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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2
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1answer
50 views

If $z$ is an $n$th root of unity, prove that $1/z$ is an $n$th root of unity

I'm not sure if how I'm going to prove this to be correct: Since $z$ is an $n$th root of unity, it means $z^n = 1$ For $1/z$ to be an nth root of unity, lets take it to the power of $n$, ...
0
votes
1answer
34 views

Field of characteristic $0$ and simple roots

Suppose $g(x)=f(x)m(x)^{-1}$ where $f(x)\in F[x]$ and $F$ has characteristic $0$ and $m(x)=(f,f')$. Then show that all roots of $g(x)$ are simple. I assume g(x) has multiple roots that is ...
3
votes
0answers
28 views

Squares of finite fields (mod p*q)

Lets say we have $\mathbb{Z}_p$, where p is prime. For each element(x) we have two squares(y) so that $y^2=x$ ie if $p=7$ for $x=4$ we have $y_1=2,y_2=7-2=5,y=\pm2 $ ok, lets have ...
0
votes
1answer
17 views

find coefficients of a polynomial given k roots

Lets say that I have k roots for a polynomial and I am trying to find the coefficients of the terms in the polynomial. (x - r1)(x - r2)(x - r3) ... (x - rk) ...
2
votes
0answers
29 views

Roots of polynomial equations

Is there anything that can be said about the roots of the polynomial $f_n(x)$ if $f_n(x) = xf_{n-1}(x) + f_{n-2}(x)$ where these are polynomials of degree $n, n-1,$ and $n-2$, respectively. In ...
1
vote
2answers
26 views

Show that $\frac{1}{x}-\sin(x)$ has exactly one root in the interval $(0,\frac{\pi}{2}]$

I have problems showing, that this function has exact one root in the interval $\left(0,\frac{\pi}{2}\right]$: $$f(x):=\frac{1}{x}-\sin(x)$$ My idea was to use the Intermediate value theorem, but ...
0
votes
3answers
55 views

Solving a radical equation for real roots

I'm attempting to solve the derivative of my function $f(x)$ for real roots. $$ \\ \begin{align*} \\ f(x) &= 3x^2 + 3\arcsin{x} \\ f^{\prime}(x) &= 6x + \dfrac{3}{\sqrt{1-x^2}} \\ \\ 0 &= ...
1
vote
1answer
51 views

Müller's Method

I have these question and I cannot solve it. Can somebody help me? Use Müller’s method to determine the roots of $$ f(x)=2x^5−2x^4+6x^3−6x^2+8x−8 $$ Choose $x_2=0.8 $, $x_0=0.808$ ...
-1
votes
3answers
31 views

Find the speed of a jet given the time of travel back and forth

The problem: A jet flew from Tokyo to Bangkok, a distance of 4800km. On the return trip, the speed was decreased by 200 km/h. If the difference in the times of the flights was 2 hours, what ...
2
votes
1answer
156 views

Why is this answer wrong? (quadratic functions)

Question: Determine the quadratic function that has the given roots (x-intercepts) and passes though the given point. $x=2+\sqrt{5}, x=2 - \sqrt{5}$. The given point is $(2,10)$. This is my ...
0
votes
0answers
29 views

How do I solve this quadratics problem? [duplicate]

The problem: A jet flew from Tokyo to Bangkok, a distance of 4800km. On the return trip, the speed was decreased by 200 km/h. If the difference in the times of the flights was 2 hours, what ...
0
votes
1answer
31 views

Which are the conditions for a biquadratic equation to have 4 different roots?

Which are the conditions for a biquadratic equation to have 4 different roots in R? I think D>0, If we have $$t=x^2$$ then t>0. Is there any other condition?
1
vote
2answers
42 views

Prove that the rounding error can contaminate half the digits of computed root

I am trying to resolve the following problem: If $b^2 \approx 4ac $ the rounding error can contaminate half the digits of the root computed with the formula: $\dfrac {-b \pm \sqrt {b^2 - 4ac}} ...
1
vote
1answer
23 views

Discriminant with non-Real result

I have the following equation: $ ax^2 + (a+1)x - a = 0 $ Where $a$ is not $0$ When calculating the discriminant $\Delta$ i get a non-real result. But what does it mean? I know that a negative ...
0
votes
1answer
42 views

Polynomial in $Q[x]$ with 2 complex roots

I need to show that for any $n$, I need to show that there is an irreducible polynomial in $Q[x]$ of degree $n$ having exactly $n-2$ real roots. As a hint I have that if $f(x) \in \mathbb{R}[x]$ is a ...
2
votes
1answer
86 views

Irreducible polynomial in $\mathbb Q[x]$ of degree $n$ having exactly $n-2$ real roots.

I need to show that for any $n$ there is an irreducible polynomial in $\mathbb Q[x]$ of degree $n$ having exactly $n-2$ real roots. I know that from a previous exercise that if $f(x) \in ...
1
vote
2answers
52 views

Polynomial $f(x)$ over $\mathbb{R}$ has $k$ distinct roots, then $f(x) + a$ too

I am trying to learn Galois theory by myself. When reading a section for applications to polynomials, I got stuck in the following exercise: If $f(x) \in \mathbb{R}[x]$ is any polynomial having ...
3
votes
0answers
25 views

Magnitudes of roots of random polynomials

I'm looking at the roots of random polynomials with integer coefficients, and constant term=leading term = 1. Using the Mathematica code ...
0
votes
1answer
55 views

Solving for $x$, ${\sqrt{7x-5}} - {\sqrt{2x}} = {\sqrt{15 - 7x}}$

could I please have some help solving this equation for $x$ ? ${\sqrt{7x-5}} - {\sqrt{2x}} = {\sqrt{15 - 7x}}$ Thank you
0
votes
0answers
25 views

Iterpolating to find the zeros of a complex function

I have an $N\times M$ grid of complex points sampled from some unknown complex function. I would like to interpolate and find the zeros of that function. I believe that this function can be well ...
1
vote
1answer
23 views

If $\tan\alpha,\, \tan\beta,\,\tan\gamma$ are roots of $au^ 3 +(2a-x)u+ y=0$

If $\tan\alpha,\, \tan\beta,\,\tan\gamma$ are roots of $au^ 3 +(2a-x)u+ y=0$ for fixed $x$ and $y$ and $\tan\alpha + \tan\beta = h$ Find $ah^3 +(2a-x)h$. Options are A) $y$ B) $-y$ C) $2a- x$ D) ...
0
votes
0answers
34 views

Analytical solution to equation $ \arctan(x)-k \arctan(x/k)=c$

For the equation:$$\arctan(x)-k \arctan(x/k)=c$$ which is part of a gasdynamics function called Prandtl–Meyer function, it is not difficult to find the solution numerically, however, I'm wondering, ...
0
votes
0answers
16 views

bounds on the non real zeros of a polynomial $P_n(z)$

There are several bounds on the complex zeros (including real zeros and nonreal zeros) in the literature. There are also several bounds on the positive/negative zeros in the literature. Are there ...
0
votes
1answer
65 views

Number of solutions to an equation $6\ln(x^2+1)-e^x = 0$

I need to determine how many solutions the equation $6\ln(x^2+1)-e^x = 0$ has. I wanted to find monotonic intervals of this function and check function's values at local extrema so I calculate ...
1
vote
0answers
41 views

Irrational roots of polynomials with integer coefficients

Consider an equation of the form $$ p(x) = 0, $$ where $p(x) = \sum_{i=0}^{n} a_i x^i$ and all $a_i \in \mathbb{Z}$. Is there an algorithm which can tell us that this equation has a root of the form ...
0
votes
3answers
41 views

How to find the range of zeroes in a polynomial

I am writing a java program that finds all of the zeroes of a polynomial by bisection. The first step, clearly, is to iterate through integers in a certain range looking for sign changes. I could ...
0
votes
0answers
47 views

How to get the proper fixed point iteration function?

When we find the approximated root of a function $f(x)$ in an interval $[a,b]$ from the fixed point iteration method, we derive a new function $g(x)$ which has a fixed point as a root of $f(x)$. Is ...
0
votes
1answer
21 views

Showing that at least one of the equations has two real roots

Let's suppose that $b_1, b_2, c_1, c_2$ are real numbers. We know that $b_1b_2=2(c_1+c_2)$. The task is to prove that at least one of the equations $x^2+b_1x+c_1=0$, $x^2+b_2x+c_2=0$ has two real ...
0
votes
1answer
17 views

How do we know that $\tan z$ or $\cos z$ don't have singularities off of the real axis?

Since $\sin z$ is bounded, $\tan z = \frac{\sin z}{ \cos z}$ has singularities when $\cos z = 0$. We know $\cos z = 0$ for $z = \frac{\pi}{2} + n\pi$ for $n \in \mathbb{Z}$, but could it not also be ...
4
votes
2answers
46 views

Show that $x+e^{-Bx^2}\mbox{cos}(x)$ has only one root over all reals ($B>0$).

Let $B>0$ and define for all $x\in \mathbb{R}$ the function $f(x)=x+e^{-Bx^2}\mbox{cos}(x)$. Prove that $f$ has exactly one root over $\mathbb{R}$. My original attempt was to show that the ...
7
votes
1answer
87 views

Can the general septic be solved by infinitely nested radicals?

I. Quintic. The general quintic can be reduced to the form, $$x^5=p+x\tag1$$ $$x = \sqrt[5]{p+x}$$ Hence by an iterative process, $$x =\sqrt[5]{p+\sqrt[5]{p+\sqrt[5]{p+\sqrt[5]{p+x\dots}}}}$$ ...
2
votes
3answers
63 views

condition for a cubic to have a repeated root

To write a condition for a cubic to have 2 real roots, can I equate the fuction to its derivative? I.e let $y=ax^3+bx^2+cx+d$ $\frac{dy}{dx}=3ax^2+2bx+c$ setting y and the derivative equal to 0 ...
0
votes
3answers
38 views

Find the value of $a$ from the equation

The roots of the equation: $ax^2-(5a+2)x+9a=0$ are equal. Find the value of $a$ given that $a>0$.
2
votes
2answers
136 views

Extracting factor from quadrinomial

As I'v learned about polynomials, I run into this quadrinomial: $$x^3+300x^2+30000x-953125 = 0$$ I've been studied how to factor this quadrinomial but didn't quite understand how it's done, here is ...
4
votes
5answers
89 views

How to find the cubic with roots: $k$, $k^{-1}$ and $1-k$?

This is the second part of a question that asks the same thing but for a quadratic, that part seemed to be fine. The next part asks you to show that: $$x^3-\frac{3}{2}x^2-\frac{3}{2}x+1=0 $$ is the ...
1
vote
0answers
17 views

Proving collinearity of the roots without finding the roots.

I was solving the polynomial $2z^3-(3-3i)z^2-(1+i)=0$ and found that they were in fact collinear! My question is, is there a way to prove that they are collinear without explicitly finding the roots? ...
0
votes
0answers
42 views

Control theory: Why doesn't the separation principle hold in nonlinear control theory?

It is widely known in control that separation principle is one of the best tool for pole placement and design of stabilizing controller in linear system. Many results also note the inability of ...
0
votes
1answer
27 views

How can I show this ratio is >1 for intervals of x,y

I come here from a substantial application in statistics where I have reason to belive that the following ratio (function) is $$f(X,Y)=\frac{1}{(2XY^2-X^2Y^2+X^2-2X+1)^{\frac{1}{2}}} \ge 1$$ for ...
1
vote
1answer
31 views

N-th roots equation

I am facing the following equation and I do not have any idea about how to solve it. $\dfrac{(n^c-1)^a}{n^{ac}}$ = $\dfrac{1}{2}$. I am free to choose $c$ (any constant). $a$ on the other hand can be ...
0
votes
4answers
66 views

Roots of complex number

Solve the equation $$z^3-1=0$$ Show that the roots are represented in an Argand diagram by the vertices of an equilateral triangle. (EDITED: Thank you for your quick respond)
3
votes
3answers
127 views

explicit solution for transcendental equation

Does anyone knows whether there is an explicit, analytical solution for transcendental equations of the form $A x + B \tanh(C x) + \coth(x) = 0$, where $A, B$, and $C$ are positive real constants?
0
votes
1answer
24 views

About uniqueness of interest yield

I am not sure this belong to this site, in case I will post it elsewhere. Let $P$ be the price of a bond, let $C_k$ the promised cash flow in year $k$. Then we define the interest yield $y$ as the ...
2
votes
0answers
44 views

Find an equation in $x$ and $k$

Find an equation in $x$ and $k$ if, $$6u-8v+2=k^2$$ $$u^{2}=1+2v^{2}$$ $$v=2xy$$ $$u=x^2+2xy-y^2$$ Since we have 4 equations, we can eliminate 3 variables. But somehow, I'm not able to find an ...
1
vote
1answer
53 views

Zeros in the polynomial ring $\mathbb{R} [X,Y]$

I know that for $p(X) \in \mathbb{R} [X]$, $a$ is a zero of $p(X) \iff (X-a)|p(X)$. But what would the statement be for $p(X) \in \mathbb{R}[X,Y]$? This question comes from an example in my ...
0
votes
2answers
71 views

Prove: p-mq | f(m) where 'm' is any integer

How to prove that $p-mq \mid f(m)$ where $m$ is any integer, $f(x) = A_0 + A_1 x + A_2 x^2 + ... + A_{n-1} x^{n-1} + A_n x^n$, $f(x)∈ ℤ[x]$, $p/q$ is a zero for $f(x)$ and $p$ and $q$ are coprime ...
1
vote
3answers
79 views

Prove that the equation $x^{2}-x\sin(x)-\cos(x)=0$ has only one root in the closed interval $(0,\infty)$.

Here's the graph (http://www.wolframalpha.com/input/?i=%28x%5E2%29-xsenx-cosx%3D0). The part I'm having trouble with is proving that the root is unique. I can use the intermediate value theorem to ...
1
vote
3answers
61 views

Determining polynomial from roots of another polynomial

I am working on an exercize and I know how to more bruteforcely solve it through pure algebra in its simplest form, but it's such a massive mess to demonstrate so I would like to see if there is ...
0
votes
2answers
72 views

Solve this equation $(2x^2-3)^2=4(x-1)^2$?

This is the way I solved: What should I do next? Should I factorize or take a t that represents something?
1
vote
2answers
28 views

How to solve a twin root equation?

I thought to factorize a sqrt(x), but I can't find out anything. I thought to multiply both sides with themselves four times, but I'm not sure that works.
1
vote
1answer
29 views

Solving equation with complex roots.

I have the following question. My problem lies in (c). Question a) Find the three roots of the equation $(w+5)(w+8)(w+9)=360$. b) Let $z_0=\sqrt{-2+6i}$, where $i^2={-1}$. Show that the solutions ...