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.

learn more… | top users | synonyms (1)

2
votes
3answers
52 views

Determine roots of a polynomial with variable exponent

I need to know the nature of the roots of the equation $$ x(x+a)^b -1 = 0 $$ when changing a and b, where $ a,b $ are natural numbers, I've looked around on the web but I was unable to find how to do ...
2
votes
2answers
52 views

Roots of a sixth degree polynomial

I have this question: The polynomial $f(x) = x^6 - ax^4 - ax^2 +1 $ has $(x-p)$ as a factor, where $a,p$ are real numbers. Show that $a = p^2 + p^{-2} - 1$ Here's my attempt: Let $u = x^2 ...
0
votes
0answers
153 views

Find the number of zeroes of a function

let $f(z)=(z^2+9)(z^2+1)(z^2-1)+z^5(z^2+4)$. How many zeroes does $f$ has in $\{z|\operatorname{Re}{z}<0\}$. I want to use the argument principle, but the integral is too long. I think I need to ...
-1
votes
1answer
42 views

not easily factored quadratic expression how to find its roots [closed]

Could you please help me and explain this issue: If a quadratic equation is not easily factored then its roots can be found using quadratic formula: If $ax^2+bx+c=0$ ($a\ne0$), then the roots are ...
3
votes
1answer
95 views

The Passare-Tsikh solution to the principal quintic

The Bring-Jerrard quintic, $$x^5+x+t=0$$ can be solved as, $$x = -\sum_{k=0}^\infty(-1)^k\frac{(5k)!}{k!(4k+1)!}\;t^{4k+1}\tag1$$ when, $$|t|<\frac{4}{5^{5/4}}\approx 0.53\dots$$ This paper ...
3
votes
0answers
395 views

Roots of a 6-degree polynomial [duplicate]

Find the roots of the equation $$2000x^6+100x^5+10x^3+x-2=0.$$ I am struggling finding a root using rational root theorem. Even if I get a root, I have to find all the roots. Please don't use ...
0
votes
2answers
61 views

Finding the complex roots of an equation.

I feel ridiculous asking this, its something I should be able to do, however I shall ask anyway. I am doing a calculation that requires me to find the roots of the equation ...
4
votes
8answers
149 views

Find the cubic equation of $x=\sqrt[3]{2-\sqrt{3}}+\sqrt[3]{2+\sqrt{3}}$

Find the cubic equation which has a root $$x=\sqrt[3]{2-\sqrt{3}}+\sqrt[3]{2+\sqrt{3}}$$ My attempt is ...
0
votes
2answers
34 views

Improper Square Root Simplification

I'm trying to simplify a ratio to modify a vector by. Basically I want to find a constant such that the xy-components of two vectors are equal: http://math.stackexchange.com/a/1330263/194115 So I do ...
0
votes
1answer
93 views

Why use methods as Newton, ridder or secant method for root finding? [closed]

Why use methods as Newton, ridder or secant method for root finding? I am bit confused for what reason someone would use these method to determine the root of a function, as it can easily be ...
11
votes
5answers
1k views

Finding cubic with golden ratio as root

I want to find a cubic such that it meets the following criteria: Has the golden ratio as its only real root Has integral coefficients Has a leading coefficient of $1$ and a final coefficient of ...
2
votes
1answer
71 views

Prove that the equation $az^3-z+b=e^{-z}(z+2)$ has two solutions in the right half-plane $\{z\in\mathbb{C}\,:\,\Re z>0\}$ when $a>0$ and $b>2$.

Prove that the equation $$ az^3-z+b=e^{-z}(z+2) $$ has two solutions in the right half-plane $\{z\in\mathbb{C}\,:\,\Re z>0\}$ when $a>0$ and $b>2$. This is an old qualifying exam ...
5
votes
3answers
721 views

Solution of a quartic equation.

Suppose that the equation $x^4-2x^3+4x^2+6x-21=0$ is known to have two roots that are equal in magnitude but opposite in sign. Solve the equation. This is what I have been thinking. Suppose ...
1
vote
2answers
48 views

How to approximately guess the roots of a function

My question is : How to approximately guess the root of a function... By root i mean is the starting point guess when used in case of Newton's method or any other root formulating methods. (Without ...
0
votes
2answers
27 views

Find all $a \in \Bbb {C}$ such that $F$ has at least one multiple root.

Let $F=X^{18}-8X^9+4A$. Find all $A \in \Bbb {C}$ such that $F$ has at least one multiple root. For each $A$ found determine how many different roots $F$ has and their multiplicity. My attempt: $F$ ...
0
votes
2answers
197 views

Paradox - minus one equals one using square roots [duplicate]

I was looking on Howard Eves's book "An Introduction to the History of Mathematics" and I stumbled upon a demonstration on how $-1 = 1$. The demonstration follows: $$ \sqrt{-1} = \sqrt{-1} $$ $$ ...
1
vote
1answer
22 views

Infinite roots of a scalar function

I've been struggling with a problem for a while, I have to proove if the following proposition is true or false: Let $f:\mathbb{R^n}\to\mathbb{R}$ be a smooth funcion (i.e $f \in C¹$). Suppose that ...
1
vote
2answers
42 views

Finding Imaginary Values of a Degree 6 Polynomial

Here's the problem: $0 = x^6 - 65x^3 + 64$. I understand to insert "z" for $x^3$, which gets roots 4 and 1. That leaves you with a $4^{th}$ degree polynomial (which I have found). I know how to ...
2
votes
1answer
40 views

Product of roots inside the unit disk

I have one polynomial $Q(z) = \sum\limits_{n=0}^{2a-1} z^n c_n$, with $c_n \in \mathbb{R}$ and $c_{2a-1}\neq0$. Using Rouché's Theorem, I could locate them as inside or outside the unit disk, with $a$ ...
6
votes
4answers
150 views

When does $(x^x)^x=x^{(x^x)}$ in Real numbers?

I have tried to solve this equation:$(x^x)^x=x^{(x^x)}$ in real numbers I got only $x=1,x=-1,x=2$ , are there others solutions ? Note: $x$ is real number . Thank you for your help .
2
votes
2answers
56 views

Zeros of the derivative of a polynomial.

If all the zeros of a polynomial $f: \mathbb{C} \rightarrow \mathbb{C}$ are real, does this tell us that the zeros of the derivative are also all real valued? i.e, if $f(z) = 0$ only has real roots, ...
0
votes
3answers
176 views

How to prove that the roots of a quartic equation are not ALL real [closed]

Given this equation: $$x^4 + x^3 - 3x^2 + 4x - 2 = 0$$ I wanna prove that not all roots are real. How can I go about achieving this?
-1
votes
8answers
311 views

Solving for n in the equation $\left ( \frac{1}{2} \right )^{n}+\left ( \frac{1}{4} \right )^{n}+\left ( \frac{3}{4} \right )^{n}=1$

Solving for $n$ in the equation $$\left ( \frac{1}{2} \right )^{n}+\left ( \frac{1}{4} \right )^{n}+\left ( \frac{3}{4} \right )^{n}=1$$ Can anyone show me a numerical method step-by-step to ...
0
votes
2answers
44 views

Explaining this inequality

In a proof I was working on today, I assumed this equation was true which lead to devastating results $$ \sqrt{\bar{x^2}} =\bar{\lvert x\rvert} $$ For instance, given the data 0 and 2, the left hand ...
2
votes
1answer
50 views

If $\alpha$ is a root of $f(t) = t^n + a_{n-1}t^{n-1} + \cdots + a_0$, then $|\alpha| \leq n \max_i |a_i|$

Let $f(t) = t^n + a_{n-1}t^{n-1} + \cdots + a_0$. Let $\alpha$ be a root of $f$. Then show that $\alpha \leq n \max_{i} |a_i|$. I could only figure it out for the special case where $|a_i| < 1$ ...
1
vote
2answers
112 views

Why $\zeta(-2) $ is not $\sum_{n=1}^{\infty}\frac{1}{n^{-2}}$? [duplicate]

Let $\zeta(s)= \sum_{n=1}^{\infty}\frac{1}{n^{s}}$ a standard formula. I'm confused if you tell me: does this series: $\sum_{n=1}^{\infty}\frac{1} {n^{s}}$ converge? I will answer you: this series ...
0
votes
1answer
65 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
49 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 ...
3
votes
1answer
43 views

Find the asymptotic behavior of solutions of the equation

Find the asymptotic behavior of solutions $y$ of the equation $$x^5 + x^2y^2=y^6,$$ which tends to $0$ when $x$ tends to $0$. My solution: if $y=Ax^n$, then $$x^5 + A^2x^{2+2n}=A^6x^{6n}.$$ If ...
0
votes
0answers
59 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
75 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
155 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
58 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
25 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
32 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
244 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
100 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
76 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
99 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
65 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
199 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
57 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
108 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
36 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
50 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
62 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
68 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
117 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, ...
-1
votes
3answers
136 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?