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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1answer
247 views

Use Intermediate Value Theorem to prove $\sqrt{s}$ exists, for $s>0$

I'm self-studying proof theory, and working on the following problem: Consider $s\in\mathbb{R}$, with $s>0$. Apply the Intermediate Value Theorem to prove the existence of $\sqrt{s}$. I ...
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2answers
342 views

Solve a quadratic matrix equation?

Given a known symmetric matrix $M$, vector $\vec{v}$ and scalars $a$ and $b$, I'm trying to solve for a scalar $x$ such that: $\vec{v}^T(M+(ax+b)I)^{-1}\vec{v} - ...
3
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2answers
442 views

geometric interpretation of quadratic equation with complex coefficients

When an equation has real coefficients and non-negative discriminant, the geometric meaning of it's roots is intersection of the function with the x-axis. I know how to get roots of quadratic ...
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2answers
639 views

What happens as you take repeated square roots, starting with 8? What does the answer approach as you take more and more square…

a) What happens as you take repeated square roots, starting with 8? b) What does the answer approach as you take more and more square roots? c) Would the answer be the same if you started with any ...
2
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1answer
948 views

Polynomial root finding

I have an univariate polynomial of some degree - how do I numerically find all of its real roots? I never thought I would ask this question - everyone knows how to find polynomial roots, right..? ...
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1answer
177 views

Factoring $1-(x+x²+x³+x⁴+x⁵)$

I calculated the generating function $G$ of the recurrence: $$F(0)=0$$ $$F(1)=F(2)=F(3)=F(4)=1$$ $$F(n)=F(n-1)+F(n-2)+F(n-3)+F(n-4)+F(n-5)$$ I got: $$G(x)=\frac{x+x²+x³+x⁴}{1-(x+x²+x³+x⁴+x⁵)}$$ I ...
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3answers
579 views

Find the root of a polynomial of degree 5

I have the following equation: $${138000\over(1+x)^5}+{71000\over(1+x)^4}+{54000\over(1+x)^3}+{37000\over(1+x)^2}+{20000\over1+x}-200000=0$$ And I need to find the real solution(s) to said equation, ...
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2answers
181 views

Exercise of complex variable, polynomials.

Calculate the number of zeros in the right half-plane of the following polynomial: $$z^4+2z^3-2z+10$$ Please, it's the last exercise that I have to do. Help TT. PD: I don't know how do it.
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1answer
52 views

$\int_0^{\infty} \lim_{m \rightarrow \infty} x_m \left( \varepsilon \right) e^{- \varepsilon} \mathrm{d} \varepsilon$

In the expression $$\int_0^{\infty} \lim_{m \rightarrow \infty} x_m \left( \varepsilon \right) e^{- \varepsilon} \mathrm{d} \varepsilon$$ Is it possible to move the integral inside the Newton's ...
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1answer
178 views

Finding roots of a fourth degree equation having arbitrary constant

The below sum is from Linear differential equations with constant coefficients. Solve. D^4 + k = 0 I have to get the general solution for it . I am stuck in finding the roots of this equation .
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1answer
213 views

Formal Derivative and Multiple roots

I am currently really stuck on the following problem: Prove that if f(x) in Fp[x] and Df = 0 (where D : Fp[x] → Fp[x] is the formal derivative) then there exists g(x) in Fp[x] such that f(x) = g(x)^p ...
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2answers
84 views

Question about roots

Let $a,b,c$ be roots of equation $x^3-6x^2+kx+k=0$,and $(a-1)^3+(b-2)^3+(c-3)^3=0$. how to compute $a,b,c,k=?$ if we do work equivalently as to find out the solution: ...
6
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2answers
456 views

Convergence of fixed point iteration for polynomial equations

I'm looking for the solution $x$ of $$x^n+nx-n=0.$$ Thoughts: From graphing it for several $n$ it seems there is always a solution in the interval $[\tfrac{1}{2},1)$. For $n=1$, the ...
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8answers
4k views

Fastest Square Root Algorithm

What is the fastest algorithm for finding the square root of a number? I created one that can find the square root of "987654321" to 16 decimal places in just 20 iterations (I'm not ready to release ...
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1answer
66 views

Consider polynomial $q(x,y)=(2x+3y)^2-1$, how to show that it has roots with arbitrary values of x,y?

Given the following polynomial $q(x,y)=(2x+3y)^2-1$. How would I show that it has roots with any large $x,y$?
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1answer
37 views

Maximal Distinct Roots in $F_q$

Let $a\in F_q[x]$, and let $r(\cdot)$ denote the number of distinct roots over $F_q$. For any $i|q$, prove that $$ \max_{\deg(a)=1}r(x^i-a)=r(x^i-x) $$
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2answers
2k views

Find the roots of a polynomial using its companion matrix

I would like to find the roots of a polynomial using its companion matrix. The polynomial is ${p(x) = x^4-10x^2+9}$ The companion matrix $M$ is $M={\left[ \begin{array}{cccc} 0 & 0 & 0 ...
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2answers
455 views

closed-form expression for roots of a polynomial

It is often said colloquially that the roots of a general polynomial of degree $5$ or higher have "no closed-form formula," but the Abel-Ruffini theorem only proves nonexistence of algebraic ...
17
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1answer
254 views

Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes

Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes. I've been thinking about this question, but ...
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1answer
136 views

Mean values theorem and countable sets

The mean values theorem says that there exists a $c∈(u,v)$ such that $$f(v)-f(u)=f′(c)(v-u)$$ My question is: Assume that $u$ is a root of $f$, hence we obtain $$f(v)=f′(c)(v-u)$$ Assume that $f$ is a ...
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1answer
53 views

Root of sum of shifted polynomials

For an arbitrary positive odd integer $k$, I would like to obtain an expression for the root $x_{root} \in \mathbb{R}$ of the following polynomial $$p(x) = \sum_{i=1}^N (x-x_i)^k,$$ where $x_i\in ...
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1answer
340 views

How to solve $x!=5^x$?

Or, more generally, $$\Gamma (x+1)=\int_0^{\infty}t^{x}e^{-t}dt=p^x$$ with $p \in \mathbb{Z}^+$ and $x \in \mathbb{C}$. Perhaps begin with $\large p^x=p^x \lim_{n \rightarrow ...
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1answer
28 views

How to get started on showing the conditions that $ax+by+cz=0$

I am looking at this question from Hardy's book, A Course of Pure Mathematics and have no idea where to begin. I was wondering, what is the first step to deriving the conditions? Question What are ...
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0answers
133 views

Given relations of coefficients and $m$ zeros of a complex polynomial, find the polynomial of degree $2n$ and $m \geq n$.

Given relations of coefficients and $m$ zeros of a complex polynomial (coefficients are complex), find the polynomial of degree $2n$ and $m \geq n$. For examples, we are finding $P(x)=C_{2n}x^{2n} + ...
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1answer
57 views

Is there an analytic solution to the following equation

I have the following general equation in $x$ $$a\cos(b - cx) - d\cos(e - fx) = 0$$ with constants $a,b,c,d,e,f$. Is there an algerbraic solution to this or only a numeric one?
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1answer
525 views

Root Locus, Meaning of the Roots?

I'm studying control theory and I encountered the root locus, I know that It plots the roots of the characteristic equation but i've some questions. What is the physical meaning of the Roots of the ...
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1answer
201 views

Change of argument of $\exp(z)-z$ on each side of a square

Show that as the positive integer $N$ tends to $\infty$, the change in argument of $e^z − z$ is bounded on $3$ sides of the square with corners $ \pm 2\pi N$ $\pm 2\pi iN$ but is unbounded on the ...
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3answers
169 views

Multiplying a square root by a non-square root

This is not something I do very often, so I'm a bit dicey on the rules. I just want to make sure that I understand things right... $$-\frac{1}{2}\cdot \sqrt{\frac{2}{5}} = -\sqrt{\frac{1}{4}}\cdot ...
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2answers
370 views

Proof $ \sqrt{1 + \sqrt[3]{2}} $ is irrational using the theorem about rational roots of a polynomial

I'm having trouble with this specific problem at the moment. The theorem states that if $n/m$ is a rational root of a polynomial with integer coefficients, the leading coefficient is divisible by m ...
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1answer
718 views

Argument principle: number of zeroes of $f(z)=\cos(z)-1 +z^2/2$ in the unit disk

I am trying to work on this old qual exam. Here is the question: Find the number of roots (counting multiplicities) of the function $$f(z)=\cos(z)-1 + \frac{z^2}{2}$$ inside the domain $\vert ...
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3answers
250 views

Common method of calculating zero places of quadratic and linear function.

Very basic stuff from school we know that we can calculate zero places of quadratic function which has form $ax^2 + bx + c$ and we assume that $a \neq 0$, now what if $a=0$? Why can't we use delta to ...
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1answer
99 views

About canonical factors for Weierstass infinite products.

I was reading the proof of below theorem (p.145 complex analysis Elias M.Stein): Given any sequence $\{a_n\}$ of complex numbers whit $|a_n| \to \infty $ as $n \to \infty $, there exists an entire ...
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1answer
227 views

Number of roots of a complex equation/ Rouche's theorem

For $n\geq2$ consider the equation $z^n+z+n=0$ for $z\in \mathbb C$. Show that if $k$ is an integer with $1\leq k \leq n$ then inside the sector $$ S_k=\left\{z\in \mathbb C: 0< Arg(z) < ...
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2answers
70 views

Exponential equation with three summands

I had a simple looking math problem the other day: Solve for $y(x) = 0$: $$ 10^{2x} - 101 \cdot 10^x + 100 = 0$$ Since I have three summands, I cannot just put them to either side of the ...
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1answer
52 views

Pick out the case(s) which ensure that the polynomial $p(\cdot)$ has a root in the interval $[0, 1]$

Please help me to solve the problem below. Consider the polynomial $p(x) = a_0 + a_1x + a_2x^2+\dots+ a_nx^n$, with real coefficients. Pick out the case(s) which ensure that the polynomial ...
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1answer
716 views

Complex Analysis - Argument Principle vs. Rouche's Theorem

The Argument Principle Suppose a function $f$ is meromorphic on an open set that contains a circle $C$ and its interior. Further assume that $f$ has no zeroes on $C$ (but may have zeroes in the ...
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2answers
127 views

Plot of a Bessel function if possible

i would like to know where i could find a plot of $$ J_{ia}(2\pi i)$$ (1) using Quantum mechanics i have conjectured that if $ a= \frac{x}{2} $ and $ i= \sqrt{-1} $ then $$ J_{it}(2\pi ...
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1answer
417 views

Complex Analysis - Location of roots of a polynomial

How many roots does the polynomial $z^4 + 3z^2 + z + 1$ have in the right-half complex plane (i.e. $Re(z) \gt 0$)? I honestly can't think of how to approach the problem as it seems different from the ...
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2answers
124 views

How to show that there exists a root of $f(x)=0$?

Let $f(x)=\sum_{k=0}^n a_k x^k$, where $a_k$'s satisfy $\sum_{k=0}^n \frac{a_k}{k+1}=0$.Show that there exists a root of $f(x)=0$ in the interval $(0,1)$.
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1answer
325 views

Root Finding Algorithm for Discrete Functions

I was recently working with functions of the form $$N - \sqrt{\frac{N}{x}}\cdot\left\lfloor \frac{N}{\sqrt{N/x}}\right\rfloor + \sqrt{\frac{N}{x}} - \left\lfloor \sqrt{\frac{N}{x}}\right\rfloor$$ ...
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0answers
66 views

Zeros of $ \frac{1}{B(xi)^{1/2}}((iA)^{ix})(ix)^{ix}+ \frac{1}{B(-xi)^{1/2}}((-iA)^{-ix})(-ix)^{-ix}=H(x)$

What would be the zeros of the following function? $$ \frac{1}{B(xi)^{1/2}}((iA)^{ix})(ix)^{ix}+ \frac{1}{B(-xi)^{1/2}}((-iA)^{-ix})(-ix)^{-ix}=H(x)$$ This function is real and I believe it is equal ...
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1answer
267 views

Multiple roots of a polynomial in two variables

Let $F\in\mathbb{C}[X,Y]$ be an irreducible polynomial and $n\in \mathbb{N}$, $n\ge1$, $p_i\in\mathbb{C}[X]$ for $0\le i\le n$, such that $$F(X,Y)=\sum\limits_{i=0}^{n}p_i(X)Y^{n-i}.$$ Let ...
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4answers
765 views

Find all roots of $\,(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$

The question is to find all complex roots of $$(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$$ and it is meant to be solved by hand. Is there any quick way to solve this using some trick that I'm not ...
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0answers
286 views

Using the duplex method to calculate square roots

I have been assigned to find out how a calculator figures out square roots, so far the shortest thing I can see is "the duplex method". But the thing is that the explanation on Wikipedia makes no ...
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2answers
531 views

Zeros of Fourier transform of a function in $C[-1,1]$

I am trying to prove the following: Let $g \in C[-1,1]$. Then the function $$G(z) = \int_{-1}^1 e^{itz}g(t)dt$$ has infinitely many zeros. I know that $G(z)$ is entire and $\lim_{x \to \pm ...
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0answers
266 views

Computing square roots modulo prime powers

I am trying to implement an algorithm that can compute the square root of a quadratic residue mod a prime power. For integers $a$ such that $p\not\mid a$ $p\neq 2$ it's relatively straightforward ...
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3answers
1k views

Proving square root of a square is the same as absolute value

Lets say I have a function defined as $f(x) = \sqrt {x^2}$. Common knowledge of square roots tells you to simplify to $f(x) = x$ (we'll call that $g(x)$) which may be the same problem, but it isn't ...
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1answer
86 views

Most effective way to solve system of non-linear equations with unique set of roots

What is the most effective way to solve a system of non-linear equations if we know for sure that they have a unique set of roots?
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2answers
90 views

Find $\lim_{x \to \alpha}[1+ax^2+bx+c]^\frac{1}{x-\alpha}$

If $\alpha , \beta$ be the roots of $ax^2+bx+c=0$. Find $$\lim_{x \to \alpha}[1+ax^2+bx+c]^\frac{1}{x-\alpha}$$ Here $\alpha +\beta=-\frac{b}{a}$ and $\alpha \beta=\frac{c}{a}$. How can I ...
0
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0answers
83 views

Isolation of zeros in the case of univariate analytic functions expressed as a bivariate function.

We know that the zeros of an analytic non-constant function are always isolated. A proof is here. Let $L(v)$ be an analytic function in $v$, where $v\in\mathbb{R}$. Let us write $L(v) \equiv L(v,p)$ ...