge f'(x^* )\ne 0$. Further, let $U(x^*):=\left\{x : |x-x^* |\le r\right\}$ for some $r>0$ and $\;\;\;\;\;\;\;\;\;\;\displaystyle ...
2
votes
1answer
175 views

Solving a logarithmic polynomial

I want to solve this equation for $x$: $${\frac{1}{\sqrt{2 \pi x}} \left(\frac{e z}{2x}\right)^x} = \epsilon$$ Is there a closed form for it, or does it have to be solved numerically? I can turn it ...
2
votes
1answer
77 views

If $f$ analytic in $|z|>1$ and $|f(z)|<|z|^n$, then $f$ has finitely many zeros in $|z|>2$.

Let $f(z)$ be analytic in $\Omega = \{|z|>1\}$. Suppose that $f$ satisfies $|f(z)| < |z|^n$ for all $z \in \Omega$ and for some n> 0. Prove that either $f$ has finitely many zeros in ...
2
votes
2answers
107 views

Polynomial whose roots are very nearly integers

Let $n\geq 6$ be an integer ; consider the polynomial $$ P_n=\big((x-1)(x-2)\ldots (x-n)\big)^2+(n+1) $$ Let $\varepsilon=\frac{n^4}{173\big((\lfloor \frac{n}{2} \rfloor)!\big)^2}$. I have checked ...
0
votes
0answers
42 views

roots of a polynom in a localization of a UFD

let $ {R} $ be a UFD, $ Q $ the localization of $ R $. I need to find all the roots in $ Q[i] $ of the polynom: $ f(x) = x^4 + \frac {4} {5+i}x^3 - \frac {6+10i} {2+3i}x^2 - \frac {12} {5+i}x + \frac ...
0
votes
2answers
46 views

Roots of a given equation

How can I show that the equation $$e^x-\ln(x)-2^{2014}=0$$ has exactly two positive roots?
5
votes
1answer
199 views

How to imagine zeros of an analytic function of several variables

Let $f(z_1,\cdots, z_n)$ be a holomorphic function of several variables in an open subset of $\mathcal C^n$. Let $Z(f)=\{ (z_1,\cdots, z_n) \: | \: f=0\}$ be the zero set of $f$. If $n=1$, the zeros ...
2
votes
2answers
95 views

Question about bisection method

We have $f(x)=(x-1)^3(x-2)(x-3)$. $a_0<1,b_0>3$. We had to show that if $\frac{a_0+b_0}{2}\ne 1,2,3$, there is one root of $f$ that we can't get it by the bisection method. I guess that this is ...
1
vote
1answer
120 views

Rational Root theorem issue

I've given my class an example: $$2x^3+3x^2+6x+4=0$$ By the rational root theorem, if there is a rational root then it should be of the form $\frac{p}{q}$ where $p$ is a factor of 4 and $q$ is a ...
1
vote
2answers
49 views

Lemma 2.5.5 Boas, Entire functions

I'm reading Boas, Entire functions, but I don't understand lemma 2.5.5, which states that $\sum_{1}^{+\infty}\frac{1}{r_{n}^{\alpha}}$ and the integral $\int_{0}^{+\infty}t^{-\alpha -1}n(t)dt$ ...
2
votes
1answer
93 views

Order of growth of $ \prod_{n=1}^{+\infty} (1-e^{-2\pi n}\cdot e^{2\pi i z})$

The order of an entire function $f$ id defined as $$ord ( f) = inf \{\lambda \geq 0 \ | \ \exists A, B > 0 \ s.t. \ |f(z)|\leq Ae^{B|z|^{\lambda}} \forall z \in \mathbb{C} \}$$ I have $F(z) = ...
1
vote
1answer
51 views

A limit involving the exponent of convergence

Let $f$ be an entire non-constant function with at least one zero. If $\{z_{j}\}_{j\in \mathbb{N}}$ are the zeros of $f$, set $$b =\inf\left\{\lambda >0 \ | \sum_{j}\frac{1}{|z_{j}|^{\lambda}}< ...
3
votes
3answers
778 views

$\alpha,\beta,\gamma$ are roots of cubic equation $x^3+4x-1=0$

If $\alpha,\beta,\gamma$ are the roots of the equation $x^3+4x-1=0$ and $\displaystyle \frac{1}{\alpha+1},\frac{1}{\beta+1},\frac{1}{\gamma+1}$ are the roots of the equation $\displaystyle ...
0
votes
1answer
24 views

primitive roots, field dimension

Let $\zeta$ be a primitive $m$-th root of $1$. Determine the values of $m$ such that: $[\mathbb Q$($\zeta$):$\mathbb Q$]$=2$. The only thing I have in mind is that $[\mathbb Q $($\zeta$):$\mathbb ...
0
votes
1answer
63 views

Rate of Convergence of Generalized Iterative Method

Consider the generalized iterative method for finding polynomial roots: $z_{k+1}=z_k +d\frac{(1/p)^{(d-1)}(z_k)}{(1/p)^{(d)}(z_k)}$ where d is a positive integer. Note that Newton's Method is a ...
6
votes
2answers
140 views

Prove that $\sqrt[2012]{2012!}<\sqrt[2013]{2013!}$

I need to prove that $\sqrt[2012]{2012!}<\sqrt[2013]{2013!}$ My attempt: Let $a=\sqrt[2012]{2012!}$ and $b=\sqrt[2013]{2013!}$ Then $\displaystyle\frac{b^{2012}}{a^{2012}}=\frac{2013}{b}$ ...
2
votes
2answers
68 views

rectangle where $\cos{z} =iz$ has exactly one solution

Determine a rectangle inside which there is exactly one solution of the equation $\cos{z} = iz$. I know the following result: Let $f$ be holomorphic in $\Omega$ with $a \in \Omega$. Let $f(a)= b$ is ...
2
votes
4answers
98 views

On the roots of a polynomial [closed]

Let $f(x)= x^3 - 3x + 1$. Show that if a complex number $a$ is root of $f(x)$, then $a^2-2$ is too.
0
votes
2answers
277 views

If two polynomials both of n degree have n identical real roots, are they equal? Proof?

CORRECTION: The polynomials don't have to be equal, but one has to be a constant multiple of the other. I ask the question because I saw this fact used in this solution to a problem: Problem: Given ...
11
votes
4answers
240 views

Coefficients of a polynomial also are the roots of the polynomial?

How many real solutions $(r_1, r_2, \cdots, r_n)$ are there such that $(r_1, r_2, \cdots, r_n)$ are the roots of the polynomials $x^{n} + r_1 x^{n-1} + r_2 x^{n-2} + \cdots + r_n$ For $n = 2, 3, 4$ I ...
6
votes
3answers
298 views

Number of integral roots of a polynomial

Let $p(x)$ be a polynomial with integral coefficients. Let $a$, $b$, $c$ be three distinct integers such that $p(a) = p(b) = p(c) = -1$. Find the number of integral roots of $p(x)$.
2
votes
1answer
60 views

uniformly bounded sequence of non constant holomorphic functions

Let $\{f_n\}_{n=1}^{\infty}$ be a uniformly bounded sequence of nonconstant holomorphic functions in a connected open set $\Omega$. Let $f \not \equiv 0 $ be a holomorphic function in $\Omega$. ...
3
votes
2answers
151 views

Behavior of $f(z)=\int_0^1\mathrm{e}^{\alpha t^2}\sin(tz)\,dt$ when $\alpha <0$

Define $$f(z)=\int_0^1\mathrm{e}^{\alpha t^2}\sin(tz)\,dt,$$ where $\alpha \in \mathbb{R}$. If $\alpha >0$ then $f(z)$ has infinitely many real zeros and at most a finite number of complex ...
0
votes
3answers
88 views

Absolut value of cubic polynomial roots lower than 1

Assume we have a cubic polynomial $ x^3 +bx^2+xc+d=0 $, with b,c,d real numbers. Let $x_1, x_2, x_3 $ be the roots, either real or complex. What is the relation of the coefficients b,c and d in ...
4
votes
2answers
116 views

meromorphic function in the unit disc with only one pole of order n

Let $f$ be meromorphic in a neighborhood of $\{|z| \leq 1\}\setminus \{1/2\}$ and have a pole or order $n$ at $1/2$. Suppose that $|f| < 3$ on $\{|z|=1\}$. Show that for any $\phi \in \mathbb{R}$, ...
2
votes
3answers
152 views

theory of equations finding roots from given polynomial

If the equation $x^4-4x^3+ax^2+bx+1=0$ has four positive roots then $a=\,?$ and $b=\,?$ $\textbf{A.}\,6,-4$ $\textbf{B.}\,-6,4$ $\textbf{C.}\,6,4$ $\textbf{D.}\,-6,-4$ we can ...
1
vote
1answer
81 views

Find the root of the polynomial?

Consider the root of the polynomial $p(x) = x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots+a_1x -1$. Suppose that $p(x)$ has no roots in the open unit disc in a complex plane and $p(-1)=0$. Show that ...
4
votes
1answer
59 views

About the zeros of $f_n(z)=\sum_{k=1}^n k^{-z}$.

Let $z$ be a complex number. Consider $f_n(z)=\sum_{k=1}^n k^{-z}$. Now I wonder : Are there infinitely many positive integer $n$ such that there exists a $z$ with $f_n(z)=0$ and $Re(z)>1$ ? I ...
1
vote
1answer
41 views

Repeated Eigenvalues in Systems of ODEs

Question is to find the general solution of the given system of equations below. $$ x' =\left(\begin{array}{rr}\frac{-3}{2} & \frac{-1}{4} \\ 1 & \frac{-1}{2}\end{array}\right)x $$ My ...
0
votes
1answer
108 views

Solve : $x^4 + 6x^3 -3x^2 + 2 = 0$

$x^4 + 6x^3 -3x^2 + 2 = 0$ To find the zeros, I tried this by Ferrari's method but got stuck where a value of 'lambda' has to be obtained.
13
votes
1answer
153 views

Prove $\sum_{i=0}^n \binom{n}{i}^2x^{n-i} = 0$ has $n$ negative roots

Let's $n \in \mathbb{Z^+}$, how to $\text{prove}|\text{disprove}$ that: the equation $\boxed{\sum_{i=0}^n \binom{n}{i}^2x^{n-i} = 0}$ has exactly $n$ distinct negative roots. My friend get ...
4
votes
2answers
67 views

Given $f \in \mathbb{Q}[x]$ irreducible. How many and which roots of $f$ are contained in $\mathbb{Q}[x]/(f)$?

It is a fact that struggle me for a while. When working with irreducible polynomial over $\mathbb{Q}$ it is natural to build the extension ${\mathbb{Q}[x]}/{(f)} $ in which "lives " one root of the ...
3
votes
2answers
117 views

Existence of holomorphic function with a sequence of zeros in the unit disc

The question is : Prove that there exists a holomorphic function $f$ on the open unit disc $\{z \in \mathbb{C} : |z| <1\}$ with the properties that $f(0) = 0$ and $f(1-1/n)=1$ for every integer $n$ ...
0
votes
2answers
173 views

A 3rd degree polynomial $P(x)$ has three unequal real roots. What is the least possible # of unequal real roots for $P(x^2)$

I got that if P(x) is a 3rd degree polynomial then P($x^2$) must be a 6th degree polynomial. I don't know how to proceed from here.
2
votes
4answers
130 views

Find the solution of the equation

Find all real solutions of this equation : $$x=\sqrt{2+\sqrt{2-\sqrt{2+x}}}$$
1
vote
1answer
64 views

Solution of equations of the form: $a^x+b^x+c=0$

Is it possible to solve equations of the form: $a^x+b^x+c=0,\;abc\neq0$ with analytical methods; if so, how is this done?
3
votes
1answer
188 views

zeros of a function holomorphic in the closed unit disc

Let $f$ be a holomorphic function in a neighborhood of the closed unit disc $\{z \in \mathbb{C} : |z| \leq 1\}$, and suppose that $\Re{(\bar{z}f(z))} > 0 $ when $|z| = 1$. Prove that $f$ has ...
2
votes
3answers
173 views

How bad, really, is the bisection method?

We know that the bisection method for root finding is slow (linear convergence), but has the advantage of always working for a continuous function, if we start with a interval which brackets the root. ...
1
vote
1answer
89 views

How to calculate the integral of a function with a root?

I have to solve this integral: $\int\limits_{-1}^1(3x^3-5x^2+12x-9)~dx$ I used Grapher (a nifty program that comes with Mac OS X) to display the curve of $f(x)=3x^3-5x^2+12x-9$ and it obviously has ...
4
votes
1answer
340 views

Roots of $f(x)=\sin(x)-ax$

How many roots are there of the function $f(x)=\sin(x)-ax$, where $a$ is a positive number? Clearly for all $a$, $x=0$ is a root; if $a>1$ that is the only root. The roots will also be symmetric ...
7
votes
5answers
216 views

Analytical solution to $a^x+b^x=x$

Maybe stupid question, but I am wondering. Is there an analytical solution to equation $$a^x+b^x=x$$ for general $a$, $b$. How should I tackle this problem, if I want to find at least one $x$. ...