# Tagged Questions

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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### 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 ...
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### 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 ...
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### 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 $\mathbb C$,...
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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 ...
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### Irreducibility of a polynomial if it has no root (Capelli) [duplicate]

Let $F$ be a field of arbitrary characteristic, $a\in F$, and $p$ a prime number. Show that $$f(X)=X^p-a$$ is irreducible in $F[X]$ if it has no root in $F$. This answer to a related question ...
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### The trigonometric solution to the solvable DeMoivre quintic?

Using the relations for the Rogers-Ramanujan cfrac described in this post, $$\frac{1}{r}-r = x$$ $$\frac{1}{r^5}-r^5 = y$$ and eliminating $r$ yields, $$x^5+5x^3+5x = y$$ This is the case $a=1$ ...
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### 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 $(-1, 1)$.
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### 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 ...
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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: $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ $2(\... 6answers 1k views ### Complex roots of$z^3 + \bar{z} = 0$I'm trying to find the complex roots of$z^3 + \bar{z} = 0$using De Moivre. Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know ... 1answer 1k views ### Why not write the solutions of a cubic this way? For the solution of the cubic equation$x^3 + px + q = 0$Cardano wrote it as: $$\sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}+\sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^2}{4} + \frac{p^3}... 0answers 168 views ### Let x_n be the (unique) root of \Delta f_n(x)=0. Then \Delta x_n\to 1 Note that by Cesaro's Theorem, we have as a consequence$$\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)$$One can argue by r_n(... 1answer 2k views ### \lambda-z-e^{-z}=0 has one solution in the right half plane Let \lambda > 1 , want to show that the equation$$\lambda-z-e^{-z}=0$$has exactly one solution in the right half plane \{z:Re(z)>0\}. Moreover, the solution must be real.I tried to use ... 6answers 725 views ### Solving \sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2} Where do I start to solve a equation for x like the one below?$$\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$$After squaring it, it's too complicated; but there's nothing to factor or to expand?... 5answers 610 views ### Finding real roots of P(x)=x^8 - x^7 +x^2 -x +15 Let P(x)=x^8 - x^7 +x^2 -x +15 , Descartes' Rule of Signs tells us that the polynomial has 4 positive real roots , but if we group the terms as$$ P(x)= x(x-1)(x^6+1) +15 $$we find that P(x) ... 1answer 77 views ### Find real roots of the equation Find all real solutions to$$\dfrac{\sqrt{x+1}}{2+\sqrt{2-x}} - \dfrac{\sqrt{x^2-x+2}}{2+\sqrt{-x^2+x+1}} = x^3-x^2-x+1$$This question is very similar to one of my previous problem, ... 1answer 103 views ### How to solve in radicals this family of equations for any degree k? Part I. Given any constant a,b, the equation in x,$$\left(\frac{x+\sqrt{x^2+4a}}{2}\right)^{k}+\left(\frac{x-\sqrt{x^2+4a}}{2}\right)^{k}=b\tag1$$is solvable in radicals for any degree k. ... 1answer 292 views ### Location of zeros of a sum of exponentials Describe the approximate locations of the zeros of the function$$ f(z) = e^{iz}+e^{-iz}+e^z $$lying outside the circle |z|=R >>1. Another prelim problem. For Rouche's theorem we need to ... 3answers 2k views ### How to tell if a quartic equation has a multiple root. Is there any way to tell whether a quartic equation has double or triple root(s)?$$x^4 + a x^3 + b x^2 + c x + d = 0$$5answers 207 views ### Real Roots and Differentiation Prove that the equation x^5 âˆ’ 1102x^4 âˆ’ 2015x = 0 has at least three real roots. so do i sub in values of negative and positive values of x to show that there are at least three real roots? the ... 1answer 211 views ### zeros of a polynomial Given P(z)=z^6+6z+10, find how many roots are in each quadrant I have already seen that P(z) has six different roots, and that none of them are real or of the form ki, k\in \Bbb R. Since ... 3answers 916 views ### complex zeros of the polynomials \sum_{k=0}^{n} z^k/k! inside balls this is a question from a Temple prelim exam, and i'm trapped in it! We have p_n(z)=\sum_{k=0}^n\frac{z^k}{k!} and we have to prove that \forall r>0 \quad \exists N\in\mathbb{N} s.t. p_n(z) ... 1answer 177 views ### n \approxeq k + 2^{2^k}(k+1). How can one get the value of k(n) from this equation? I am trying to find approximation for this sum. Asymptotic approximation of sum \sum_{k=0}^{n}\frac{{n\choose k}}{2^{2^k}} Doing following way. Let a_k(n) = \frac{n\choose k}{2^{2^k}}. I tried to ... 5answers 17k views ### How to find the roots of x^4 +1 I'm trying to find the roots of x^4+1. I've already found in this site solutions for polynomials like this x^n+a, where a is a negative term. I don't remember how to solve an equation when a ... 1answer 81 views ### How to solve cubic equations with given coefficients? I have a large data set that requires a cubic equation to be solved for each point. There are too many points to use Goal Seek (numerical Excel method) on them all. For example:$$y=7\cdot 10^{-7}x^... 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 ... 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 ... 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$n = 1, 2, 3, \cdots. For example, \begin{align*... 3answers 410 views ### How can I prove\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}…}}}}=2$[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? 9answers 15k 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 ... 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 ... 3answers 399 views ### All roots of the quartic equation$a x^4 + b x^3 + x^2 + x + 1 = 0$cannot be real Problem Prove that all roots of$a x^4 + b x^3 + x^2 + x + 1 = 0$cannot be real. Here$a,b \in \mathbb R$, and$a \neq 0$. Source This is one of the previous year problem of Regional ... 3answers 342 views ### Solve$2000x^6+100x^5+10x^3+x-2=0$One of the roots of the equation$2000x^6+100x^5+10x^3+x-2=0$is of the form$\frac{m+\sqrt{n}}r$, where$m$is a non-zero integer and$n$and$r$are relatively prime integers.Then the value of$m+n+...
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 $1$)? ...