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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61 views

Finding zero of function which is a real number

Is there an easier way of finding or approximating the x-axis-intersect of this function: $$ 0=x^3-3x^2+x+3 $$ The approximate solution is: $$ x=-0.76929 $$ and the precise solution is: $$ x=1 - ...
6
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1answer
103 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 ...
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1answer
54 views

Root of equation, solvability

I was trying to solve the following equation for t $$(P\cdot l \cdot \exp(-l\cdot t) + R \cdot l \cdot \exp(-l \cdot t))/t + (P \cdot \exp(-l \cdot t) + R \cdot (\exp(-l \cdot t) - 1))/t^2 = 0 $$ ...
0
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1answer
52 views

Application of Rouché's (Rouche's) Theorem to a Polynomial

Here is my question: State Rouché's theorem. How many roots of the polynomial $p(z) = z^8 + 3 z^7 + 6 z^2 + 1$ are contained in the annulus {$1 < |z| < 2$}? The statement is fine. I then ...
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1answer
29 views

Simple Pole Search

How do I find poles of: $H(z) = \dfrac{z^3}{z^3+\alpha}$. I know I must find the z values that do $z^3 = -\alpha$. I know how to do it in Matlab (with "residuez" function) but, how can i solve this ...
0
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1answer
51 views

Set of Solutions of A Quadratic Equation with Coefficients in $\{0,1,\cdots , \ p-1\}$

I was just playing with quadratic equations and this interesting question came into my mind. Say I have a set of quadratic polynomials $S=\{f_{(b,c)}(x)=x^2+bx+c:b,c\in \{0,1,\cdots, p-1 \}\}$ where ...
3
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2answers
81 views

Cubic polynomial - radical expression of roots

Let $f=X^3+X^2-2X-1$ be a polynomial with the three roots $x_1,x_2,x_3$ with $x_1=2\text{cos}(\frac{2 \pi}{7})$. We define $z:=(x_1-x_2)(x_1-x_3)(x_2-x_3)$. I want to find a radical expression for ...
2
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1answer
50 views

Find roots of a function

$f$ is a function defined on the whole real line which has the property that $f(1+x)=f(2-x)$ for all $x$. Assume that the equation $f(x)=0$ has $8$ distinct real roots. Find the sum of these roots. I ...
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3answers
47 views

Find the rational roots of $x^{3}-{2x^{2}\over 3}+3x-2$

I need to find the rational roots of $$x^{3}-{2x^{2}\over 3}+3x-2$$ I thought about using descartes´ rational root theorem but I need to have integers as coefficients of my polynomial: can I work with ...
2
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1answer
48 views

Finding the scope of a parameter where a polynomial can have roots

I have this problem- lets say I have a polynomial which has real parameters as coefficients and I'm looking for the scope of the parameters where the polynomial can have real roots. e.g $x^2+kx+k$ we ...
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0answers
19 views

Is $\min \deg$ in a dirichlet subring interesting or is it always $1$?

Let $s \in C$. Let $D = A[[n^{-X}]]$ be a subring of the formal (or absolutely converging on a region; whatever is needed) Dirichlet series with base ring $A$. Define a minimal Dirichlet series for ...
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1answer
21 views

relation between the number of real roots of the derivative and the original polynomial

If the derivative of polynomial has n real roots then can we conclude that the original polynomial has to have n+1 real roots?
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1answer
46 views

conclusion about roots for positive derivative of a polynomial

If the derivative of a polynomial is always positive then what can we conclude about the number of real roots the original polynomial?
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3answers
74 views

Relation between the roots of $x^2+x+1$ and its derivative

If $f(x)$ is a polynomial in n degree and has $n$ real roots then is it necessary that $f'(x)$ has to have $n-1$ real roots? If this is so then $x^2+x+1$ has no real roots but the derivative of the ...
0
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1answer
45 views

it it possible to solve these equation for their root.

I am trying to solve an equations such as the roots of $$k*x(11*x + 1) + d*x(11x + 1)$$ has to match the roots of this function $$x^2 + 0.1x + 6 + k*x(11*x + 1) + d*x(11x + 1)$$, where I have to ...
1
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1answer
135 views

Tangent at average of two roots of cubic with one real and two complex roots

I was able to easily prove that the tangent at the average of two roots of a real cubic polynomial passed through the third root of the function. But I have only done this for functions with three ...
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2answers
55 views

$x^3+3x^2+4x+5=0$ and $x^3+2x^2+7x+3=0$, how many common roots they have?

My attempt, Equate both, at the end you will get $x^2-3x-2=0$ That means $x=-1$ and $x=2$. But what after that. Please provide solutions as well.
5
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2answers
53 views

Disk with root in center with no other roots in polynomial

Say we have a polynomial $p$ with roots $r_1,r_2...r_n$, I'm looking for a way to find a disk which, if placed on the center of any root, does not contain any other root (multiple roots considered as ...
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0answers
45 views

Prove that eventually Hannah and the other swimmer will settle into a pattern where they pass each other (Please refer to the context in my question)

From the 2014 Mathcamp quiz: Hannah is about to get into a swimming pool in which every lane already has one swimmer in it. Hannah wants to choose a lane in which she would have to encounter the other ...
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1answer
47 views

Finding a disk containing all roots of a complex polynomial

I'm trying to list all roots of a polynomial so I found this paper, in Part 9 on page 29 it gives a simple recipe to find all the roots. But there is this remark: We have assumed throughout the ...
0
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0answers
26 views

Bounding the Number of Roots of Integer Polynomial

Let $P(x)$ be a non constant polynomial in $\mathbb{Z[x]}$. Let $n$ be the number of roots of $P(x)^2-1 = 0$. Show $n \le \deg P+2$.
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4answers
133 views

Solve the equation $x^x=10^9$.

The main question was to solve $x\log_{10}{x}=9$. I reduced it to this equation. This is $x$ Degree equation. How to solve this? I know this can be solved by newton's method. But I am not getting how ...
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0answers
39 views

Is any of this true about infinite series of functions?

Let $f_n^+(x)$ be a sequence of non-negative functions $f_n^+: X \to \Bbb{R}_{\geq 0}$, such that each $f_n^+$ has countably many zeros. Then if $f(x) = \sum f_n^+(x)$ converges point-wise, the ...
5
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1answer
80 views

“Polynomials” with non-integer exponents

Are there some books or articles regarding "polynomials" with non-integer (real) exponents, i.e., $$f(x)=a_1x^{e_1}+a_2x^{e_2}+\dots+a_nx^{e_n},$$ where $e_1,e_2,\dots$ are any real numbers (and $x$ ...
0
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0answers
42 views

Why is this root close to $\frac{2\pi}{\text{ZetaZero[1]}}$?

Why is the root, from the following algorithm close to $\frac{2\pi}{\text{ZetaZero[1]}}$? Let: $$y=\frac{2}{5-x}$$ Solve for $x$ in the equation: $$\frac{1}{x+y}=\exp(x-y)$$ Let again: ...
1
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1answer
47 views

Do there exist $a_k$ and $b_k$ so the equation $\sum\limits_{k=1}^{n} (a_k \sin(kx) + b_k \cos(kx)) = 0$ has no roots?

Do there exist real numbers $a_1, a_2, ..., a_n$ and $b_1, b_2, ..., b_n$ such that the equation $$\sum\limits_{k=1}^{n} (a_k \sin(kx) + b_k \cos(kx)) = 0$$ has no solutions?
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1answer
69 views

Describe the graph of f if the graph of its integral its given

Describe the graph of $f$ if the graph of its integral $g(t) = \int_{0}^{t} f(s) ds $ is: graphic of g graphic of f I analyze the derivative and the sign of the derivative and try to find ...
2
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2answers
50 views

Prove that $p(z) = 2z^5 + 6z - 1 $ have roots (in two sets)

Prove that $p(z) = 2z^5 + 6z - 1 $ have one real root in $(0,1)$ and four root in $\left\{ z \in \mathbb{C} : 1<|z|<2 \right\}$. I suppose that we should use Rouché's theorem but I have no ...
1
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1answer
77 views

Complex Analysis: Isolated Singularities, Poles, and Residues

I was given the following question. Show that the isolated singularities of the function $f(z) = \frac{z}{z^4+4}$ are poles. Determine the order of each pole and find the corresponding ...
2
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1answer
41 views

How to show that it holds $|z|<2\max_{0\le k<n}|a_k|^{\frac{1}{n-k}}$ for any root of $X^n+\sum_{k=0}^{n-1}a_kX^k$?

Let $z\in\mathbb{C}$ be a root of the complex polynomial $$f=X^n+\sum_{k=0}^{n-1}a_kX^k$$ I want to show that it holds $$|z|<2\max_{0\le k<n}|a_k|^{\frac{1}{n-k}}$$ Proof: For $s>1$, consider ...
44
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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 ...
2
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2answers
137 views

Show the Equation $2x-1-sinx=0$ has Exactly One Real Root

Question : Use the Intermediate Value Theorem and Mean Value Theorem to show that the queation $2x-1-sinx=0$ has exactly one root. My answer : Since we cannot compute the $y$ when $x=0$, we ...
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4answers
54 views

Technique to simplify algebraic calculations on roots of polynomial

I was once told about a technique to simplify algebra on the roots of a polynomial. So if you want to find $\alpha^3+\beta^3+\gamma^3$, where $\alpha,\beta \text{ and } \gamma$ are roots of ...
5
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4answers
110 views

What is the minimum value of $abc$

If the roots of the equation $$ax^2-bx+c=0$$ lie in the interval $(0,1)$, find the minimum possible value of $abc$. Edit: I forgot to mention in the question that $a$, $b$, and $c$ are natural ...
4
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2answers
66 views

Evaluate $a+b+c+d$

If $a$, $b$, $c$, and $d$ are distinct integers such that $$(x-a)(x-b)(x-c)(x-d)=4$$ has an integral root $r$, what is the value of $a+b+c+d$ in terms of $r$? I tried to analyze graphically by ...
0
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2answers
57 views

Elliptic Curves and “roots”

Given elliptic curve $\omega$ in $\mathbb{R}^2$ such that $y^2 = x^3 + ax + b$, how can you find how many solutions (and what they are) of $x^3+ax+b$ have a $y$ value of $0$; or as they call it, a ...
12
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3answers
171 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 ...
3
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2answers
62 views

Find the value of $\left | b-c \right |$

Given that $a, b, c \in \mathbb{Z}$, $a>10$ and $$(x-a)(x-12)+2=(x+b)(x+c)$$ Find the value of $\left | b-c \right |$ NOTE: The answer to this problem (as given on the last page of my book) is ...
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1answer
26 views

How to prove a nonlinear tracendent equation has two positive roots?

How to show (but do not use numerical software such as Mathematica, Matlab...etc.) that this equation \begin{equation} \frac{u (83811 u-88223)+18076}{396-3276 u}-\frac{10 \log ...
2
votes
1answer
35 views

Finding Complex Zeros

I have to find how many zeros $3e^z - z$ has in $abs(z) < 1$. I know a function has a zero of order m if $f(z) = (z-z_0)^mg(z)$, where $g(z)$ does not equal 0. I was thinking of maybe applying ...
8
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1answer
61 views

Existence of root of a polynomial over $\mathbb F_p$.

I came accross the following question and I can't find an easy proof of this fact : Let $p\geq 17$ be a prime number such that $p\equiv 1 \pmod 4$. Show that for any $z\in \mathbb ...
2
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3answers
73 views

Find amount of roots

We are given equation $$\frac {e^x}{x^2} = a$$. Task is to find how many solutions equation would have depending on values of a. Let's illustrate a(x): It's easy to conclude that there aren't no ...
2
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2answers
74 views

Zeroes of polynomial

$$c_1,c_2 \text{ are polynomial's }g(x)=x^2+ax+b \text{ roots } \Leftrightarrow \begin{cases} g(c_1)=c_1^2+ac_1+b=0 \\ g(c_2)=c_2^2+ac_2+b=0 \end{cases}$$ Prove that for every polynomial with integer ...
0
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1answer
41 views

Quadratics and roots

The question I am trying to solve is this: $4 x^2 - 3 x - 3 = 0$ has roots $p, q$. Find all quadratic equations with roots $p^3$ and $q^3$. I was able to answer this question by simply finding the ...
2
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0answers
24 views

Extensions of the Hermite Bielher and Hermite-Kakeya Theorem

A stable polynomial is one with zeros in the upper half plane or lower half plane. Interlacing polynomials are polynomials with only real zeros, where between every two zeros of one polynomial lies a ...
1
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1answer
39 views

Constant function with maximum modulus [duplicate]

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is ...
6
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2answers
201 views

All roots of polynomial inside the open unit disc

I know from here that for a polynomial $p(z)=a_0+a_1z+...+a_nz^n$ with $0<a_0\leq a_1\leq...\leq a_n$ all roots are in the closed unit disk. What condition do we need to get that all roots are in ...
2
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1answer
106 views

Minimum Modulus Principle for a constant fuction in a simple closed curve

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is ...
0
votes
0answers
24 views

Counting the roots of nonlinear systems of equations

I have a "nice" function (vector field) $$\mathbf{f}: \mathbb{R}^n \rightarrow \mathbb{R}^n$$ and I need to find how many roots (zeros) it has in a certain domain (hopefully prove that it has at most ...
3
votes
2answers
120 views

Real zeros of the zeta function

How does one show that the negative even integers make up all the real zeros of the zeta function? That is, how does one show that there are no real zeros on the interval [0,1]? I am aware that you ...