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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4answers
35 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
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2answers
18 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
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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
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1answer
100 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
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1answer
32 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 ...
1
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0answers
53 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
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1answer
54 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
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2answers
187 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
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2answers
177 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
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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
44 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
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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
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2answers
38 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
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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
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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
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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
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3answers
59 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 $f(x) = 0$.
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0answers
23 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
98 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
57 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
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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 ...
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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
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1answer
25 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
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0answers
48 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
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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
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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
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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
60 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 $(x-a)^2+y^2=(y-b)^2+x^2=a^2+b^2$

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
27 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
36 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
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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
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2answers
51 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
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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 ...
0
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0answers
32 views

Finding a mirror point on a parabola

What is the height of the ball at a point of 3 metres beyond where it was thrown, measured horizontally? How far is the ball from where it was thrown when its height has this value again? ...
1
vote
1answer
35 views

Finding all roots of multivariate polynomial using Newton's method

I read that it is possible to find a solution to a nonlinear system of equations using Newton method and Jacobian matrix. But if I understood correctly, this finds just one solution, and which one ...
0
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3answers
50 views

Cubic equation (polynomial)

A cubic polynomial with real coefficients, $a x^3 + b x^2 + c x + d$, has either three real roots, or one real root and a pair of complex conjugate ones. If the latter happens, what is the explicit ...
0
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1answer
17 views

How to guess initial intervals for bisection method in order to reduce the no. of iterations?

SO, A function $f(x)$ is given to me and but the initial intervals are not given. I need to find the root of the equation using Bisection method. Sometimes when I randomly guess the initial interval ...
1
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0answers
35 views

Analytic closed-form solution

I have the following equation: $$\left(\frac{x}{\cosh(x)}\right)^2-x\tanh(x)+\ln\cosh(x)=0$$ and I would like to know if there is some analytic closed form solution. WA gives me two non-zero ...
0
votes
2answers
61 views

Calculating $a_1^4+a_2^4+a_3^4$ of the roots of a polynomial

We have a polynomial $f=X^3+19X^2+12X+3\in\mathbb{C}[X]$ with roots $a_1,a_2,a_3$. What is $a_1^4+a_2^4+a_3^4$? And how do I know that these roots are all different? Edit: How can I show that ...
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3answers
84 views

Functions Mapping Integers to Zero?

I am looking for functions such that: $z∈$ Z ⇔ $f(z)=0$ That is to say, functions that map from Z to the zero set. One example is $f(z)=\sin(πz)$. EDIT: To narrow the possible group of ...
2
votes
0answers
30 views

Finding the roots of a polynomial with limited information

Let $\ f(x) \in \mathbb{R}[x]$ be a 7-th degree polynomial, such that. $$ f(0)=0 \land f(i)=-3i $$ $$ f'(0)=0 \land f'(i)=-21 $$ Find all the complex roots of $ f(x)-3x^7$. Find all possible ...
3
votes
1answer
33 views

Let $a,b,c,d$ be distinct integers such that the equation $(x-a)(x-b)(x-c)(x-d)-9=0$ has an integer root $r$,then find the value of $a+b+c+d-4r.$

Let $a,b,c,d$ be distinct integers such that the equation $(x-a)(x-b)(x-c)(x-d)-9=0$ has an integer root $r$,then find the value of $a+b+c+d-4r.$ As $r$ is the integer root of the equation ...