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
488 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 ...
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, ...
45
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 ...
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
978 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 ...
36
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 ...
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 ...
4
votes
2answers
678 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 $(-1, 1)$.
5
votes
1answer
833 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
424 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 ...
24
votes
3answers
768 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
143 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
430 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
406 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?
6
votes
1answer
214 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
1answer
2k 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 ...
2
votes
6answers
5k views

How to find the roots of $x^4 +1$

I'm trying to find the roots of $x^4+1$. I've already found in this site solutions for polynomials like this $x^n+a$, where $a$ is a negative term. I don't remember how to solve an equation when $a$ ...
38
votes
4answers
939 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 ...
25
votes
3answers
367 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, ...
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
590 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: $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ ...
6
votes
2answers
765 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 ...
7
votes
4answers
332 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
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 ...
3
votes
1answer
377 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
649 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 ...
3
votes
5answers
407 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( ...
2
votes
1answer
79 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) = ...
3
votes
1answer
120 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
323 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
1answer
65 views

Minimum Modulus Principle for a constant fuction in a simple closed curve

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is ...
1
vote
3answers
349 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 ...
119
votes
1answer
3k views

Rational roots of polynomials

Can one construct a sequence $(a_k)_{k\geqslant 0}$ of rational numbers such that, for every positive integer $n$ the polynomial $a_nX^n+a_{n-1}X^{n-1}+\cdots +a_0$ has exactly $n$ distinct rational ...
30
votes
4answers
854 views

$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x)$, then all the roots of $p_k(x)$ are real

$p_0(x)=a_mx^m+a_{m-1}x^{m-1}+\dotsb+a_1x+a_0(a_m,\dotsc,a_1,a_0\in\Bbb R)$ is a polynomial, and $$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x),\qquad n=1,2,\dotsc$$ then, there exist $N\in\Bbb N$, such ...
22
votes
2answers
509 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 ...
6
votes
3answers
650 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 ...
15
votes
2answers
2k views

Why does the discriminant of a cubic polynomial being less than 0 indicate complex roots?

The discriminant $\Delta = 18abcd - 4b^3d + b^2 c^2 - 4ac^3 - 27a^2d^2$ of the cubic polynomial $ax^3 + bx^2 + cx+ d$ indicates not only if there are repeated roots when $\Delta$ vanishes, but also ...
2
votes
1answer
94 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 ...
10
votes
4answers
213 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 ...
9
votes
2answers
517 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 ...
8
votes
2answers
415 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
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
506 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 ...
5
votes
1answer
193 views

Limits of the solutions to $x\sin x = 1$

Let $x_n$ be the sequence of increasing solutions to $x\sin{x} = 1$. Define $$a = \lim_{n \to \infty} n(x_{2n+1} - 2\pi n) $$ and $$b = \lim_{n \to \infty} n^3 \left( x_{2n+1} - 2\pi n - \frac{a}{n} ...
4
votes
4answers
927 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
208 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
465 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
249 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
2answers
77 views

Most Efficient Method to Find Roots of Polynomial [duplicate]

I am designing a software that has to find the roots of polynomials. I have to write this software from scratch as opposed to using an already existing library due to company instructions. I currently ...