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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133
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1answer
4k 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 ...
54
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, ...
49
votes
5answers
2k 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 ...
45
votes
5answers
1k views

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 ...
39
votes
5answers
3k 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 ...
39
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 ...
31
votes
4answers
919 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 ...
28
votes
2answers
781 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 ...
26
votes
3answers
405 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, ...
26
votes
1answer
494 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 ...
25
votes
4answers
4k views

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?
25
votes
3answers
827 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 ...
25
votes
2answers
704 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
534 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 ...
18
votes
6answers
3k 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 ...
18
votes
4answers
783 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
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 ...
18
votes
2answers
690 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 : ...
17
votes
3answers
2k 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 ...
17
votes
3answers
385 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 : ...
17
votes
1answer
256 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 ...
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 $1$)? ...
16
votes
2answers
646 views

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} - ...
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 ...
15
votes
3answers
638 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: $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ ...
15
votes
1answer
1k 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 ...
14
votes
2answers
366 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 ...
14
votes
1answer
307 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 ...
13
votes
2answers
738 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 ...
13
votes
1answer
206 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
1answer
152 views

Prove $\sum_{i=0}^n \binom{n}{i}^2x^{n-i} = 0$ has $n$ negative roots

Let's $n \in \mathbb{Z^+}$, how to $\text{prove}|\text{disprove}$ that: the equation $\boxed{\sum_{i=0}^n \binom{n}{i}^2x^{n-i} = 0}$ has exactly $n$ distinct negative roots. My friend get ...
13
votes
3answers
146 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 ...
12
votes
3answers
193 views

How do we solve $a \le b^{r}-r$ for $r$?

Given two values $a$ and $b$, how should one go about solving the following inequality for $r$: $$a \le b^r -r .$$ Applying $\log_b$ on both sides of the inequality doesn't help me much since that ...
12
votes
2answers
439 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 ...
12
votes
1answer
3k 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 ...
12
votes
2answers
91 views

Function such that zeros$=$order of the derivative

Does there exist a function $f\in C^n(\mathbb{R},\mathbb{R})$ for $n\ge2$ such that $f^{(n)}$ has exactly $n$ zeros, $f^{(n-1)}$ has exactly $n-1$ zeros and so on ? Where $f^{(n)}$ is the nth ...
12
votes
2answers
125 views

Fully factored integer polynomials with constant differences

Given a degree $d$, it is possible to construct a pair $(F,\delta),$ where $F$ is a polynomial in $\mathbb{Z}[X]$ and $\delta$ a non-zero integer, such that $F(X)$ and $F(X)+\delta$ both split into ...
12
votes
1answer
205 views

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

We have the nice radical identity involving $d = 163$, $$-\sqrt{ 44- \sqrt{ 44 - \sqrt{ 44-x}}}=x,\quad\quad x = 2-2\sum_{n=1}^{27}\cos\left(\frac{2\pi\, t_1(n)}{163}\right)=-6.15824\dots$$ where ...
12
votes
3answers
173 views

Why is this polynomial a monomial?

Let $p$ be a polynomial of degree $n$ such that $|p(z)| = 1$ for all $|z| = 1$. Why is it that $p(z) = az^n$ for some $|a| = 1$? I've noticed that we could easily prove this by induction if we ...
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, ...
11
votes
6answers
732 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 ...
11
votes
3answers
1k views

How to solve polynomial equations in a field and/or in a ring?

I'm studying for my exam, and I stuck on solving polynomials in a field and/or in a ring. Let me give you some examples: (1) Solve equation $x^2+4x+3=0$ in field $\mathbb{Z}_5$, $\mathbb{Z}_8$ and in ...
11
votes
2answers
137 views

Solve $x^7-5x^4-x^3+4x+1=0$ for $x$

Solve for $x$ $$x^7-5x^4-x^3+4x+1=0$$ This equation has been bugging me since the past few days. I have found, using the Rational Root Theorem that $x=1$ is a root of this equation. However, ...
11
votes
2answers
531 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 ...
11
votes
1answer
210 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 ...
11
votes
1answer
600 views

Can the real and imaginary parts of $\dfrac{\sin z}z$ be simplified?

I have calculated the real and imaginary parts of $\dfrac{\sin z}z.$ I've obtained $$\begin{eqnarray} \frac{\sin z}z&=&\frac{\sin(x+iy)}{(x+iy)}\\ &=& ...
11
votes
4answers
232 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 ...
10
votes
7answers
4k views

How to solve an exponential equation with two different bases: $3^x - 2^x = 5$

Can anyone tell me how to solve this equation $$3^x - 2^x = 5$$ other than graphically? I'm stunned. I don't know what to do in the first step.
10
votes
4answers
389 views

sum of reciprocals of derivative of polynomial at its roots

If $P(x)$ is a polynomial of degree $n > 1$ with only simple roots $a_1,\ldots,a_n$, is it true that $\frac 1{P'(a_1)} + \cdots + \frac 1{P'(a_n)} = 0$, and, if so, what is the proof? I ...
10
votes
5answers
425 views

Finding real roots of $ P(x)=x^8 - x^7 +x^2 -x +15$

Let $ P(x)=x^8 - x^7 +x^2 -x +15 $, Descartes' Rule of Signs tells us that the polynomial has 4 positive real roots , but if we group the terms as $$ P(x)= x(x-1)(x^6+1) +15 $$ we find that $ P(x) $ ...