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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6
votes
3answers
466 views

Square roots — positive and negative

It is perhaps a bit embarrassing that while doing higher-level math, I have forgot some more fundamental concepts. I would like to ask whether the square root of a number includes both the positive ...
44
votes
5answers
1k views

Why are the solutions of polynomial equations so unconstrained over the quaternions?

An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb ...
11
votes
1answer
2k views

Derivation of asymptotic solution of $\tan(x) = x$.

An equation that seems to come up everywhere is the transcendental $\tan(x) = x$. Normally when it comes up you content yourself with a numerical solution usually using Newton's method. However, ...
48
votes
4answers
3k views

Are the solutions of $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$ correct?

Problem: Find $x$ in $$\large x^{x^{x^{x^{ \cdot^{{\cdot}^{\cdot}} }}}}=2$$ Trick: $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$, so, $x^{(x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}})}=x^2=2$, and, ...
15
votes
1answer
741 views

Showing that the roots of a polynomial with descending positive coefficients lie in the unit disc.

Let $P(z)=a_nz^n+\cdots+a_0$ be a polynomial whose coefficients satisfy $$0<a_0<a_1<\cdots<a_n.$$ I want to show that the roots of $P$ live in unit disc. The obvious idea is to use ...
15
votes
3answers
2k views

Why is it so hard to find the roots of polynomial equations?

The question that follows was inspired by this question: When trying to solve for the roots of a polynomial equation, the quadratic formula is much more simple than the cubic formula and the cubic ...
35
votes
5answers
2k views

Continuity of the roots of a polynomial in terms of its coefficients

It's commonly stated that the roots of a polynomial are a continuous function of the coefficients. How is this statement formalized? I would assume it's by restricting to polynomials of a fixed ...
5
votes
1answer
803 views

Why not write the solutions of a cubic this way?

For the solution of the cubic equation $x^3 + px + q = 0$ Cardano wrote it as: $$\sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}+\sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^2}{4} + ...
5
votes
2answers
350 views

Polynomial $p(a) = 1$, why does it have at most 2 integer roots?

The question that I am trying to answer is : Suppose is $p(x)$ is a polynomial with integer coefficients. Show that if $p(a) = 1$ for some integer a then $p(x)$ has at most two integer roots. I have ...
23
votes
3answers
736 views

Galois groups of polynomials and explicit equations for the roots

Lets say I have calculated the galois group of some polynomial and I also have the subgroup structure. What's an effective procedure to turn the group into equations for the actual roots of the ...
8
votes
0answers
132 views

Let $x_n$ be the (unique) root of $\Delta f_n(x)=0$. Then $\Delta x_n\to 1$

Note that by Cesaro's Theorem, we have as a consequence $$\frac{x_n}n\to 1$$ Consider $$r_n(x)=e^{-x}-\sum_{k=0}^n (-1)^k\frac{x^k}{k!}$$ and $$f_n(x)=(-1)^{n+1}e^{-x}r_n(x)$$ One can argue by ...
3
votes
6answers
325 views

Solving $\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$

Where do I start to solve a equation for x like the one below? $$\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$$ After squaring it, it's too complicated; but there's nothing to factor or to ...
5
votes
2answers
379 views

Roots of the incomplete gamma function

Is there any way that one can describe all the roots of the incomplete gamma function $\Gamma(n,z)$, for $n\in \mathbb{N}$, analytically?
3
votes
1answer
1k views

Find all real zeros of $f(x)=2x^3+10x^2+5x-12$

Hey guys I'm having a little trouble with one problem: Find all real zeros of $$f(x)=2x^3+10x^2+5x-12.$$ I got $x=-4,(2x^2+2x-3)$. I'm just having trouble using the quadratic formula to get ...
25
votes
3answers
350 views

Patterns of the zeros of the Faulhaber polynomials (modified)

Faulhaber polynomial of order $p \in \Bbb{N}$ is defined as the unique polynomial of degree $p+1$ satisfying $$ S_{p}(n) = \sum_{k=1}^{n} k^p $$ for $n = 1, 2, 3, \cdots$. For example, ...
6
votes
2answers
666 views

How was Euler able to create an infinite product for sinc by using its roots?

In the Wikipedia page for the Basel problem, it says that Euler, in his proof, found that $$\begin{align*} \frac{\sin(x)}{x} &= \left(1 - \frac{x}{\pi}\right)\left(1 + \frac{x}{\pi}\right)\left(1 ...
16
votes
5answers
4k views

Roots of Legendre Polynomial

I was wondering if the following properties of the Legendre polynomials are true in general. They hold for the first ten or fifteen polynomials. Are the roots always simple (i.e., multiplicity $1$)? ...
15
votes
4answers
550 views

Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$

Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$. Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ Here's my idea: $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ ...
4
votes
6answers
3k 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 ...
1
vote
1answer
313 views

Multiple choice question - number of real roots of $x^6 − 5x^4 + 16x^2 − 72x + 9$

The equation $x^6 − 5x^4 + 16x^2 − 72x + 9 = 0$ has (A) exactly two distinct real roots (B) exactly three distinct real roots (C) exactly four distinct real roots (D) six distinct real roots
10
votes
6answers
625 views

Complex roots of $z^3 + \bar{z} = 0$

I'm trying to find the complex roots of $z^3 + \bar{z} = 0$ using De Moivre. Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know ...
7
votes
4answers
323 views

How prove this $\displaystyle\lim_{n\to \infty}\frac{n}{\ln{(\ln{n}})}\left(1-a_{n}-\frac{n}{\ln{n}}\right)=-1$

let equation $x^n+x=1$ have positive root $a_{n}$. show that $$\displaystyle\lim_{n\to \infty}\dfrac{n}{\ln{(\ln{n}})}\left(1-a_{n}-\dfrac{n}{\ln{n}}\right)=-1$$ some hours ago,it prove that ...
3
votes
5answers
382 views

How to find the number of real roots of the given equation?

The number of real roots of the equation $$2 \cos \left( \frac{x^2+x}{6} \right)=2^x+2^{-x}$$ is (A) $0$, (B) $1$, (C) $2$, (D) in finitely many. Trial: $$\begin{align} 2 \cos \left( ...
6
votes
1answer
185 views

Location of zeros of a sum of exponentials

Describe the approximate locations of the zeros of the function $$ f(z) = e^{iz}+e^{-iz}+e^z $$ lying outside the circle $|z|=R >>1$. Another prelim problem. For Rouche's theorem we need to ...
3
votes
2answers
452 views

Proof the Legendre polynomial $P_n$ has $n$ distinct real zeros

I need a proof to show that the inequality $m < n$ leads to a contradiction and $P_n$ has $n$ distinct real roots, all of which lie in the open interval $(-1, 1)$.
3
votes
1answer
112 views

$n \approxeq k + 2^{2^k}(k+1)$. How can one get the value of $k(n)$ from this equation?

I am trying to find approximation for this sum. Asymptotic approximation of sum $\sum_{k=0}^{n}\frac{{n\choose k}}{2^{2^k}}$ Doing following way. Let $a_k(n) = \frac{n\choose k}{2^{2^k}}$. I tried to ...
3
votes
1answer
305 views

How do iterative methods applied to the companion matrix of a polynomial $p(\lambda)$ relate to $p$ itself?

A few days ago, I had a vague question in my mind about "matrix methods" for finding roots of a polynomial. Now I can ask at least a semi-precise question, thanks to the post How to calculate complex ...
1
vote
3answers
293 views

If $(2x^2-3x+1)(2x^2+5x+1)=9x^2$,then prove that the equation has real roots.

If $(2x^2-3x+1)(2x^2+5x+1)=9x^2$,then prove that the equation has real roots. MY attempt: We can open and get a bi quadratic but that is two difficult to show that it has real roots.THere must be an ...
22
votes
2answers
483 views

How to show that a root of the equation $x (x+1)(x+2) … (x+2009) = c $ can have multiplicity at most 2?

How to show that a root of the equation $$x (x+1)(x+2) ....... (x+2009) = c $$ can have multiplicity at most 2 , and to find the value of $ c $ for which this is possible. I proceeded by using the ...
33
votes
4answers
862 views

AM-GM-HM Triplets

I want to understand what values can be simultaneously attained as the arithmetic (AM), geometric (GM), and harmonic (HM) means of finite sequences of positive real numbers. Precisely, for what points ...
6
votes
3answers
616 views

What we can say about $-\sqrt{2}^{-\sqrt{2}^{-\sqrt{2}^\ldots}}$?

Problem: How we can strictly prove $-\sqrt{2}^{-\sqrt{2}^{-\sqrt{2}^\ldots}}$ can't be 2? Can $-\sqrt{2}^{-\sqrt{2}^{-\sqrt{2}^\ldots}}$ have the value expressed by complex numbers? (See below, in ...
2
votes
1answer
76 views

Game of polynomials

Written on a blackboard is the polynomial $x^2+x+2014$.Calvin and Peter take turns alternatively (starting with Calvin) in the following game. During his turn, Calvin should either increase or ...
9
votes
2answers
459 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 ...
17
votes
6answers
2k views

Can $x^3+3x^2+1=0$ be solved using high school methods?

I encountered the following problem in a high-school math text, which I wasn't able to solve using factorization/factor theorem: Solve $x^3+3x^2+1=0$ Am I missing something here, or is indeed a more ...
16
votes
3answers
1k views

On applying the quadratic formula to a first-degree equation

You're probably thinking, "Why?" Please let me explain... It is (very) well-known that $$ \forall (a,b,c,x) \in \mathbb{C}^* \times \mathbb{C}^3: ax^2 + bx + c = 0 \Leftrightarrow x = \frac{-b \pm ...
11
votes
2answers
488 views

Number of roots of $x^a-1=0$ with $a \in \mathbb{C}$

It is well known that $x^2-1=0$ has two roots in $\mathbb{C}$, namely $\pm 1$. In general $x^n-1=0$ has exactly $n$ roots in $\mathbb{C}$. But what happens when $n$ is non integer (rational or real or ...
4
votes
4answers
889 views

Finding all complex zeros of a high-degree polynomial

Given a large univariate polynomial, say of degree 200 or more, is there a procedural way of finding all the complex roots? By "roots", I mean complex decimal approximations to the roots, though the ...
11
votes
1answer
206 views

Solving $x^2+bx^{1+\varepsilon}+c =0$

Let $x \in \mathbb{R}$. Is it possible to find the roots of $x^2+bx^{1+\varepsilon}+c =0$ where $b,c \in \mathbb{R}$ and $\varepsilon$ is small. I am guessing that an explicit expression might not be ...
5
votes
1answer
425 views

Finding all roots of polynomial system (numerically)

I want to numerically find all the roots of a system of polynomials (n equations in n variables). Since I can compute the Jacobian for the system (analytically or otherwise), I can use the Newton ...
4
votes
2answers
188 views

Relation between root systems and representations of complex semisimple Lie algebras

I'm trying to understand the machinery of root systems for the purpose of classifying complex semisimple Lie algebras. During this process i lost the overview, espacially when it came to highest ...
1
vote
3answers
611 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 ...
10
votes
1answer
2k views

Using Vieta's theorem for cubic equations to derive the cubic discriminant

Background: Vieta's Theorem for cubic equations says that if a cubic equation $x^3 + px^2 + qx + r = 0$ has three different roots $x_1, x_2, x_3$, then $$\begin{eqnarray*} -p &=& x_1 + x_2 ...
9
votes
2answers
543 views

Complex Zeros of $z^2e^z-z$

Can anyone give me a hint on showing (in a relatively elegant way, as I know the answer from WolframAlpha), that the complex valued function $z^2e^z-z$ has at most 2 roots with norm less than 2? ...
7
votes
2answers
446 views

Approximating roots of the truncated Taylor series of $\exp$ by values of the Lambert W function

To everyone: don't bother writing up another answer, i'm giving this bounty Antonio's answer. It just doesn't let me yet (24 hours delay). If you map the nth roots of unity $z$ with the function ...
6
votes
1answer
295 views

Why are primitive roots of unity the only solution to these equations?

I was led by this question to the following problem: Find $n$ complex numbers $\lambda_1\dots\lambda_n\in\mathbb{C}$ that satisfy $$\begin{align} \sum_i\lambda_i & =0\\ \sum_i\lambda_i^2 ...
5
votes
1answer
136 views

Zeroes of a Particular Function

I'm looking for the zeroes of $f(k) = e^{\sqrt{k}}[\frac{s}{k} - \frac{d}{\sqrt{k}}] - 1$ on the set $k > 0$. Is there a nice way to describe the set of solutions for given $s$ and $d$? Thanks!
4
votes
1answer
560 views

$\lambda-z-e^{-z}=0$ has one solution in the right half plane

Let $\lambda > 1$ , want to show that the equation $$\lambda-z-e^{-z}=0$$ has exactly one solution in the right half plane $\{z:Re(z)>0\}$. Moreover, the solution must be real.I tried to use ...
4
votes
3answers
894 views

How to tell if a quartic equation has a multiple root.

Is there any way to tell whether a quartic equation has double or triple root(s)? $$x^4 + a x^3 + b x^2 + c x + d = 0$$
4
votes
3answers
293 views

Is there an algorithm to find the roots of high-order polynomials?

It is not generally possible to determine the roots of a polynomial whose grade is bigger than 4 in terms of roots and basic operations. But I heard, that it is possible to give a criteria whether a ...
3
votes
2answers
111 views

Root of the function $f(x)=xe^x-R$

How can we find the root of the function $f(x)=xe^x - R$ for a general R where $R>=-1/e.$ I don't have any idea as to how to even approach this. Came across this problem during my self-study in ...