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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2answers
34 views

if $f(x)=x^2$ and $g(x)=x\sin x+\cos x$ then number of points where $f(x)=g(x)$?

The question is if $f(x)=x^2$ and $g(x)=x\sin x+\cos x$ then number of points where $f(x)=g(x)$? My approach:- $$f(x)=g(x)\implies x^2=x\sin x+\cos x\implies x^2-x\sin x-\cos x=0$$ Let $$h(x)=x^2-x\...
2
votes
1answer
30 views

Show that this sum of polynomials has no zeros with positive real part

Let $0 < \lambda_1 \leq \ldots \leq \lambda_n $ and $k_1, \ldots, k_n> 0$. Let further $$ \begin{align} P(x)&:=\prod_{i=1}^n (x+\lambda_i) = (x+\lambda_1)\cdot \ldots \cdot (x+\lambda_n) \\...
4
votes
2answers
50 views

The Set $x:\left |x+\frac{1}{x}\right|>6=?$

The question is that ,the Set $x:\left |x+\frac{1}{x}\right|>6$ equals what intervals of $x$? My approach:- I tried to solve the inequality and get interval for $x$'s value as follows:- $$\left|\...
4
votes
4answers
76 views

Show that over $\mathbb{Z}_7$, $x^5-1$ has no roots other than 1.

Show that over $\mathbb{Z}_7$, $x^5-1$ has no roots other than 1. I know I can iterate through each member of $\mathbb{Z}_7$ and show that each one, when raised to the 5th power, is not equal to 1 ...
-2
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1answer
46 views

Find the roots of the equation - $z^2 +12jz+64 = 0$

Just needing a little guidance. This is what I've done so far and I'm not sure if I'm doing it right. Using quadratic formula: $$z^2+12jz+64=0$$ $$ z= \frac{-12j ±\sqrt{(12j^2-4\times1\times64)}}{2\...
-5
votes
1answer
59 views

Proof that $0^0 \neq 1$ [closed]

Suppose that $t = \sqrt{t}^{\sqrt{t}}$, then, it follows that; $$ t^{\sqrt{t}} = \sqrt{t}^{t} \\ \frac{1}{2}t\ln{\left(t\right)} = \sqrt{t}\ln{\left(t\right)} \\ \ln{\left(t\right)}\left[\frac{1}{2}t ...
2
votes
1answer
39 views

A non-constant polynomial with odd-integer co-efficients and of even degree , has no rational root?

Let $f(x)$ be a non-constant polynomial in $\mathbb Z[x]$ with odd-integer co-efficients and even degree ; then is it true that $f$ has no rational root ?
0
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1answer
25 views

Construct a Sinusoidal Equation for an Irregular Period

I would like to be able to construct a sinusoidal function of limited domain given a set of real roots, assuming that the function is graphically centered on $y=0$. I expect that this would ...
0
votes
1answer
53 views

How to show that $a^3+b^3+c^3+d^3\geq abc+abd+acd+bcd$ if $a,b,c,d>0$

How can I prove that if $a,b,c,d>0$ then $$a^3+b^3+c^3+d^3\geq abc+abd+acd+bcd?$$ I think there is some simple proof but I can't remember... is this a special case of some general inequality? ...
0
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1answer
20 views

Solving a characteristic Polynomial of the Hilbert Matrix

I need to find the eigenvalues of the following characteristic polynomial but I can't seem to successfully find the roots of the equation: $P[λ]$ = $λ^5$ - $563/315λ^4$ + $0.3476λ^3$ - $0.0038λ^2$ ...
0
votes
6answers
50 views

Roots of quadratic equation are given by $b \pm \sqrt{b^2 - c}$

I was reading slides about the cancellation error in quadratic equations and it's written: The roots of the quadratic equation: $$x^2 - 2bx + c = 0$$ with $b^2 > c$ are given by $b \...
2
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1answer
35 views

Number of zeroes in the disk $D(0; \frac{1}{4})$ - $f(z)=z^4-5z+1$

Question : How many zeroes the polynomial $f(z)=z^4-5z+1$ are there in the disk $D(0; \frac{1}{4})$? Rouché's theorem : Let $D \subset \mathbb{C}$ a domaine and $f,g: D \to \mathbb{C}$ two ...
0
votes
1answer
30 views

Rouché's theorem - Application??

If $f$ is analytic at the interior of the circle $C(0;1)$ and if $|f(z)|<1$ for $|z|=1$, show that there exists a only point $z_0$ at the interior of this circle such that $f(z_0)=z_0$ Rouché's ...
2
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1answer
14 views

Polynomial $1+3z^m+5z^n$ ($1<m<n$) - Annulus $\frac{1}{3}< |z| < 1$

Rouché's theorem : Let $D \subset \mathbb{C}$ a domaine and $f,g: D \to \mathbb{C}$ two holomorphic functions in $D$. Let $C$ a closed path contained in the interior of $D$. If $|f(z)+g(z)| < |f(z)|...
2
votes
1answer
28 views

Show that if $a>e$, the equation $az^n=e^z$ admit $n$ roots in the unit disk - Rouché theorem

Rouché theorem : Let $D \subset \mathbb{C}$ a domaine and $f,g: D \to \mathbb{C}$ two holomorphic functions in $D$. Let $C$ a closed path contained in the interior of $D$. If $|f(z)+g(z)| < |f(z)|+|...
3
votes
4answers
86 views

Find an irreducible polynomial of degree $21$ in $\mathbb{F}_2[x]$

Note: this is from a paper on Galois Theory, so I believe the technique will come from Galois... Possibly. I am going to reduce it to the monic case: $x^{21}+a_{20}x^{20}+... a_0$ where all $a_i$ ...
0
votes
3answers
49 views

Showing that a function does not have two distinct roots

I have a function $f(x)= x^3+\frac 32 x^2+\lambda$ where $\lambda$ is any real number and need to show that the function does not have two distinct roots in the interval $[0,1]$. I know I am to use ...
0
votes
1answer
36 views

Showing a function does not have two distinct roots

I have the function $f(x)=x^3-(3/2)x^2+k$ where $k$ is any real number and I am to show that the function does not have two distinct roots in the interval $[0,1]$. I am in need of help applying the ...
1
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1answer
50 views

Can the root locus of a minimum phase plant become unstable?

I have a discrete system for which the root locus equation is given as: $$A(z) + K\cdot B(z) = 0$$ They are such that $A(0) = 1, B(0) > 0$, and $K>0$. $\frac B A$ is minimum phase and a ...
2
votes
2answers
59 views

Prove that a periodic nested radical converges to the largest real root of the corresponding polynomial

Let $a_k$ be a sequence of fixed positive integers, $k \in [1,n]$. Consider the periodic nested radical: $$x=\sqrt{a_1+\sqrt{a_2+\cdots+\sqrt{a_n+x}}}$$ We can transform this nested radical into the ...
5
votes
0answers
56 views

“gapped” polynomial leads to ring-shaped roots

Given a polynomial $$P(z)=\sum_{n=0}^N a_n z^n$$ with real coefficients distributed as a gaussian curve $a_n=\frac{1}{\sigma\sqrt{2\pi}}e^{(n-b)^2/2\sigma}$ ($b>0$). The sum of all the polynomial ...
0
votes
0answers
29 views

Closed form roots of sum of exponential functions

Do anyone know a way to solve an equation like the following (over the complex numbers)? $1+2^z+3^z=0$ I certainly cannot. I've tried by hand, and by mathematica, but I can't figure it out. Thanks in ...
0
votes
0answers
54 views

Comparison between roots of two polynomials

Let $m,n,p$ be natural number greater than $2$. Consider $$f(x)=(x-p+1)(x-m+1)(x-n+1)-x(2x-m-p+2)$$ We also have $g(x)$ which is obtained by changing $m$ to $m+1$ and $n$ to $n-1$ in $f$, i.e. $$g(x)=...
1
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1answer
62 views

Why this book says that $ 2^{1/2} = ±\sqrt{2} $?

Shouldn't it be: $ 2^{1/2} = \sqrt{2} $ ? I know the problem is that there they are working with complex numbers, but I still don't understand. The book is in the link, page 113, when they move ...
1
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1answer
37 views

Real solutions of $x^5+ax^4+bx^3+cx^2+dx+e=0$

If $2a^2<5b$,prove that the equation $x^5+ax^4+bx^3+cx^2+dx+e = 0$ has at least one complex root。Thanks.
0
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1answer
43 views

Simple question about finding roots of a polynomial

What am I doing wrong here? This is the denominator of one of my problems and I need to find the roots, so: $6i-z^2+1 \to z=\sqrt{1+6i}$ and $z=-\sqrt{1+6i}$ $\therefore$ $(z+\sqrt{1+6i})(z-\sqrt{...
1
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2answers
58 views

'Elegant' ways on solving for roots for this cubic function?

I have this interesting cubic equation, $$ x^{3} - 80\alpha x^{2} + (1744\alpha^{2}-81)x + (3240\alpha-5760\alpha^{3}) = 0 $$ where $\alpha$ is some constant. I went about the method of Cardano, ...
0
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2answers
37 views

integrate the following function

please integrate for me the following function, it comes to me in the exam, and I put my answer on it, but I don't know if it's true or not I did analyze the to and I multiply it to and the ...
2
votes
1answer
28 views

Find all the roots of this polynomial

I'm currently stuck with the following problem: Find all the roots of the equation $$1-\frac{x}{1}+ \frac{x(x-1)}{2!}-...+(-1)^n \frac{x(x-1)...(x-n+1)}{n!}=0$$ I can sort of see that the roots ...
1
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3answers
55 views

How to solve a polynomial $P(x) = 0$

At the moment we just learnt the factor theorem of polynomials and how if $x-a$ is a factor of $P(x)$, then $P(a) = 0$. We're then taught to find the roots of a polynomial its best to check the ...
1
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0answers
33 views

Isolating roots of polynomial system

I would like to isolate the regions which contain the roots of a system of two bivariate cubic polynomials. I thought I would project the solutions onto $x$ and $y$ axis by means of resultant ...
0
votes
2answers
39 views

Why is the product of the roots of $p(z) = 1 + z + az^n$ equal to $1$?

This should be very straightforward to show, but I am having some issues doing so. We have: $$1 + z + az^n = a(z-z_1)(z-z_2)\dots(z-z_n),$$ where $z_1$, $z_2$, $\dots , z_n$ are the roots of $p$. ...
2
votes
1answer
64 views

When to use Newtons's, bisection, fixed-point iteration and the secant methods?

I've been introduced more or less to these methods of finding a root of a function (a point where it intersects the $x$ axis), but I'm not sure when they should be used and what are the advantages of ...
0
votes
0answers
29 views

Showing fields are algebraically closed

Let K be a field, and let P be separable and irreducible over K. Let L be a splitting field of P over K. I want to show that the fields K(u) and K(v) are isomorphic, where u and v are roots of P, ...
0
votes
0answers
35 views

Iterative methods to find roots

I'm trying to do optional exercises for my numerical methods class. I'm stuck in this one right now: Consider the function $f(x)=-e^{-2x}+3x$. a) Prove that $f$ has an unique real root. ...
0
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3answers
89 views

Proving there exists such a polynomial

I'm having trouble proving the following statement: For all primes $p$, there exists a non-constant polynomial $f(x)\in \mathbb Z_p[x]$ such that f(x) does not have a root in $\mathbb Z_p$ What I ...
3
votes
1answer
42 views

Use an expression for $\frac{\sin(5\theta)}{\sin(\theta)}$ to find the roots of the equation $x^4-3x^2+1=0$ in trigonometric form

Question: Use an expression for $\frac{\sin(5\theta)}{\sin(\theta)}$ , ($\theta \neq k \pi)$ , k an integer to find the roots of the equation $x^4-3x^2+1=0$ in trigonometric form? What I have ...
0
votes
1answer
24 views

Integer values of a rational function

How does one analytically determine the integer values of a rational function $f(x)$$=$$\frac{40-8x}{8x+2}mod1$ where $x$ is an element of the rationals? I just gave the function listed as an example,...
1
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4answers
90 views

Cubic polynomial with three (distinct) irrational roots

I am looking for an equation $$x^3+ax^2+bx+c=0, \qquad a, b, c \in \Bbb Z,$$ of degree $3$ that has $3$ different roots. For an equation of degree $2$ it is easy---for example $x^2-2=0$---but I ...
-1
votes
1answer
32 views

show that $|x_1 -x_2|=\frac{\sqrt{\Delta}}{|a|}$, where $\Delta$ is the discriminant and $x_1,x_2$ the roots of a second degree polynomial equation [closed]

Suppose that $P(x)=ax^2+bx+c, a,b,c\in\mathbb{R}, a\not=0$ has two real roots $x_1,x_2$. Show that $|x_1-x_2|=\frac{\sqrt{\Delta}}{|a|}$, where $\Delta$ is the discriminant.
0
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1answer
32 views

Show that if $w^3=1$, then $1+w+w^2=0$

Find the cube root of unity. Hence, show that if $w^3=1$, then $1+w+w^2=0$. The cube root of unity is $1, -\frac{1}{2}+\frac{\sqrt{3}}{2}$ and $ -\frac{1}{2}-\frac{\sqrt{3}}{2}$. If $w^3=1$ then $...
17
votes
1answer
736 views

Can we permute the coefficients of a polynomial so that it has NO real roots?

Let $P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\ldots+a_{0}$ be an even degree polynomial with positive coefficients. Is it possible to permute the coefficients of $P(x)$ so that the resulting polynomial ...
1
vote
2answers
43 views

Confusion about nth root of a number

Let's say we want to find the 6th roots of 64. Then according to the method from my textbook: $z^6=64$ $z^6=64+0i=64[\cos(0)+i\sin(0)]=64cis(0)=64cis(0+2k\pi)$ Then by De Moivre's theorem: $z=\...
1
vote
0answers
26 views

Zeroes of infinite product function?

Given an entire function $\prod_{n=0}^∞ E_n (\frac{z}{z_n})$ , show that $(z_n )_n∈N$ is a complete list of the zeroes of this function in which each zero appears as many times as its multiplicity. ...
0
votes
2answers
206 views

How to use the property $\sqrt{a}\sqrt{b}=\sqrt{ab}$ with caring that it not always holds?

I was solving a question from my book. If $\alpha$, $\beta$ are the roots of $pt^2+qt+q=0$, $p \neq 0$ and $q \neq 0$ then show that, $\sqrt{\frac{q}{p}}+\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\...
10
votes
3answers
101 views

Mental $n-$th root of $N$

It has been a while since I started thinking about this problem: a fast method to evaluate (in an approximate way) mentally the $n-$th root of a number $N$. I'm talking about great numbers, because ...
0
votes
2answers
44 views

Using the roots of $ z^5=1$, or otherwise, write $z^4+z^3+z^2+z+1$ as the product of two quadratic expressions with real coefficients.

Using the roots of $z^5=1$, or otherwise, write $z^4+z^3+z^2+z+1$ as the product of two quadratic expressions with real coefficients. Hence find the exact value of the product $\cos(2\pi/5)\cos(4\pi/5)...
3
votes
1answer
51 views

Zeros of analytic function accumulating to the boundary

By $\mathbb D$ denote the open unit disc in $\mathbb C$. Suppose that $f : \overline{\mathbb D}\to\mathbb C$ is analytic on $\mathbb D$ and continuous on $\overline{\mathbb D}$. Assume now that there ...
2
votes
2answers
50 views

Finding number of roots using Rolle's Theorem, and depending on parameter

I need to count the number of real solutions for $ f(x) = 0 $ but I have an $m$ in there. $$ f(x) = x^3+3x^2-mx+5 $$ I know I need to study $m$ to get the number of roots, but I don't know where to ...
3
votes
0answers
59 views