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)

0
votes
0answers
7 views

Small roots of multivariate polynomials

I'm looking for a program/algorithm (or even "theory"?) which checks if a given multivariate polynomial, say $P(x_1, \dots, x_n)$ has a \textit{real} root in some given region, say a closed ...
0
votes
1answer
17 views

Bounding the Roots of a Complex-Valued Function

Roots: $Z_1$= $\frac{v(1+ \alpha)+ \sqrt{v^2(1+\alpha)^2 -4 \alpha}}{2}$ $Z_2$= $\frac{v(1+ \alpha)- \sqrt{v^2(1+\alpha)^2 -4 \alpha}}{2}$ It is clear that $|Z_2| \leq|Z_1|$ However I'm stuck on ...
3
votes
4answers
73 views

Show that $8x^4 −16x^3 +16x^2 −8x+k = 0$ has at least one non-real root for all real $k$. Find the sum of the non-real roots

Show that $8x^4 −16x^3 +16x^2 −8x+k = 0$ has at least one non-real root for all real $k$. Find the sum of the non-real roots. Since this polynomial looks so symmetric, I think factoring it might ...
1
vote
1answer
32 views

Finding the number of solutions of the equation $2x^5- 6x^3 + 2x = 4x^4 - 6x^2 +1$ in the interval $I = [-2, 2]$

I have to find the number of solutions for the following equation on the interval $I=[-2,2]$ $$2x^5- 6x^3 + 2x = 4x^4 - 6x^2 +1$$ Now I know I have to put them all on one side and then use the ...
0
votes
1answer
51 views

Show that no ring containing R can contain a root of g(x) = 3x +1

Show that if $R = \mathbb Z_6$ and $g(x) = 3x + 1 ∈ R[x]$, then $R[x]/(g(x)R[x])$ does not contain a root of $g(x)$. More generally, show that no ring containing $R$ can contain a root of $g(x)$. ...
1
vote
0answers
17 views

Does the resolvent cubic of the quartic equation always have at least 1 positive real root

I have written some code to solve for the roots of a 4th order polynomial, and in the process, I noticed that the resolvent cubic always has at least one positive real root. I can't find anywhere ...
2
votes
2answers
50 views

Roots of trigonometric function.

I have a function: $ f(x) = (ax+b) \cdot \sin(x) + (cx+d) \cdot \cos(x) + e$ for which I want to determine the roots. I know that for $ax \cdot \sin(x) + cx \cdot \cos(x)$ the roots are ...
0
votes
1answer
39 views

why are the Bisection and Newton Method for finding roots complementary to each other?

my lecture note states that the bisection and newton method for finding roots are most of the time complementary to each other but I can not figure out why. I have basic understanding of both of the ...
3
votes
1answer
52 views

Closed form for the sole positive root of the polynomial ${x^\alpha } + {x^{\alpha - 1}} + \cdots + {x^3} + {x^2} -p$, $p > 0$

For a study I'm making about the minimum and maximum radial values of bounded orbits in a central force system with general power law forces, I came across this special polynomial equation: ...
1
vote
4answers
53 views

finding real roots by way of complex

I was given $$x^4 + 1$$ and was told to find its real factors. I found the $((x^2 + i)((x^2 - i))$ complex factors but am lost as to how the problem should be approached. My teacher first found 4 ...
3
votes
2answers
20 views

Finding a root approach with a polynomial

So, i'm solving last's year's exams in Mathematical Analysis and i've found one interesting. It says: The equation $e^{-4x}=5x^2$ has one root close to (nearby) 0. By approaching $e^{-4x}$(close to ...
1
vote
0answers
21 views

Zeros of this function?

Let $$f(z)=\gamma + z^{\beta_2-\beta_1}$$ where $\gamma\in \mathbb{R}$, $\beta_1\in \mathbb{Z}$, $\beta_2 \in \mathbb{Z}$ and $\beta_2 > \beta_1$. The variable $z$ takes complex values. Is there a ...
7
votes
1answer
105 views

Roots of iterations of polynomials

Let $f \in \Bbb Q[X]$ a polynomial, and let denote by $f^n$ the composition $\underbrace{f \circ \cdots \circ f}_{n \text{ times }}$. Let $R(f^n) \subset \Bbb C$ the roots of $f^n$. I'm interested in ...
1
vote
2answers
38 views

Find multiple roots of a three-dim system

Consider the three equations $$ y-x^2=0,\quad z+xy=0,\quad -y-z+x^2-xy+y^2+z^2-x^4=0. $$ How can I find multiple roots of this? Is it allowed to reduce the system as far as possible and then to find ...
2
votes
0answers
72 views
+50

how to show this function has zeros interlacing and including those of Riemann zeta

Let $\chi (t) = H \left( - \frac{i}{2} (2 t - 1) \right) = \dfrac{4 i \pi \zeta (t) \left( \left( \ddot{\Psi} \left(\frac{t}{2} \right) - \ddot{\Psi} \left( \frac{1}{2} - \frac{t}{2} \right)\right) ...
0
votes
1answer
55 views

Is it possible to solve the following equation without using the Rational Root Theorem?

Given $f(x)=x^4+2x^3+2x^2-2x-3$, where $x-1$ is a factor of $f(x)$, how is it possible to solve $f(x)$ without the Rational Root Theorem? Here's my progress: $$f(x)=x^4+2x^3+2x^2-2x-3$$ ...
11
votes
2answers
189 views

Is $x$ irrational when $2^{x}+3^{x}=6$?

Is $x$ rational or irrational when $2^{x}+3^{x}=6$. How to show that?
2
votes
2answers
179 views

How to find $x$ when $2^{x}+3^{x}=6$?

$$2^{x}+3^{x}=6$$ How to find the real number x? I mean it's approximately $1.19$ bur can we write $x$ as the form of $a, b, c$ when $a^{x}+b^{x}=c$ in general. Maybe an infinite sum?
3
votes
1answer
29 views

Rational Points on Fibonacci-like Sequence of Polynomials

Let $\{a_n\}$ be a sequence of polynomials in $\mathbb{Q}[x,y]$ with $a_0=0,a_1=1$, and $$a_n=xa_{n-1}+ya_{n-2}$$ The first few look like $$a_3:y+x^2$$ $$a_4:2xy+x^3=x(2y+x^2)$$ $$a_5:y^2+3x^2y+x^4$$ ...
4
votes
3answers
45 views

Find the two values of $k$ for which $2x^3-9x^2+12x-k$ has a double real root.

Find the two values of $k$ for which $2x^3-9x^2+12x-k$ has a double real root. I've found one method which is to equate $$2x^3-9x^2+12x-k=2(x-r)^2(x-c)$$ Expanding and equating coefficients I ...
0
votes
1answer
72 views

Solution to the equation $x^3-3=2\sqrt{x+2}$

Solve the equation $x^3-3=2\sqrt{x+2}$. I have tried to let $t=\sqrt{x+2}$ then we have $$\begin{cases} x^3-3&=2t \tag 1\\ t^2 &=x+2 \end{cases}$$ But I've stuck here... Any help ...
3
votes
2answers
40 views

All zeroes of monic cubic $x^3+ax^2+bx+c$ are negative reals and $a\lt3$. Range of $b+c$?

$a,b,c$ are real numbers. I have to find the range of values of $b+c$. So, I started off by assuming $\alpha , \beta , \gamma$ as the roots. This gives us $\alpha \beta \gamma = -c$ and ...
0
votes
1answer
42 views

Prove that the equation $sin(x) = ax + b$ has at least one real root

I came across a question earlier this day, that I did not manage to solve. I have been asked to prove that the equation $\sin(x) = ax + b$ has at least one real root, for all $a, b$, where: 1) $a$ is ...
1
vote
1answer
19 views

Roots of polynomials combined with Trigonometric Functions

If $$ f(x) = x^2 + ax + d \cos x $$, where $a$ is an integer and $d$ is a real number, what are all possible values of the tuple $(a,d)$ such that $f(x)$ and $f(f(x))$ have the same set of real roots? ...
0
votes
1answer
22 views

Simplifying Rational Expressions in a Finite Field Extension

In Dummit and Foote's textbook one of the exercises is: Let $\theta$ be a root of $x^3-2x-2$ over $\mathbb{Q}$. Compute $\frac{1+\theta}{1+\theta+\theta^2}$ in $\mathbb{Q}(\theta)$. My approach ...
1
vote
2answers
49 views

magnitude of vector in algebra

I am trying to solve the following equation for x, in plain algebra this was easy $ y = x - \frac{1}{ x} $ $ x^{2} - yx - 1 = 0 $ $ x = \frac{-y \pm \sqrt (y^{2} + 4)}{2} $ However, throwing ...
1
vote
3answers
60 views

Find $\alpha^3 + \beta^3$ which are roots of a quadratic equation.

I have a question. Given a quadratic polynomial, $ax^2 +bx+c$, and having roots $\alpha$ and $\beta$. Find $\alpha^3+\beta^3$. Also find $\frac1\alpha^3+\frac1\beta^3$ I don't know how to proceed. ...
0
votes
1answer
52 views

Find a root of f(x) = 0, arccos & arcsin

Can someone please help me with this question? Let $f(x) = 2\arccos(\frac{x}{2}) + 6\arcsin(\frac{3}{2x}) - 2 \pi$ Find a root of $f(x) = 0$, that is a point x where $f(x) = 0$.
1
vote
0answers
26 views

Companion matrix of bivariate polynomial

A polynomial in one variable can be expressed as a companion matrix, of which the eigenvalues are the roots of the polynomial and which can be found by using e.g. QR decomposition or power iteration. ...
4
votes
1answer
99 views

root pattern of second degree polynomial

I'm considering the following 2nd degree polynomial for the case where the roots are complex conjugate. $ P(z) = z^2 + (f^2 + f q -2)z + (1 - f q) = (z - z_1) (z - z^*_1) $ where f and q are real ...
4
votes
3answers
44 views

What is the extraneous solution of $\sqrt a=a-6$?

What is the extraneous solution of $$\sqrt a=a-6$$ The roots are $9$ and $4$. So I'm assuming that $4$ is the extraneous solution because when you plug it in to the equation you wind up with $2=-2$. ...
4
votes
3answers
58 views

Is this quadratic pointing up or down? How do I know?

The equation is $-2x^2 + 4x + 30 = 0$. I simplified it to $-2(x^2 - 2x - 15)$. To know if it points up, I need to look at $ax^2$, and if $a > 0$ it is up and if $a$ is $< 0$ it is down. ...
1
vote
3answers
29 views

Which form of this quadratic do i use to solve intercept and range?

So my equation is: $-2x^2 + 4x + 30 = 0$ If I use this form to look at my y intercept, it will be 30. However, once I simplify it to: $x^2 - 2x - 15$, then my y intercept will be $-15$. Which one do ...
-1
votes
3answers
41 views

Solve for $x $(quadratic)

$$ 0=0.001 + \frac{-0.0018 x+0.009 x^2}{\left(\sqrt{0.04 - x^2}\right)^3}$$ Can't seem to figure out a way how to.
1
vote
1answer
28 views

What is the equation of the bottom half of the parabola $x + (y - 2)^2 = 0$?

A parabola has the equation: $$x + (y - 2)^2 = 0$$ I can't find the $y$ without getting the equation into some weird recursion.
2
votes
0answers
52 views

The convergence of the fixed-point iteration for solving a cubic equation

I have a third-grade polynomial of the form $Ax^3+Bx+C$ and I want to find its roots. I cannot use Gauss to guess the first root and it is not trivial, so I try this: $0=Ax^3+Bx+C$ and for a given ...
1
vote
0answers
49 views

Fastest way of find roots of polynomial defined over a finite field

Suppose we have polynomial $G(x)$ of degree $d$, where $d$ is a large value (e.g. $10^6$). The polynomial is defined over a finite field $\mathbb{F}_p$ for a large prime number $p$ (e.g. $p$ is ...
0
votes
1answer
21 views

Testing if a polynomial has roots within a radius/range

Is there a way to test if a high-order polynomials has any roots within a radius r of a specified point? I need this so that I can find all the complex roots of the following system for arbitrary ...
1
vote
2answers
26 views

Finding roots of cubic (trig)

The question is By putting $x$ $=$ $\frac 23 cos (\theta)$ Find the exact roots of the equation in terms of $\pi$ $$ 27x^3 - 9x = 1 $$ What I have attempted: $$ ...
4
votes
3answers
45 views

Compute all roots of $(-8)^{\frac{1}{3}}$

$$(-8)^{\frac{1}{3}}$$ The problem states to compute all roots of the complex number above. Below is my attempt, but my inquiries are if I did it right and why it doesn't match Wolfram. Wolfram only ...
0
votes
2answers
70 views

Compute all the roots (complex number problem)

$$ (-1+i)^{\frac{1}{3}} $$ Below is what I've attempted, but I'm not 100% positive if it's right. Also, and more importantly, how do I know if I've computed ALL of the roots of a complex number? ...
2
votes
1answer
61 views

Roots of $x^3-x+1$

I am trying to find nice explicit formulas for the roots of the polynomial $x^3-x+1$. Is there some clever way to write down the roots in a reasonably easy way? I found the roots, but my expressions ...
-1
votes
1answer
43 views

$a,b,x,y$: real number and satisfy $(x-a)^2+y^2=(y-b)^2+x^2=a^2+b^2$ [on hold]

I would appreciate if somebody could help me with the following problem: Q: $a,b,x,y$: real number and satisfy $$(x-a)^2+y^2=(y-b)^2+x^2=a^2+b^2$$ Show that $$(x,y)=\Bigg( \left(\frac{1}{2} ...
2
votes
2answers
28 views

Quartic with $4 $ equidistant roots

Today I got the problem $(x^2 -1)(x^2 -4)=k$, and I have no idea how to prove this algebraically. $K$ is a real, non-zero number that makes the equation have $4$ distinct real equidistant roots. Solve ...
0
votes
0answers
24 views

Solving $\mathbb{E}_X[\log (a + X)] = b$ to $a$

I'm trying to solve the equation $$ \mathbb{E}_X[\log (a + X)] = b$$ to $a$ where $b>0$ and $X$ is a positive random variable distributed according to $P(X)$. The solution can be written as a ...
0
votes
2answers
37 views

Sum of Reciprocal of a sum of square roots

I am trying to find a closed form expression of the following sum: $\sum_{k=1}^{N}\frac{1}{\sqrt{k}+\sqrt{k+3}},\; N>1.$ I tried to determine whether methods used for evaluating more conventional ...
0
votes
1answer
37 views

Partial fraction integration with unclear roots

Let's look at a simple example like $\frac{1}{x^3+2x+1}$ here. We know that the denominator has a real root between $0$ and $-1$ (could go closer, but that's not the point). By the concept of slope of ...
1
vote
2answers
52 views

Set of solutions to $\sqrt{x^2}=-x$

The question is: The set of all real numbers x such that $$\sqrt{x^2}=-x$$ consists of: A: zero only B: nonpositive real numbers only C: positive real numbers only D: all real ...
0
votes
0answers
16 views

Counting zeros of trigonometric functions of functions

There is not any context for this problem, it is a general question: In General: If you are given a trigonometric function of a function $\sin(f(x))$ with an arbitrary function f(x), is there any ...
2
votes
1answer
43 views

When is it more appropriate to use multiple branches for roots?

Often times, more often then I believe should be, I will see a question under the tag algebra-precalculus that asks about odd solutions, which many answers will note to be extraneous solutions, when ...