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.

learn more… | top users | synonyms (1)

2
votes
0answers
102 views

Rational Non-Integral Root

Prove by contradiction that the following equation with integral coefficients can not have a rational but non integral root. $x^{n}+p_{n-1}x^{n-1}+p_{n-2}x^{n-2}+\cdots +p_{0}=0$
3
votes
1answer
136 views

Find the value of $m + n + r$

One of the roots of the equation $2000x^6+100x^5+10x^3+x-2=0$ is of the form $\frac{m+\sqrt{n}}r$ ,where $m$ is non zero integer and $n$ and $r$ are relatively prime numbers.Then the value of $m+n+r$ ...
1
vote
2answers
173 views

Show quartic polynomial has no real solutions

To show a lower bound for the runtime of an algorithm, I need to show that $$ 3 x^4 - \frac{64}{5} x^3 + \frac{192}{5} x^2 - \frac{192}{5} x+ 12 > 0 $$ for all real numbers $x\in \mathbb{R}$. ...
0
votes
1answer
106 views

Is this polynomial solvable by radicals?

The polynomial $p(x) = x^6-9x^4-4x^3+27x^2-36x-23$. has at least one (real, irrational) root that is expressible by radicals (can you find it?). Are all the roots of $p$ expressible by radicals and ...
0
votes
1answer
85 views

Substitution to linear + nth power form

Given an arbitrary polynomial: $$a_0 + a_1x + a_2x^2 ... a_nx^n$$ Does there exist a series of substitutions (or single substitution if you choose to combine them) that leaves this function in the ...
1
vote
4answers
147 views

Multiplicity of zeros

Can you explain me how to get the multiplicity of a zero? In particular, I would ask you how to determine the zeros' multiplicity of $$\cos(\frac{\pi}{2}z)$$ I suppose they are $z = 2k+1, k \in ...
3
votes
1answer
46 views

Why is the RSA exponentiation function a permutation (i.e. a bijection) over $\mathbb{Z}^*_N$

Why is the RSA exponentiation function a permutation (i.e. a bijection) over $\mathbb{Z}^*_N$? My doubt was specifically why, when raising to the power of the decryption key d we get a unique number ...
0
votes
1answer
41 views

Integer root of an equation

I saw this question somewhere, have a doubt whether it's correct. Suppose $a_1, a_2 \cdots a_{2n}$ are distinct integers. The equation $(x-a_1)(x-a_2)...(x-a_{2n})-(-1)^n(n!)^2$=0 has an integer ...
4
votes
3answers
199 views

If $x\in\mathbb R$, solve $4x^2-40\lfloor x\rfloor+51=0$.

If $x\in\mathbb R$, solve $$4x^2-40\lfloor x\rfloor+51=0$$ where $\lfloor x\rfloor$ denotes the integer part of the number. $\lfloor x\rfloor\le x$ and $\lfloor x\rfloor=x-\{x\}$, where $\{x\}$ ...
4
votes
2answers
83 views

Newton iteration method

i need some help here. My function is $f(x) =x^{3}$ . I was asked to find the number of iterations that are needed to reach the precission $10^{-5}$ if $x_{0} = 0.9$ I was wondering if there is a ...
3
votes
1answer
147 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
65 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
62 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
65 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
141 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
89 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
134 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
129 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
188 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
83 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
225 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
78 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
108 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
47 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
249 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
102 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
128 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
53 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
94 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
959 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
66 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
141 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
71 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
349 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
248 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
375 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
64 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
92 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
125 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
201 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
84 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
44 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 ...