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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0
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1answer
48 views

Can we find sufficient conditions in which this equation have only three distinct real roots

Let us consider the polynomial equation: $$ξ₁x⁸+ξ₂x⁷+ξ₃ x⁶+ξ₄ x⁵+ξ₅ x⁴+ξ₆ x³+ξ₇ x²+(ξ₈-1) x+ξ₉ =0$$ where $ξ_{i}$ are real coefficients. My question is: Can we find sufficient conditions in ...
2
votes
0answers
47 views

Are all complex zeros of ${\frac {\zeta \left( s+1 \right) }{\zeta \left( s-1 \right) }}\pm\, 2\,\pi\frac{2-s}{s\,(s+1)}$ on the critical line?

From this question, it is easy to derive that a zero of $\xi(a+s)\pm \xi(a+1-s)$ should occur when: $$\displaystyle{\frac {\zeta \left( s+a \right) }{\zeta \left( s-a \right) }}=\pm{ \frac {{\pi ...
0
votes
0answers
27 views

Multiplicity of roots of non-polynomial

Define $f: \mathbb{C} \to \mathbb{C}$ by $f(z) = z^{11} + 4e^{z+1} - 2$ and $D := \{z \in \mathbb{C}: 1 < |z| < 3\}$. The question is to show that $f$ has $11$ different roots in $D$. I already ...
0
votes
2answers
62 views

Simulating an orbit - numerically solving $M(E) = E + \sin(E)$

Well for a given kepler orbit (which is a ellipse) $0 \leq e < 1$. There are several functions to describe the motion of an object. $$r(\nu) = \frac{a (1 - e^2)}{1 + e \cos(\nu)}$$ Where $a$ is ...
0
votes
1answer
46 views

Comparing Square Roots

How do you compare square roots? Of course, the positive square root of 49 is greater than the positive square root of 36. However, what if you were to have $\pm\sqrt{49}$ ? $\pm\sqrt{36}$? Would it ...
1
vote
1answer
53 views

For which values of a parameter an equation has one Real root

The following equation is given $$\log_{x-1}(x^2+2ax) - \log_{x-1}(8x-6a-3)=0$$ And I am trying to find for which values of $a$ it has only one root, which is real. It is obvious that $$x-1>0 ...
1
vote
0answers
24 views

Look for Max in function

I need to show that the follwing function has just got a minimum and no maximum. I know what it looks like and it is pretty obvious but i can't find a way to explain. The question implicates we might ...
2
votes
0answers
30 views

Linear combination of matrix elements

Consider the following sequence of problem: With $A \in \mathbb{R}^{n \times m}$, $m>n$, and $x \in \mathbb{R}^m$, I am looking to linearly combine (non-trivially) the elements of the vector $Ax$ ...
4
votes
3answers
226 views

Factors in a cubic equation

I have no idea how to go about this. Any Hint? Suppose that $(x-3)$ is a factor of $$kx^3 - 6x^2 + 2kx - 12.$$ Solve for $k$.
5
votes
1answer
60 views

Struggling with an Application of Rouche's Theorem

Prove that the zeros of the polynomial $p(z)=z^n+c_{n-1}z^{n-1}\cdots + c_1z+c_0$ all lie in the open disc centered at $0$ with radius $$R=\sqrt{1+\vert c_0\vert^2+\vert c_1\vert^2+\cdots + \vert ...
3
votes
1answer
71 views

Properties of distribution of zeros of polynomial

Polynomial $p_n(z) = (1 + \frac{z}{n})^n - 1$ has a property that all its zeros lie on the circle of radius $n$. It is easy to see because $$\frac{z}{n} = e^{\frac{i2\pi k}{n}} - 1$$ So we can "fit" ...
7
votes
2answers
89 views

Show that $p_n(a)\neq 0$ if $|a|=n$

I am working the next problem: Consider the polynomials $$ p_n(z)=\sum_{j=0}^{n}\frac{z^j}{j!} $$ For $n \geq 2$, show that if $a \in \mathbb{C}$ is such that $|a|=1$ or $|a|=n$, then ...
2
votes
2answers
51 views

Show that $f=x^3+7x+5$ has no roots in $\mathbb {Q}(\sqrt[4]{2})$

Show that $f=x^3+7x+5$ has no roots in $\mathbb {Q}(\sqrt[4]{2})$. I'm given a hint: suppose $\alpha$ is a root of $f=x^3+7x+5$ and $\alpha\in\mathbb{Q}(\sqrt[4]{2})$, compute ...
-1
votes
1answer
133 views

Radical solution to a polynomial quartic equation

Consider the following quartic equation: $$x^4 + rx^3 + r^2x^2 + r^3x + r^4 - 1 = 0$$ By Lodovico Ferrari solution, this equation must possess four radical solution provided that $r$ is a rational ...
3
votes
3answers
54 views

Zeroes of polynomials and their sum

Let $a, b$ are zeroes of the polynomial $x^2-10cx-11d$ and $c,d$ are the zeroes of the polynomial $x^2-10a x-11b $ where $a,b,c, d$ are distinct reals then $a+b+c+d=?$
4
votes
0answers
97 views

Roots of a polynomial

I am working the next problem: Consider the polynomials $$ p_n(z)=\sum_{j=0}^{n}\frac{z^j}{j!} $$ For $n \geq 2$, show that if $a \in \mathbb{C}$ is such that $|a|=1$ or $|a|=n$, then ...
0
votes
0answers
20 views

Old and recent results concerning the number of real roots for a polynomial function

Let $$p(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots +a_0$$ be a real polynomial function with real coefficients. My question is: I want to make a list of old and recent results concerning the number of real ...
0
votes
0answers
32 views

What will happen if there is a way predicting at a least one root of $p_{n}(x)=0$ without calculator?

let $p_{n}(x)$ be a polynomial of degree $n$ defined as follow : $p_{n}(x)=x^n +a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+.....a_{0}$ which : $a_{n-1},a_{n-2},.....,a_{0}$ are non nul real numbers coefficients. ...
0
votes
0answers
7 views

On the zero set of a smooth function which is linear on the first variable.

Let $f:C^n\times R^n\rightarrow C$ be a smooth function which is linear in $z\in C^n$. (1) For any $z\in C^n$, $f(z,R^n)$ is compact and there exists $x_z\in R^n$ satisfying $f(z,x_z)=0$. (2) There ...
2
votes
2answers
41 views

Concerning Roots of the cubic equation $f(x)=x^3+x^2-5x-1$ and the Greatest Integer (or Floor) function

The Question I got into a rather tight corner with this question. It says: Let $\alpha, \beta, \gamma$ be the roots of $f(x)=0$, where $f(x)=x^3+x^2-5x-1$. Then, the value of ...
3
votes
0answers
49 views

Sum of zeros of $P(x)$

I asked this question here before too, but vaguely, hopefully, this time will be a better attempt: There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero ...
1
vote
1answer
59 views

Finding the real root of the polynomial $2x^3-3x^2+2 $

I want to get exactly roots of this equation... $2x^3-3x^2+2 = 0$ I try to solve it but can not find the solution. wolframealpha just give me aproximation.. I know the real root is $-1< root ...
0
votes
1answer
70 views

How can I prove this statement about square root?

Introduction In computer science there is a field called Formal Methods and Specifications. In this field software designers design softwares by specifying their functionalities in formal methods, ...
0
votes
3answers
92 views

If a quadratic equation $ax^2+bx+c=0$ has more than two roots, then $a=b=c=0$ [closed]

If a quadratic equation $ax^2+bx+c=0$ has more than two roots, then it is an identity i.e. it is true for all values of $x$ and $a=b=c=0$. What is a proof of this?
1
vote
1answer
50 views

Can we write “fractional root” symbol in math?

Fractional exponents are legit but I have never seen fractional roots, so I just wonder if we can write fractional roots such as this: It sometimes can be convenient to think about too.
19
votes
4answers
364 views

Probability of $ax^2 + bx + c = 0$ having real solutions

$a$, $b$, $c$ are random integer numbers between $1$ and $100$ (including $1$ and $100$, and uniformly distributed). What is the probability that the equation $ax^2 + bx + c = 0$ has real ...
0
votes
0answers
54 views

Finding the roots of weighted sum of $\tanh$

How can this equation be solved analytically? Is it possible, why? Is there a better alternative than the Newton approximation? $k\tanh^2(ax+b) + l\tanh^2(cx+d) + m = 0 $ with $a,b,c,d,k,l,m \text{ ...
3
votes
6answers
93 views

Solving $e^\frac1x = x$ non-graphically?

This question has come up twice in different tests and the instructions always point out that it should be solved using a graphic calculator. Fair enough, the answer is ≈ 1.76322...(goes on forever?). ...
10
votes
1answer
225 views

Solving a special Quartic Equation.

Solve for $x$ $$(x^2-4)(x^2-2x)=2$$ I have tried the Rational Root Theorem and found that there are no rational roots. Further, the polynomial $p(x)=(x^2-4)(x^2-2x)-2$ is irreducible since ...
1
vote
1answer
33 views

Show that a real polynomial of degree more than or equal to $3$ is reducible

Let $f \in \mathbb R [x]$ and suppose that $\deg(f) \geq 3$. Then $f$ is reducible. Proof: By the Fundamental theorem of algebra there are $\lambda _j \in \mathbb C$ such that $$f(x) = (x-\lambda_1) ...
1
vote
1answer
21 views

existence of roots for a general function over the complex plane

For some more general functions other than polynomials, are there any fixed conditions for the existence of roots in a general sense? For instance, function like $z\mathrm{sin}z-1$
6
votes
4answers
159 views

Can $\sin(\pi/25)$ be expressed in radicals

I suspect that sin(pi/25) is not expressible in elementary forms in radicals because it is the root of some quintic (or rather cos(pi/25) is). Can anyone prove that that particular quintic has no ...
0
votes
1answer
32 views

How to find out if a sequence with exponentiation in fraction is convergent

I need to find the convergence of this function: $\sum^{\infty}_{x=1}{\frac{(x+1)^{x^2}}{x^{x^2}2^x}}$ Now my problem is, I have no clue how to do this (I tried the root-test and it did not work ...
0
votes
0answers
11 views

Finding common roots to a variable number of functions

I am trying to solve the following problem. Given $a\in\mathbb R^n$, $u\in\mathbb{R}^n$, $m\in\mathbb{N}^\star$, Find the/some common roots $(t_1,...,t_m)$ of the $\frac{m(m-1)}{2}$ ...
3
votes
4answers
248 views

Solving equations with exponentials and a non-exponential term.

I know how to solve exponential equations. Just use logarithms, e.g., $$ 2^x-3=0 \\ 2^x=3 \\ x=log_23 \\ $$ But on a recent math test I found an equation of the form: $$ 2^{n-3}=\frac {20}{n} $$ ...
5
votes
4answers
107 views

Root of $(x+a)^{x+a}=x^{x+2a}$ and $e$

Let us denote solution to the equation $$(x+a)^{x+a}=x^{x+2a}$$ with $X_a$. ($a$ is a non-zero real number) Prove that: $$\lim_ {a \to 0} X_a = e$$ This is something that ...
5
votes
1answer
131 views

Roots of a Cubic Polynomial with Elementary Symmetric Polynomial Coefficients

Let $R_n$ be a set of $n$ distinct nonzero rational numbers. Let $e_k$ be elementary symmetric polynomials over $R_n$---i.e. $e_0=1$, $e_1 = \sum_{1\le i\le n} r_i$, $e_2 = \sum_{1\le i<j\le n} r_i ...
0
votes
2answers
81 views

How to find the positive and negative roots of a function?

Iam trying to solve the following question: Find all numbers $a$, such that the equation $x^2-ax-a = 0$ has one positive root and one negative root. I've tried it already but I cannot seem to ...
4
votes
1answer
52 views

Trigonometric root of a polynomial

If $4\cos^2 \left(\dfrac{k\pi}{j}\right)$ is the greatest root of the equation $$x^3-7x^2+14x-7=0$$ where $\gcd(k,j)=1$ Evaluate $k+j$ I tried factorizing the equation but it wasn't ...
3
votes
3answers
115 views

Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots.

Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots. My attempt: Consider the polynomial $g=(x-1)(1+x+x^2+x^3+\cdots+x^n)$ As $f\mid g, g$ all the roots of $f$ are roots of $g$. This means I ...
0
votes
3answers
86 views

What are the values of $p$ so that equation $x^3+(p-2)x^2+(5-2p)x-10=0$ has exactly $2$ real roots…

I found this question in a maths-group in Facebook- What are the values of $p$ so that equation $x^3+(p-2)x^2+(5-2p)x-10=0$ has exactly $2$ real roots........ I think we do not count repeated roots ...
4
votes
1answer
78 views

Number of complex roots of a degree 6 polynomial

Given some degree 6 polynomial $f(x) \in \mathbb{Q}[x]$, is there any invariant of the polynomial (depending on the coefficents) that will tell you if this polynomial has 6 complex roots or just 2 ...
1
vote
6answers
53 views

AS Maths simplify

A question in my AS Exam this morning was simply, Simplify $(5\sqrt5)^3$ I tried $(5\sqrt5 \cdot 5\sqrt5)^2$ and ended with $5^4$. Is that correct? I think it's completely wrong, but it'd be awesome ...
1
vote
0answers
38 views

Two-term asymptotic approximation for roots of a polynomial (dominant balance)

I'm trying to find the roots to the following equation: $t^5 - \epsilon t^3 + \epsilon^3 = 0$ as $\epsilon \rightarrow 0$. From expansion $t= \epsilon^{\alpha}t_1 + \epsilon^{2\alpha}t_2 + ...
2
votes
2answers
35 views

Zeros of weighted sum of two Bessel functions

Just a simple and very tentative query to alleviate my seemingly futile internet digging: is there anything known on the structure of the entire function given by \begin{equation} ...
1
vote
3answers
76 views

Distinct roots of $z^n-z$

How would we prove that for any positive integer $n$ the complex roots of $z^n-z$ are all distinct? In the case that $n=1,2,3$ I have factored it directly but how can we do it in general?
0
votes
0answers
28 views

Roots of serie $\sum_{i=1}^N \frac{a_i + b_i \log(c_i - x)}{c_i - x}$

I encountered the following serie during an optimization problem: $\sum_{i=1}^N \frac{a_i + b_i \log(c_i - x)}{c_i - x}$ I need to find the roots of that serie, but I cannot find the way to find a ...
-1
votes
1answer
31 views

Prove that the line integral on $\beta$ of $f'(z)/f(z) = (A-B)/2 \pi i$ using Rouche's Theorem

Suppose that $\alpha$ is a regular closed contour. $f$, our function, lacks zeros and poles on $\beta$ and if A=the number of zeros of f inside $\beta$ (a zero of order n is counted n times) and B= ...
0
votes
1answer
27 views

If $p>0$ demonstrate that the $1/2\pi i$ the line integral of $z^p f'(z)/f(z)$ is $\sum (z_k)^p$

This is basically a deviation of Rouche's Theorem from what I can tell. My first instinct was to do this via induction in which we know that $p=0$ we would have Rouche's theorem. But it gets ...
2
votes
1answer
38 views

Approximating non-rational roots by a rational roots for a quadratic equation

Let $a,b,c$ be integers and suppose the equation $f(x)=ax^2+bx+c=0$ has an irrational root $r$. Let $u=\frac p q$ be any rational number such that $|u-r|<1$. Prove that $\frac 1 {q^2} \leq |f(u)| ...