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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Existence (and uniqueness) of root of function with positive and strictly increasing derivative

Consider a differentiable function $f:\mathbb{R}\to\mathbb{R}$, and assume that: $f(a)<0$ for some $a\in\mathbb{R}$. $f'(a)\geq 0$ and $f'(x)$ is strictly increasing in $(a,+\infty)$. My question ...
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Proof that $y+y^2=x+x^2+x^3$ has no integer root (except $x=0$). [duplicate]

Proof that $y+y^2=x+x^2+x^3$ has no integer root (except $x=0$). $LHS = (y-x)(y+x+1) = x^3$, and $gcd(y-x, y+x+1)$ seems to be $1$. So there are two integers $a,b$ that could make $$y-x=a^3, y+x+1=b^...
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7 votes
1 answer
181 views

Is this statement about roots of polynomials well-known?

Here is the statement : Let $P$ be a non constant polynomial of $\mathbb{C}[X]$ which has at least two distinct roots. If $P''$ divides $P$ hence all the roots of $P$ belong to the same (real) line. ...
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1 vote
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How to find the roots of a linear combination of transcendental functions.

I have run across a situation where I need the roots of a function of the following form: $$f(t) = A sin(wt) - Be^{-Ct}$$ I started to try a series approach, but that quickly got into unfamiliar ...
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Can a general quintic be solved using Inverse Beta Regularized function?

Tyma Gaidash has recently posted solutions to some quintics in terms of Inverse Beta Regularized function. He also found the closed form for the equation $\cos x=x$ using the same Inverse Beta ...
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Finding positive roots of quadratic quasi-polynomials

Are there any good exact or approximate closed-form expressions for the strictly positive roots of $x-\ bx^{v}\ +\ c$ that work for any $0 < v < 1$ (when the roots exist)? Using the iterative ...
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Let $f:\mathbb{R}\rightarrow \mathbb{R}$ with $f(x) = xm, m>1$ . Write Newton’s method for approximating root $x^∗ = 0$ of $f$ starting with $x_0= 0$. [closed]

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ with $f(x) = xm, m > 1$ Write down Newton’s method for approximating the root $x^∗ = 0$ of $f$ starting with initial guess $x_0= 0$. Express the $n^{th}$ ...
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4 votes
1 answer
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+100

Generalizing Ramanujan cubic denesting formula to higher powers

We have the following theorems for denesting radicals of degree 2 and 3 : Denesting theorem for degree 2 : If $\alpha, \beta$ are the roots of the equation, \begin{equation} x^2-ax+b = 0 \end{equation}...
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On the number of roots of $p(z,\bar z)$

Let $p$ be a polynomial with complex (or even real) coefficients in the variables $z$, $\bar z$, where $\bar z$ is the conjugate of $z$. What can we say on the number of complex roots of $p$? Clearly $...
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quadratic function cutoff analysis,some ideas?

I have a parabola of the form $f(x)=ax^2+bx+c$ What condition must the coefficients a,b,c, meet, so that the parabola intersects the x-axis at two points? a) in the negative part of x b) on the ...
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Reasoning behind the rejection of possibility of infinite degree polynomial to show that cos x cannot be written as a polynomial in sinx

https://math.stackexchange.com/a/3954/961436 . In the given solution for showing cosx cannot be written in terms of a polynomial in sinx it is said there that : " "By squaring we see that ...
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-1 votes
1 answer
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Number of zeros of the complex function in the unit disk

$ \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\VP}{V.P.} \DeclareMathOperator{\e}{e} \DeclareMathOperator{\AC}{AC} \DeclareMathOperator{\BB}{B} \...
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What is nature of root of the polynomial $x^5 -10x + 20?$

Problem The polynomial $x^5 -10x + 20$ has a. both positive and negative real roots b. only positive real roots c. only negative real roots d. at least two complex roots My Approach Tried to solve ...
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Proving that cubic polynomial has no real roots in some range (0, a) [closed]

Suppose we have the univariate cubic polynomial $$f(x) = a_3 x^3 + a_2 x^2 + a_1 x + a_0 , \quad x \in (0, z_1), $$ where $a_j = g_j(z_1,z_2)$ are rational functions of some real positive constants $...
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How is the restructuring $x=\frac{c}{b+\sqrt{b^2-c}}$ of the Quadratic Formula for $x^2-2bx+c=0$ done?

I am studying how computers are solving mathematical problems to avoid errors but there is a mathematical formulation I don't understand. So lets say we have an equation $x^2-2bx+c=0$ where $c=1$, $b=...
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Finding zeroes for function $f(t) = e^{k(t-1)} -t$ for $k> 0$ analytically

I tried using Lambert W function the following way $$e^{k(t-1)} -t=0$$ $$e^{k(t-1)}=t$$ $$-ke^{-k} = -kte^{-kt}$$ $$W(-ke^{-k}) = W(-kte^{-kt})$$ $$-k = -kt \implies t = 1$$ but this only gives me one ...
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3 votes
2 answers
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Question regarding an algorithm for solving quadratic equations

I'm reading currently Martin Hanke Burgeois' Book about numeric analysis. It is in German, but my question is the following and I think it's not that hard: In a subsection dedicated to finding zeros ...
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Finding brackets of roots (if any): any sure fire methods for "nice" functions?

Here is a simple question about root finding or, rather, bracket finding. I was trying to find better methods than just sampling an interval with 1000s of sample points in order to find a bracket of a ...
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How to find modular roots of $x^{22}-2x^{11}-x+2$ (to show it has more than $22$ solutions by CRT).

Consider a polynomial $P$ defined by $P(x)=x^{22}-2x^{11}-x+2,$ how to show that there exists an integer $n\geq1$ such that the equation $P(x)\equiv0$ modulo has more than $22$ solutions modulo $n?$ *...
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Can roots of quintic polynomials be a solution of radicals? [closed]

Assuming integer co-efficients, does there exist a solution to a quintic polynomial that is a solution of radicals? I understand that there is no general formula to solve any quintic, but that doesn't ...
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Condition for which the real roots of the polynomial is none

https://math.stackexchange.com/a/1302643/1021792 in this solution given it was being said as coefficients are all positive hence the equation $x^6 + 4x^5 + 5x^4 + 4x^3 +2x +1 = 0$ has no real number ...
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3 votes
3 answers
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When is a multivariable polynomial the zero polynomial

It is known that if a single variable polynomial $p$ with degree at most $n$ has at least $n+1$ zeroes then it must be the $0$ polynomial. Is there an easy to use variant of that for multivariate ...
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What kind of operation is cube root extraction?

I came across this question in a random test and the correct answer was marked as "Binary Operation". I am pretty sure that to find the cube root of a number you only need that number alone ...
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Relation between higher and lower weight ans simplicity

I'm studying the article of Kac "Lie Superalgebras" (1974) and in several times he use the fact that if $L$ is a semisimple Lie Algebra, $V$ its faithful, irreducible and finite-dimensional ...
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2 votes
2 answers
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If equation $f(x)=0$, and it has roots $α$ of degree m, then prove equation $h(x)=f(x+f(x))-f(x)=0$ has roots of degree $2m-1$.

If equation $f(x)=0$, and it has roots $\alpha$ of degree $m$, then prove equation $h(x)=f(x+f(x))-f(x)=0$ has roots of degree $2m-1$. When $\alpha$ is root of degree $m$, that means $$ f(\alpha)=f'(...
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3 votes
2 answers
82 views

Finding the number of roots of $f(z) = z^5 + z^3 + 3z + 1$ in the unit disk

Suppose we have $$f(z) = z^5 + z^3 + 3z + 1$$ Find how many roots this function has in the open unit disc $\{z : |z| < 1\}$. Here's what I think about it: I tried to split $f$ into two functions $...
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Forming equation from given roots ($ -\alpha, -\beta $) where $ \alpha,\beta$ are the two roots of $ \ ax^2+bx+c=0$

If this equation $ \ ax^2+bx+c=0 $ has two roots $ \alpha,\beta$ then form an equation which has the roots $ -\alpha,-\beta $ Solution (given): Here, \begin{align} ax^2+bx+c=0 →\alpha,\beta \end{align}...
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4 votes
1 answer
109 views

Let $g(z)=1+e^z+e^{\alpha z}$ and $A=\{\Re(z) \mid g(z)=0\}$. Prove that $\overline{A}=[a,b]$.

Let $g:\mathbb{C} \to \mathbb{C}$ and $\alpha\in \mathbb{R}\setminus \mathbb{Q}$ such that $g(z)=1+e^z+e^{\alpha z}$ and let $A=\{\Re(z) \mid g(z)=0\}$.(Note that $\Re(z)$ is the real part of a ...
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1 vote
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When does the system $x^n-1=y^n-1=x(y-1)^2+y(x-1)^2=0$ have only the trivial solution $(1,1)$ over the prime field of order $p$?

Let $F_p$ be the prime field of order $p$, $\overline{F_p}$ be its algebraic closure, and $n$ be an integer such that $\gcd(n,p)=1$. Consider the following three polynomials in $F_p[x,y]$: $$ p_1(x,y)=...
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2 votes
1 answer
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How many zeros does $\sin(z)-100z^9$ have with $|z|<1$?

I have to find out how many zeros $\sin(z)-100z^9$ has with $|z|<1$ My approach was to use Roché's Theorem in the following way: Let $g(z) = \sin(z)-100z^9$ and $f(z) = -100z^9$ For $|z| = 1$ we ...
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If there exists a complex root $\alpha$ of $ f(x)$ with multiplicity > 1,then $f(f(x)) $ also has a complex root with multiplicity > 1for polynomial f

Let $f(x)$ be a nonconstant polynomial with real coefficients. If there exists a complex root $\alpha$ of $ f(x)$ with multiplicity greater than 1, then prove that $f(f(x)) $ also has a complex root ...
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1 vote
2 answers
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Summation of roots [closed]

In this post, Calculate summation of square roots we are shown how to sum square roots. My question is, can we get similarly simple expressions if instead of square roots we choose some other exponent....
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+50

Lagrange inversion theorem of $x^r(x+k)$ to generalize the W Lambert function

Motivation: $2$ branches of Lambert $\text W_k(z)$ is a limit of the inverse of $x^n(x+c)$ which is expressible in terms of FoxH in Mathematica. $\text W_0(x)=\text W(x):$ $$-\lim_{a\to0}e^{\frac{(-x)^...
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1 vote
1 answer
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Does the minimal polynomial and characteristic polynomial have same roots over F, for a linear operator on vectorspace V over the field F?

Actually my question is that whether the minimal polynomial and the characteristic have the same root over the field of the vectorspace or do they have the same root over any extension field of F. For ...
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-1 votes
2 answers
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Can quadratics with complex coefficients have more than two roots?

So I came across this quadratic, $z^2(1-2i)+6iz+-2i-1$, and used the quadratic equation to get its roots, $z_1 = \frac{2-i}{5}, z_2 = 2-i$. However, when I used this in a limit it came out differently....
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2 votes
2 answers
42 views

Method checking to get the range of the three variables under two constraints

Suppose a,b,c are real numbers and $a+b+c= 6$ , and $ab+bc+ca = 9$ , also $a<b<c$ find range of a , b, c . My method was on eliminating c we get $a^2 + b^2 -6a - 6b +ab + 9$ = 0 so making a ...
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1 vote
0 answers
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Finding polynomials for given sets of roots

Lets consider polynomials of degree 4, with coefficients in $\mathbb{Q}$ and the ratios of their roots being roots of unity. Moreover, let those roots to be distinct. Now, consider the following given ...
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2 votes
1 answer
50 views

Sequence of zeros of polynomials converges to $\pi$

Let $x_n\in\mathbb{R}$ be the zero closest to $\pi$ of polynomial: $$ p_n(x):=\sum_{k=0}^n \frac{(-1)^k}{(2k)!}\left(\frac{z}{2}\right)^{2k}. $$ I want to prove that $x_n\to \pi$ as $n\to\infty$, $\...
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1 answer
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Approximating $\sqrt[r]x$

So here's a question. What is the best way to approximate this: $$\sqrt[r]x$$ Here is a few method I found, but I am not sure which is faster: brute force. The most straight forward and easiest. Just ...
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-1 votes
0 answers
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How to find sum of roots and product of roots with the roots given

Find the sum of roots and product of roots of a quadratic equation which has the roots -1 and 9. i often have trouble understanding the question and sometimes overthink very easy questions. Any help ...
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Zeros of $f(x)=x^3-x$.

Zeros of the $f(x)=x^3-x$ are $x=-1, x=1$ and $x=0$. We can compute it by solving this equation : $x^3-x=0$ $x^2=1$ $x=1$ and $-1$ But when we divide both part of equation by x , we say that $x$ is ...
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sum over $\frac{x^{k^2}}{k!}$ from $k=0$ to $\infty$

One question that I recently am the following: suppose I got the series $ F(x)=\sum_{k=0}^{\infty}\frac{x^{ak^2}}{k!} $ where $a \in \mathbb{C}.$ I used Mathematica and check that this $F$ can not be ...
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Can one show that $\sum_{k=10}^{50} a_k \cos(k \theta)$ has at least four zeroes on $[0,2\pi]$ for $a_k \in \mathbb{R}$?

This is a complex analysis puzzle that seems tricky. How can one show that for $a_k \in \mathbb{R}$, $\sum_{k=10}^{50} a_k \cos(k \theta)$ has at least four zeroes on $[0,2\pi]$? A hint is to consider ...
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2 votes
1 answer
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Explanation/Reference request for necessary and sufficent conditions for polynomial roots to lie inside unit circle

In the book Applied Econometric Time Series by Walter Enders (third edition, page 30) there is a discussion about the characteristic polynomial of the homogeneous part of an n-th order difference (...
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1 answer
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Modifying the Ridders method with a good first estimate

I am using the ridder's method to approximate a numerical function. It has been working very well for my needs but I'm feeling that it is inefficient in that it is taking more iterations than ...
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3 votes
1 answer
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Is $ f(x,v) = \sum_{n=0}^\infty {x^n \over \Gamma(v n +1) } > 0 $ for all real $x$ and $0<v<1$?

Is $ f_1(x,v) = \sum_{n=0}^\infty {x^n \over (n!)^v } > 0 $ for all real $x$ and $0<v<1$ ? Lets start simple and take the case $v = 1/2$ and notice the inverse ratio of taylor coefficients is ...
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Extensions by Adjoining elements and Extensions by quotient of a Principal Ideal

Extensions can be constructed 2 ways to get an extension with roots of a polynomial Adjoining an element to a field - i.e. $F(\sqrt 2)$ is an extension of $F$. You can also build a tower of ...
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  • 359
3 votes
0 answers
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Finding roots of a polar trigonometric equation

I have an equation which I am using to describe the squared distance from a polar point $(r_1, \theta_1)$ to a function $g(\theta)$ which is $C_1$ smooth over the period $0-2\pi$. $$r = r_{1}^{2}+g\...
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1 answer
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$p_1 (x)=x^3-2020x^2+b_1 x+c_1$ , $p_2 (x)=x^3-2021x^2+b_2 x+c_2$ [closed]

The question is from a competition exam: $$ p_1(x)=x^3-2020x^2+b_1x+c_1 $$ I First tried to establish a relation between the third or unknown and unequal roots ...
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0 answers
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Will pertubation expansion always 'tell you if expansion sequence is wrong'

I'm currently doing a pertubation methods course and seeing all these cases (singular roots, repeated roots) etc where, to find roots of algebraic equations, we must expand in an expansion sequence ...
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