Questions tagged [roots]

Question 1

I wanted to solve this problem: $$\sqrt x + \sqrt{x+42}=\frac{7}{\sqrt x}$$ I multiplied it by $\sqrt x$ to simplify the fraction. It resulted in me having $$\sqrt x \cdot \sqrt x + \sqrt x \cdot \...

Question 2

Let $\alpha$ be a real root of $$x^7 - 3 x^4 - x^3 + 3 x + 1 = 0.$$ How can we prove that $\alpha>-\frac 13$ without using the Intermediate Value Theorem ? It is easy to prove $\alpha > -\frac ...

Question 3

Let $k$ be an integer greater or equal three, and consider the $k-1$ roots of the hypergeometric equation \begin{equation} \, _2F_1\left(1-k,1;\frac{3}{2};x+1\right)=0. \end{equation} Now take these ...

Question 4

I recently learned about the rational root theorem, which states that the possible rational roots of a given polynomial are in the form $\frac{p}{q}$, where $p$ is a factor of the constant and $q$ is ...

Question 5

I am somewhat confused about a result in my online class. The result says: let $k$ be a field of characteristic $0$. An irreducible polynomial of $k[X]$ can't have any multiple root in $k$ (and in ...

Question 6

In my alg 2 class today my teacher explained properties of polynomials and how to find their roots. One thing he emphasized was that if you know one complex root, you instantly know the other, because ...

Question 7

For a research paper, I am currently writing, I have to deal with two polynomial equations given by $$ \lambda^3 + \lambda + a = 0, \mbox{where} \ a > 0 \ \ \ \ \ (1) $$ and $$ \lambda^3 + \...

Question 8

What is the longest integer sequence $\{u_1,\dots,u_N\}$ such that $\sum\limits_{k=1}^nu_kx^k$ has $n$ distinct real roots for all $1\le n\le N$ ? My best effort so far is $N=7$, with $\{48,4,-64,-5,...

Question 9

Problem: I am dealing with a quintic univariate polynomial $P \in {\Bbb R} [x]$ whose coefficients are real and depend on a parameter $k$, with $-\infty < k < +\infty$: $$ P(x; k) = x^5 + a_4(k) ...

Question 10

I am comparing the variance sizes of two financial stochastic processes (related to fractional Brownian motion), simply using the method of taking differences. Ultimately, I need the roots of the ...

Question 11

I was reading the paper The Algebraic Degree of Geometric Optimization Problems and here they say that when given an expression like $$\frac{x-a_1}{\sqrt{d_1}}+\ldots+\frac{x-a_n}{\sqrt{d_n}}=0$$ we ...

Question 12

I am trying to use polynomial roots to approximate this ratio $R(p)$: $$R(p)=\Re\left(\frac{\underset{k\to \infty}{\text{lim}}\left(\left(H_k^{(s)}\right)^{1/p}+\left(\frac{k^{1-s}}{s-1}\right)^{1/p}\...

Question 13

I understand the definition of the order of convergence: if $(x_n)$ is a sequence that converges to a root $r$, its order of convergence is the largest positive constant $\alpha$ such that $$ \lim_{n\...

Question 14

Disclaimer. Volunteers looking for someone to help are invited to move on to the next question. (See comments.) I was looking for an easy method to count the number of (strictly) positive roots of a ...

Question 15

Is the following conjecture true: For large $n$, the largest root of $\displaystyle\sum\limits_{k=0}^n\cos\left(\frac{k\pi}{2}\right)\binom{n}{k}x^k$ is approximately $\dfrac{n}{\pi}$ for odd $n$, or ...

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Questions tagged [roots]

Does $\sqrt x\cdot\sqrt{x}=x$ or $\sqrt{x}\cdot\sqrt{x}=|x|$

How can we prove that $\alpha>-\frac 13$ without using the Intermediate Value Theorem? [closed]

Pattern spotted for zeros of polynomials arising from hypergeometric roots

Explain constraints for rational root theorem

Why this result (which seems to be a consequence of a previous result) is stated in my Introduction to Galois Theory classnotes?

How to prove that conjugate roots always come in pairs [duplicate]

How to establish that the two polynomial equations $\lambda^3 + \lambda + a = 0$ and $\lambda^3 + \lambda - a = 0$ are unstable, where $a > 0$?

What is the longest integer sequence $\{u_1,\dots,u_N\}$ such that $\sum\limits_{k=1}^nu_kx^k$ has $n$ distinct real roots for all $1\le n\le N$?

Number of real roots of a quintic polynomial with real coefficients

Is there a specific name or closed-form solution for equations of the form $a^x - b^x = c$? [duplicate]

Eliminating all roots from expression doesn't change zeros?

Using the analytic continuation of the Riemann zeta function to approximate some polynomial roots.

Order of convergence of Newton Raphson Method (modified)

The number of positive roots of a real polynomial all of whose roots are real [closed]

Conjecture: The largest root of $\sum_{k=0}^n\cos\left(\frac{k\pi}{2}\right)\binom{n}{k}x^k$ is approximately $\frac{n}{\pi}$ or $\frac{2n}{\pi}$.

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