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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3answers
18 views

Prove that $f(x)=m$ has three distinct real roots for $m\in(0,8)$

We have $f:\mathbb{R}\rightarrow\mathbb{R},f(x)=x^5-5x+4$ and we need to show that $\forall m\in(0,8)$, $f(x)=m$ has three distinct real roots. How can I prove it?
-4
votes
4answers
71 views

Suppose that $\alpha$ root of the equation [on hold]

Suppose that $\alpha$ root of this equation: $$x^4+x^2-1=0$$ Find the value of $$\alpha ^{6}+2\alpha ^{4}$$ "I want the way, not the roots of the equation." I tried, but I couldn't find any thing.
0
votes
1answer
35 views

When solving trigonometric irrational equations does the condition of existence of the radicand under an even root matter?

Hi everyone I would like to ask a thing about the following equation: $$\cos(x) + \sqrt[4]{1 - \frac{4}{3}\cos(2x) - \sin^4(x)} = 0$$ It is trigonometric and irrational, the root's index is 4 (even ...
2
votes
1answer
30 views

Does $\zeta(s)^2 \pm \zeta(1-s)^2$ have roots at the $\rho$s?

Maybe a strange (or stupid) question, but does $$\zeta(s)^2 \pm \zeta(1-s)^2$$ also have roots equal to the non-trivial zeros ($\rho$) ? At first sight you would expect so, however when I tried to ...
0
votes
0answers
40 views

How many roots are there? [on hold]

I tried to find out how many roots there is for this function: $$ xe^{\frac{2}{x}}=a \quad (a\in\mathbb{R}) $$ Can anyone help by solve it/give hints?
0
votes
1answer
14 views

Expansion and factorization to determine roots of equations

Let $(a,c)$ be the roots of the equation $x ^ 2 + ax - b = 0$. Let $(b,d)$ be the roots of the equation $x ^ 2 + cx + d = 0$. Find all the possible real values for $a, b, c, d$. NOTE: I have made ...
0
votes
3answers
34 views

Find the roots of equation involving $\arctan x$

I try to find the roots of the equation: $$y=x-2\arctan\left(x\right)$$ I know that one of them is $(0,0)$ but there are two others that should solve $$\dfrac{x}{2}=\arctan\left(x\right).$$ Is ...
-1
votes
1answer
46 views

Proof exist infinitely many $n$ such that $f_{n}(x)$ has two integers roots

The two integer sequence $\{a_{n}\},\{b_{n}\}$ such $$a_{n+1}=a_{n}+1,2b_{n+1}=a_{n}+2b_{n}$$ Define function $f_{n}(x)=x^2+a_{n}x+b_{n}$, if there exisit $k$ such $f_{k}(x)=0$ has two ...
1
vote
2answers
37 views

What type of equation is this? How to solve it?

$$m^4+a^4=0$$ , the answer I obtained is $$0+i1,0-i1$$ but the answer is given as a/sqrt(2)-a/sqrt(2),a/sqrt(2)+a/sqrt(2)
0
votes
1answer
42 views

suppose n is a natural number , prove equation $x^n+nx-1=0$ exist an unique real positive root $x_n$

suppose n is a natural number prove : equation $x^n+nx-1=0$ exist an unique real positive root $x_n$ ; and when $a>1$,$\sum_{n=1}^{\inf}x^a_n$ converges.
1
vote
1answer
35 views

How do I calculate the values of $\zeta(0.5+ie^x)$ for large $x$ ?

In wolfram alpha the values of $$\zeta(0.5+ie^x)$$ closed to zero then How do I know the real values of $\zeta(0.5+ie^x)$ for large real number $x$ ? Thank you for any help
1
vote
2answers
51 views

Why $ax^2+bx+c = a(x-r)(x-s)$, where $r$, $s$ are the roots?

When I was reading about math, I came across the following - Suppose the roots of the quadratic $ax^2+bx+c$ are $r$ and $s$. Then $ax^2+bx+c = a(x-r)(x-s)$ for all values of $x$. Is there ...
1
vote
0answers
25 views

Zeros of derivative of composition of polynomials

Let $f(x),g(x)$ be polynomials such that their derivatives $f'(x),g'(x)$ have $n$ and $m$ real roots. What is the possible minimal/maximal numbers of real roots for the polynomial $(f(g(x))'$? My ...
2
votes
3answers
36 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
39 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
45 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 ...
0
votes
1answer
36 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
62 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
2answers
376 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
44 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 ...
3
votes
8answers
114 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
23 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
46 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
30 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
678 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
32 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
23 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
51 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
17 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
27 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
23 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
124 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
36 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
2answers
74 views

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

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
297 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
27 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 ...
3
votes
1answer
48 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
87 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
38 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
45 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
24 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
48 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
29 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
22 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
218 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
56 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
65 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" ...