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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144
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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 ...
75
votes
5answers
3k 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 $\mathbb C$,...
69
votes
4answers
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Are these solutions of $2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$ correct?

Find $x$ in $$ \Large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$$ A trick to solve this is to see that $$\large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}} \quad\implies\quad 2 = x^{\Big(x^{x^{x^...
61
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5answers
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Polynomials such that roots=coefficients

Here is my question : Are there monic polynomials with degree $\geq 5$ such that they have the same real all non zero roots and coefficients ? Mathematically, prove or disprove the existence ...
59
votes
6answers
5k 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 ...
44
votes
4answers
1k 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 ...
41
votes
2answers
1k views

Only 12 polynomials exist with given properties

Prove that there are only 12 polynomials that have all real roots, and whose coefficients are all $-1$ or $1$. (zero coefficients are not allowed, and constant polynomials do not ...
35
votes
2answers
1k views

Why does this distribution of polynomial roots resemble a collection of affine IFS fractals?

Consider the following spectacular image, created by Sam Derbyshire and described in John Baez's article "The Beauty of Roots": In this image are plotted all the complex roots of all polynomials of ...
33
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4answers
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$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x)$, 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 ...
30
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4answers
576 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, \begin{align*...
30
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1answer
640 views

Distribution of roots of complex polynomials

I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$ uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000 of these polynomials are ...
27
votes
6answers
2k views

Function with no roots

Given a non-constant function $f(x)$, is it possible for it to have no zeroes (neither real nor complex)? Say for example, $f(x)=\cos x-2$, does a complex solution exist for this because for real $x$,...
27
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4answers
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Is it possible for a quadratic equation to have one rational root and one irrational root?

Is it possible for a quadratic equation to have one rational root and one irrational root? Yes, a pretty straightforward question. Is it possible?
27
votes
10answers
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Is There A Polynomial That Has Infinitely Many Roots?

Is there a polynomial function $P(x)$ with real coefficients that has an infinite number of roots? What about if $P(x)$ is the null polynomial, $P(x)=0$ for all x?
27
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3answers
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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 ...
27
votes
3answers
979 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 ...
27
votes
2answers
712 views

Something strange about $\sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$ and its friends

This post can be generalized to, $$\begin{align} \sqrt{ 2+ \sqrt{ 2 + \sqrt{ 2-x}}}=x&,\quad\quad x = -2\cos\left(\frac{8\pi}{9}\right)=1.8793\dots\quad\quad\quad \\ \\ \sqrt{ 4+ \sqrt{ 4 + \sqrt{...
26
votes
2answers
907 views

Countability of the zero set of a real polynomial

This is the question from my calculus homework: Is it possible for a polynomial $f\colon\, \mathbb{R}^{n}\to \mathbb{R}$ to have a countable zero-set $f^{-1}(\{0\})$? (By countable I mean countably ...
22
votes
2answers
607 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 ...
21
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1answer
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Showing that the roots of a polynomial with descending positive coefficients lie in the unit disc.

Let $P(z)=a_0+a_1z+\cdots+a_nz^n$ 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 ...
20
votes
2answers
869 views

Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$

Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f(x)=0$ on $\mathbb{R}$\ $(-1,1)$. Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$ My attempt : ...
20
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3answers
533 views

Probability of $ax^2 + bx + c = 0$ having real solutions

$a$, $b$, $c$ are random integer numbers between $1$ and $100$ (including $1$ and $100$, and uniformly distributed). What is the probability that the equation $ax^2 + bx + c = 0$ has real solutions? ...
19
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6answers
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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 ...
18
votes
7answers
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Find all five solutions of the equation $z^5+z^4+z^3+z^2+z+1 = 0$

$z^5+z^4+z^3+z^2+z+1 = 0$ I can't figure this out can someone offer any suggestions? Factoring it into $(z+1)(z^4+z^2+1)$ didn't do anything but show -1 is one solution. I solved for all roots of $...
18
votes
4answers
846 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 ...
18
votes
5answers
7k 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 $1$)? ...
17
votes
3answers
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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 \...
17
votes
1answer
735 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 ...
17
votes
3answers
442 views

The function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$.

Show that if $f\in \mathcal{C}^3$ and $2\cdot\pi$ periodic then the function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$. My attempt : f is $2\pi$ periodic and $\mathcal{C}^3$, we have : $$\lim_{...
17
votes
1answer
264 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 I'...
16
votes
3answers
410 views

How can I prove $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}…}}}}=2$ [duplicate]

How can I prove $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}}}=2$$ I don't know which method can be used for this?
16
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2answers
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Adriaan van Roomen's 45th degree equation in 1593

Adriaan van Roomen proposed a 45th degree equation in 1593(see this book, picture reference as follows): $$ \begin{gathered} f(x) = x^{45} - 45x^{43} + 945x^{41} - 12300x^{39} + 111150x^{37} - \color{...
15
votes
2answers
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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 ...
15
votes
3answers
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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: $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ $2(\...
15
votes
1answer
339 views

Can we prove that the solutions of $\int_0^y \sin(\sin(x)) dx =1$ are irrational?

Can we prove that the solutions of $$\int_0^y \sin(\sin(x)) dx =1$$ are irrational? Wolfram Alpha gives two approximate sets of solutions as $\{4.58+2\pi k|k\in\mathbb{Z}\}$ and $\{1.69+2\pi k|k\in\...
15
votes
1answer
268 views

If $\gamma\in \Bbb A$ then there exists a $\pm$quadratic coefficiented polynomial for which $\gamma$ is a root.

$\mathbf{1.\space Proposition}$ Let $\gamma$ be a solution to the equation: $$ \sum_{i=0}^n \rm a_i\rm x^i=0, \rm a_i\in\Bbb Z, a_n=1. $$ Then, there exists a polynomial $p\in \Bbb Z[x]$ ...
14
votes
2answers
842 views

Can we introduce new operations that make quintics solvable?

I have heard from various sources that the typical arithmetic operations (addition, subtraction, multiplication, division, rational exponentiation) are not sufficient to express in general the roots ...
14
votes
2answers
874 views

Is Vieta the only way out?

Let $a,b,c$ are the three roots of the equation $x^3-x-1=0$. Then find the equation whose roots are $\frac{1+a}{1-a}$,$\frac{1+b}{1-b}$,$\frac{1+c}{1-c}$. The only solution I could think of is by ...
14
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1answer
3k 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, ...
14
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1answer
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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 +...
14
votes
4answers
315 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 ...
14
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5answers
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Prove $\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$

If $a,b,c$ are non-negative numbers and $a+b+c=3$, prove that: $$\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$$ Here's what I've tried: Using Cauchy-Schawrz I proved that: $$(3a + b^...
13
votes
4answers
1k views

How to show that a polynomial does not have real roots

How to show generally that a polynomial does not have real roots. Well, for eg lets take the polynomial $x^8-x^7+x^2-x+15$ . Here the power($n=8$) is even so it can have real roots or it might not ...
13
votes
3answers
642 views

How would you find the exact roots of this equation?

My friend asked me what the roots of $y=x^3+x^2-2x-1$ was. I didn't really know and when I graphed it, it had no integer solutions. So I asked him what the answer was, and he said that the $3$ roots ...
13
votes
1answer
255 views

Fitting a closed curve on the roots of ${x \choose k}-c$

Let $${n \choose k} = \frac1{(n+1) \operatorname{B}(n-k+1, k+1)}.$$ be the generalized binomial coefficient, and here $\operatorname{B}$ is the Beta function. Let $f_{k,c}(x)$ be the following ...
13
votes
2answers
566 views

Number of real positive roots of a polynomial?

Consider the polynomial $$f(x)=x((1+x^n)^n+a^n)-a(1+x^n)^n,$$ where $n\geq 2$ is a positive integer and $a$ is a positive real number. I'm interesting in deducing the number of positive real roots ...
13
votes
2answers
393 views

Does this polynomial have all its roots both distinct and real?

Recently, I wondered about the following problem: let $n\geq 5$ and let $$ P_n(x)=(x-1)(x-2)\ldots (x-n)-1 $$ Is it true that $P_n(x)$ has $n$ distinct real roots for any $n\geq 5$ ? I checked it ...
13
votes
2answers
387 views

Regularity of root spacing of $G(z)=\sum_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$

Define, on $\mathbb{C}$: $$G(z)=\sum_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$$ A domain colored portrait of $G(z)$ (boxes are supposed to be negative signs): suggests that the roots of $G(z)$ are ...
13
votes
1answer
137 views

Why is this approximation of polynomial root so accurate?

I have an engineering problem where I have to find the smallest positive real root of a polynomial in $x$: $$Ax^5+Bx^3 - C = 0$$ Instead of solving numerically, I want simple approximative formulas ("...
13
votes
3answers
170 views

$\sum_i x_i^n = 0$ for all $n$ implies $x_i = 0$

Here is a statement that seems prima facie obvious, but when I try to prove it, I am lost. Let $x_1 , x_2 \dots x_k$ be complex numbers satisfying: $$x_1 + x_2 \dots + x_k = 0$$ $$x_1^2 + x_2^2 \...