Questions tagged [roots]

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 the (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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23 views

If $f:[\alpha,\beta]\to\mathbb{R}$ has positive derivative and $f(a)\cdot f(b)<0$, then is there a unique root?

Reading Zorich, mathematical analysis II, pag 38 (introduction on Newton's method) I found this sentence: My question is: is the convexity really required to say only that there is a unique point $a\...
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11 views

Polynomials and multiple roots

Let $F$ a field with $\operatorname{char} F = 0$ and let $p(x) \in F[X]$, with $\deg (p(x)) \leq n$. Show that $(x-c)^r$ divide $p(x)$ iff $p^{(k)}(c) = 0$, for each $k < r-1$ and $p^{(r)}(c) =0$, ...
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63 views

How to proof that $f(-1) = -\infty$ for an infinite series?

This question extends question Prove that pole of infinite series “explodes” to +∞ as I wonder if this proof technique still holds when the infinite series is defined as: $$f(x) = \sum\limits_{k=1}^{\...
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30 views

Approximate solution to a transcendental equation in the limit of a variable

I have the following transcendental equation: $$2\cot{x}=\frac{kx}{hL}-\frac{hL}{kx}\tag1$$ I would like to inquire whether an approximate solution to $(1)$ can be developed in the limit $h\...
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18 views

Tough polynomial form problem

Find all real values of $a$ for which the equation $(x^2 + a)^2 + a = x$ has four real roots. I played around a bit with a graphing calculator and suspect that $(-\infty, -\frac{3}{4})$ are solutions,...
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1answer
19 views

Is there an analog of Sturm sequences for finite fields?

In finite fields, is there anything analogous to Sturm sequences for counting the number of roots of a polynomial in a given interval? Alternatively, showing that there are zero roots in a given ...
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1answer
52 views

Improving the Cauchy's bound on the absolute values of the roots of the monic polynomial $x^n=m \times \sum_{k=0}^{n-1} x^k$

Given a polynomial $x^n=m \cdot \sum_{k=0}^{n-1} x^k$ (for all $m,n \in \mathbb{N}, m \geq 2,n\geq 2$), the numerical computation of roots for different $n$ and $m$ shows, that the absolute values of ...
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24 views

Numerically find the half-iterate of a quadratic

Let's say we have a function $f(x)=n+mx+lx^2$, and want to find another function $h(x) = a+bx+cx^2 $ such that $h(h(x)) = f(x)$. Here's what you get when composing h on itself: $$h(h(x)) = a+ab+b^{2}...
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Prove that if $m= \sin x+ \sin\frac{x}{2}$ has $2$ real roots $x, y$ on $(0, 2\pi]$ then $\frac{1}{x}+ \frac{1}{y}> \frac{3}{2m}$ .

Prove that if $m= \sin x+ \sin\dfrac{x}{2}$ has $2$ real roots $x, y$ on $(0, 2\pi]$ then $$\frac{1}{x}+ \frac{1}{y}> \frac{3}{2m}$$ I want to see other solution(s) that is not as same as one ...
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1answer
20 views

Integer coefficients of cubic equation imply integer roots

Problem: Let $a,b,c$ be three integers for which the sum $ \frac{ab}{c}+ \frac{ac}{b}+ \frac{bc}{a}$ is integer. Prove that each of the three numbers $ \frac{ab}{c}, \quad \frac{ac}{b},\quad \frac{bc}...
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Searching simple identities between power sums of roots?

The Newton-Girard identities relating Vieta coefficients $e_k$ with power sums $P_k$ of roots of a polynomial, namely,$$ke_k = \sum_{i = 1}^k (-1)^{i - 1}e_{k-i}P_k$$ {with $e_1$ = $P_1$, and $e_k$ = ...
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29 views

Prove that pole of infinite series “explodes” to $+\infty$

The answer to the question at Proving that pole of infinite series "goes to" $+\infty$ or $-\infty$ wonders me how to do it if I cant pull out the numerator from the summation because it depends on $k$...
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32 views

Proving that pole of infinite series “goes to” $+\infty$ or $-\infty$

Lets define the infinite series: $$f(x) = \sum\limits_{k=1}^{\infty} \frac{x^s}{(1-x^k)^2}, \quad s \in \mathbb{N}_0$$ For me, it is clearly to see that it has two poles: at $x = -1$ and at $x = 1$. ...
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21 views

Absolute summability of summation of integer exponents of polynomial roots (BIBO stability)

Let $\lambda _{i} = g_{i}\angle\theta_{i}$ be the roots of a M degree polynomial $S= \sum _{n=0}^{\infty} \left ( \left | \sum _{r=1}^{M} \left ( K_{r} \lambda _{r}^{n}\right )\right | \right )$ , ...
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1answer
34 views

Prove solution to $\sum\limits_{k=1}^{\infty}\frac{x^k}{x^{2k}-2x^k+1} = 0$.

Numerically I found a solution to the following equation at about $x = -0.4112$. $$\sum\limits_{k=1}^{\infty}\frac{x^k}{x^{2k}-2x^k+1} = 0 \quad x \in \mathbb{R}, -1 < x \leq 0$$ Now, I want to ...
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63 views

Counting the roots of $f(z)=z^4+z^3-2z^2+2z+4$

I have a polynomial $f(z)=z^4+z^3-2z^2+2z+4$, and I want to find the number of roots in the first quadrant. I'm trying to use the argument principle (or Rouche), and I could try to make my contour the ...
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35 views

Roots of $x^n+x+1$

I am trying to prove that $x^n+x+1$ does not have any real roots if n is even, and that it has just one root if n is odd. My attempt: When n is odd, I can use Bolzano's theorem to prove that there is ...
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39 views

Can the roots of a linear combination of polynomials all have negative real part?

I have $N$ monic polynomials of degree $(N-1)$, with complex unknown $x\in \mathbb{C}$: $\text{Pol}_{i}(x)$. I take a linear combination of those polynomials, obtaining another polynomial of degree $(...
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1answer
25 views

Root of nested geometric series

How, to solve: $$ \begin{align} \sum\limits_{k=1}^{\infty} \frac{1}{1-x^k} x^k = 0 \quad x \in \mathbb{R} \end{align} $$ Is it even possible? There is a root at $x = 0$, but the graph shows that ...
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Position of roots of a polynomial after the coefficients perturbation

Problem: Given two polynomials of equal degree: \begin{equation} p_1(x)=x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_0,\\ p_2(x)=x^n+(a_1+\epsilon)x^{n-1}+a_2x^{n-2}+\cdots+a_0. \end{equation} where $\...
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20 views

Root locus method, deriving the position of the asymptote centroid

In the Root Locus Method, the linear asymptotes are centered at a point on the real axis given by $$ \sigma_A=\frac{\sum_{j=1}^n(-p_j)-\sum_{i=1}^M(-z_i)}{n-M}, $$ where $p_j$ are the $j$th open-loop ...
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80 views

Solving $\arctan(x)+\arctan(2x)=\frac{\pi}{3}$. Why do I get an extra root?

I have this equation $$\arctan(x)+\arctan(2x)=\frac{\pi}{3}$$ that ends up with two roots but when I graph the equation online the graph only intercepts the x-axis once. So where is my problem? $$\...
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1answer
55 views

I found an unusual (at least for me) answer in Wolfram Alpha I don't understand

Sorry in advance, is my first question and i found something weird in wolfram The problem: Given a function f, and a chain of links with lengths (h0, h1, h2...), calculate the points where every ...
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1answer
37 views

How to show that $Re\left(\sqrt[6]{z}\right)$ is $\frac{\sqrt{2\sqrt{2} +4}}{2}?$

The complex number $z = -4\sqrt{2} + 4\sqrt{2}i$ is given. Show that $Re\left(\sqrt[6]{z}\right)$ is $\cfrac{\sqrt{2\sqrt{2} +4}}{2}.$ The previous part of the question asks to convert the $z$ into ...
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35 views

How to find square root of a number in base $B$?

How to find the square root of a number in base $B$? For example, the square root of $4$ in base $2$ will be $100^{\frac{1}{2}} = 10$.
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79 views

Lower bound on the location of the root in $(0,1)$ of the trinomial $z^m + z^l -1$ involving the degrees

I will begin with my original problem: Let $r : [0,1) \to (0,\infty )$ a continuous, increasing function such that $\lim_{h\to 1} r(h) = \infty$. In my case $r$ is explicitly given. For fixed $x\in (0,...
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4answers
121 views

Find minimal value of $\left(2-x\right)\left(2-y\right)\left(2-z\right)$

Let $x,y,z>0$ such that $x^2+y^2+z^2=3$. Find minimal value of $$\left(2-x\right)\left(2-y\right)\left(2-z\right)$$ I thought the equality occurs at $x = y = z = 1$ (then it is easy), but the fact ...
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36 views

Root of a function involving an integral

I need to dynamically solve an equation for a function $g(t)$ which is multiplied by another expression involving an integral of $g(t)$. Specifically, I have \begin{align} \tag{1} \sigma^{target}(t)...
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3answers
37 views

Prove that the equation $x+\ln x=5$ has at least one root, and find it with precision of 2 digits

I need to prove that the equation $x+\ln x=5$ has at least one root, and to find that root in precision of two digits after the decimal point. I know how to prove the root's existence using the ...
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1answer
57 views

Why is $u^{1/2}=-5$ not a solution?

\begin{align} u+2u^{1/2}-15&=0\\ (u^{1/2}+5)(u^{1/2}-3)&=0\\ \end{align} why is $u^{1/2}=-5$ not regarded as a solution, even though $(u^{1/2})^2=(-5)^2=(25)$? Is it because $\sqrt{u}$ can ...
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1answer
50 views

Prove f(x):x^3+7x^2+17x+5 has only one real root [closed]

Prove that $f(x):x^3+7x^2+17x+5$ has only one real root help! We haven't covered calculus in class yet so if I could get a solution that doesn't involve calc that would be fantastic.
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18 views

Numerical decomposition of polynomials into second-order terms

I am trying to numerically decompose real univariate polynomials into a product of second order polynomials. I have attempted to do this using Bairstow's method, and while this works fine for most ...
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1answer
47 views

How can I simplify Po-Shen Loh's method of solving quadratics?

My question is, is there a way to eliminate the need for fraction arithmetic and rationalizing denominators necessitated by Po-Shen Loh's alternative to the quadratic formula while still being able to ...
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Finding primitive roots modulo n [duplicate]

2 Questions What is the rule/theorem to find a primitive root $r$ for $n=18$? How do i check that $r = 2$ is a primitive root modulo $n=19$? (I know you can find $\text{ordn}(r)$ and $\phi(19)$, but ...
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4answers
109 views

Polynomial $x^3-2x^2-3x-4=0$

Let $\alpha,\beta,\gamma$ be three distinct roots of the polynomial $x^3-2x^2-3x-4=0$. Then find $$\frac{\alpha^6-\beta^6}{\alpha-\beta}+\frac{\beta^6-\gamma^6}{\beta-\gamma}+\frac{\gamma^6-\alpha^6}{\...
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1answer
11 views

How to derive the condition? (I am not sure it is correct.)

Given a polynomial with real coefficients $\alpha x^2 + \beta x + a^2 + b^2 + c^2 - ab- bc - ca $ has imaginary roots, how do we prove $ 2(\alpha - \beta) + ((a - b)^2 + (b - c)^2 + (c - a)^2) > 0 $...
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34 views

How do I prove that a double sum has a neat closed form?

Let $\theta$ be an integer bigger or equal to three, let $\lambda^M \ge 0$ . In addition to this let $\left\{ \zeta_j \right\}_{j=1}^\theta$ be roots of a following polynomial equation $x\cdot (x-1) \...
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3answers
63 views

Is there some theorem related to the triangle inequality that says $\sqrt{x+y}\leq \sqrt{x} + \sqrt{y}$?

I know that the triangle inequality says $|a+b| \leq |a| + |b|$. But what about square roots?
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75 views

How do I solve the equation $ 2x^3+6x^2-12x-4=0$ [closed]

I tried to group the terms like this $$ (2x ^ 3 + 6x ^ 2) - (12x + 4) = 0 $$, but I couldn't give a common factor.
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35 views

Please help me find the zeros of the following function

Let $g(z)=e^{z^2}-\frac{z}{e^{1+2z}-1}$. Clearly, $g\not\equiv 0$ since the function $\frac{z}{e^{1+2z}-1}$ is meromorphic and also $e^{z^2}$ is entire. But, what can be said about the zeros of $g(z)$?...
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14 views

Find Roots of $f(\theta) = \theta^{\delta-1}(1-\theta)^{\beta-1}-\tau$

Let $$f(\theta) = \theta^{\delta-1}(1-\theta)^{\beta-1}-\tau$$ The first term in $\theta$ proportional to a beta density. I am interested in finding the roots of $f(\theta)$. According this post, ...
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2answers
75 views

Solution of $(1-x)^n = x$ : Rate of convergence of $x \to 0$ as $n \to \infty$?

The equation $(1-x)^n = x$ has a solution in $x' \in (0,1)$ and indeed the solution $x' \to 0$ as $n \to \infty$. (Consider $f(x) = (1-x)^n$ noting that $f(0) = 1$ and $f(1) = 0$. As $n$ increases, $...
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2answers
96 views

Integer solutions to multivariate polynomial

I'm looking for a way to show that an equation like $$15(1+x+x^{2})(1+y+y^{2})=16(xy)^{2}-1$$ has no solutions with x and y odd positive integers. The 16 and 15 can be changed but I'm trying to ...
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2answers
51 views

Relation between the roots and coefficient.

Let Let a, b and c be the roots of the equation $$x^3 +3x^2-1=0$$Then what is the value of expression $a^2b+b^2c+c^2a$. I got it done by evaluate the sum and difference of $a^2b+b^2c+c^2a$ and $ab^...
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3answers
154 views

if the polynomial $x^4+ax^3+2x^2+bx+1=0$ has four real roots ,then $a^2+b^2\ge 32?$

if such that the polynomial $$P(x)=x^4+ax^3+2x^2+bx+1=0$$ has four real roots. prove or disprove $$a^2+b^2\ge 32?$$ I have solve this problem: if the polynomial $P(x)$ at least have one real root,...
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4answers
52 views

Any $(x, y, z)$ can satisfy the $5x^2+2y^2+6z^2-6xy-2xz+2yz<0$?

Please tell me whether there any $(x, y, z)$ which can satisfy the $5x^2+2y^2+6z^2-6xy-2xz+2yz<0$ ? No process or just solve it by calculator are both fine. Thank you.
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1answer
37 views

If this equation in x has real roots find the value of a and b.

$$ x^2+2(1+a)x+(3a^2+4ab+4b^2+2)=0$$ I tried to make an inequality using the discriminant and I simplified it to get $$a^2+2ab+b^2+\frac{1}{2}≤0$$ But I don't know how to solve this.
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1answer
43 views

How to find real positive roots of high order polynomials with large coefficients? Can I scale it down while conserving the zeros?

I'm trying to solve high order polynomials (~100) with really large coefficients. In my earlier post, I actually confirmed that these specific sets that I'm working with can only have one real ...
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0answers
33 views

Finding some root of a sparse sum of sinusoids

I am looking for a decent method for finding some root of the following function: $$c_0+\sum_{n=1}^{N}c_n sin{(\pi(a_n x + b_n))}$$ where for any $n$, $a_n, b_n, c_n$ are all known, and are all ...
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1answer
48 views

Why does this function only have 3 rational solution?

Why does the function $f(x,y) = x^3 - x - y^2$ only have the three rational roots $(0,0)$, $(1,0)$, $(-1,0)$?

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