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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1answer
25 views

Express $c$ and $d$ in terms of $m$ where $c$ and $d$ are zeroes of $f$ where $m > -2$

Let $$f(x) = x^2 - mx -(6m^2+25m+25)$$ where $m > - 2$ It can be shown that $f(x)$ has two zeroes. Suppose we have $c,d \in \mathbb R$ s.t. $c < d$ and $f(c) = f(d) = 0$, express $c$ and $d$ ...
0
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3answers
66 views

Finding the root of an indefinite polynomial

$0 = (a-n) x^{n-1} + ax^{n-2} + ax^{n-3} + \cdots + ax + a$ What is $x$ in terms of $a$ and $n$? I don't even know what this form of polynomial is called.
0
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2answers
27 views

Squareroot of complex number to the square $\sqrt{z^2}$

I have to calculate $\sqrt{z^2}$ an I am confused about how to procede. I thought about introducing $z=|z|\exp(i\phi+2\pi k) \implies z^2=|z|^2\exp(2i\phi+4\pi k)$. Hence, $$\sqrt{z^2}=\sqrt{|z|^2\...
6
votes
2answers
265 views

Ways to find irrational roots of an n degree polynomial

I am trying to write a program to find the roots a given polynomial of degree N, with the form $$ A_{0}X^{N}+A_{1}X^{N-1}+A_{2}X^{N-2}+A_{3}X^{N-3}+...+A_{N} $$ I know that if there are rational ...
0
votes
2answers
32 views

Number of real roots of $\frac{a_1}{a_1-x}+\frac{a_2}{a_2-x}+…+\frac{a_n}{a_n-x}=2016$ for $0<a_1<…<a_n$?

Does it have exactly $n$ roots? Would replacing the R.H.S. of the equation with any other real number change the outcome? I can show that the equation has no complex roots. But how to find the number ...
1
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0answers
43 views

For what value of $(a+b)$ will all roots of $f(x)=x^4-8x^3+ax^2+bx+16$ be positive?

I was thinking of using Descartes' rule of signs, from which I find there are at most 2 positive roots and 2 negative roots of the given equation. Also, $f(\infty)>0$ and $f(0)>0$ imply that ...
1
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2answers
35 views

Find the stopping distance

(original question, see edits below for full context) After much frustration, I have figured out a function which maps velocity during acceleration/deceleration for my project. $$\text{velocity} =s+\...
1
vote
1answer
76 views

Are all the zeros of $1-a_2x^2+a_4x^4-a_6x^6+\cdots$ real for $a_{2n}>a_{2(n+1)}$ with $a_{2n+1}=0$ and $a_{2n}>0$?

This question is related to a previous question of mine. I was not pleased about the conditions I provided there. I had something different in mind but I failed in stating it. So here are the ...
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2answers
64 views
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4answers
867 views

Understanding non-solvable algebraic numbers

Background We know from Galois theory that the zeros of a polynomial with rational coefficients whose Galois group is solvable can be expressed in a formula that involves rational powers of the ...
0
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1answer
43 views

Comparing the roots of two increasing functions

For any $0 \leq x \leq y \leq 1$, define $f(y;x):=\frac{y^2}{2}-\frac{2 y^3}{3}+\frac{y^4}{4} - \frac{x^2}{2} + \frac{x^3}{3}$ and $g(y;x):=\frac{y^2}{3}-\frac{2 y^3}{4}+\frac{y^4}{5} - \frac{x^2}{3} +...
-1
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0answers
31 views

Solve Equation with max integer [on hold]

Solve please $\dfrac{\left[\sqrt{x-[x ]}\right]}{(x+3)(x+4)}\ \geq0$ edit
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0answers
26 views

zeros of special sequence of polynomials

While answering this question, I make one question. Define a sequence of polynomials as \begin{align} p_{n}(x)=\sum_{r=0}^{\lfloor (n+1)/2\rfloor} (-1)^{r}\binom{n+1-r}{r} x^{n-r}. \end{align} I used ...
3
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0answers
35 views

how to find the the maximum of an implicit function

I have an implicit function and I would like to find the value of $h$ that maximizes $R$, i.e, I want to find $h$ that satisfies $\frac{\partial R}{\partial h} = 0$. The function is, $C=\frac{A}{1+\...
1
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3answers
73 views

Prove that $x^4+2x^2-6x+2=0$ when $x\in\mathbb{R}$ has exactly two solutions

Show that $x^4+2x^2-6x+2=0$ when $x\in\mathbb{R}$ has exactly two solutions. I first showed that the IVT guarantees that there exists at least one zero in $(0,1)$ and at least one zero in $(1,2)$. I ...
12
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4answers
1k views

Why can a quartic polynomial never have three real and one complex root?

It seems that a quartic polynomial (degree $4$) either can have $0$ real, $1$ real, $2$ real, or $4$ real roots, and the rest is complex roots. Why can't it have $3$ real roots and $1$ complex?
2
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0answers
20 views

How to identify properties of the zeroes of this polynomial? [on hold]

If $f_0(x)=1$, and $f_{n+1}=\frac{d}{dx}((x^2-1)f_n(x))$, prove that every $f_n$ has exactly $n$ distinct zeroes, all located in the interval $(-1,1)$. It's got me stumped, so any help/pointers would ...
2
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3answers
82 views

Find $x_1^n+x_2^n$ on any quadratic equation, general case.

I have a simple quadratic (with $x^2$) equation, x can Be complex too: $$x^2+x+1=0$$ But it could be any equation, the equation above is just an example. I need to compute $x_1^{10}+x_2^{10}$, but ...
8
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5answers
578 views

Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$

I've tried solving $\sqrt[3]{3} =a + b* \sqrt[3]{2}+c* \sqrt[3]{4}$, but there is no obvious contradiction, even when taking the norms/traces of both sides. I can't think of another approach. This is ...
1
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4answers
164 views

Are there real solutions to $x^y = y^x = 3$ where $y \neq x$?

I need to solve the following equation for (x,y) $$x^y = y^x = 3$$ Everytime I run a numerical method for this problem, I get $$ (x,y) = (1.82546...,1.82546..) $$ I expect there to be a solution ...
1
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0answers
35 views

Surjectivity on the image of a annulus

I'm trying to prove the Fundamental Theorem of Algebra as it is done in Birkhoff and MacLane. Unfortunately, I don't have access to the book, only to a sketch. Therefore, I'm filling the gaps myself. ...
0
votes
1answer
20 views

Find the relative width of a guitar fret

There is an equation to find the position of a fret on a guitar fretboard, given the length of a string is given by \begin{eqnarray} d = s – \frac{s}{2 ^ {(n / 12)}}, \end{eqnarray} where $d$ is the ...
0
votes
1answer
37 views

Why has the equation positive root?

Let $x \in \mathbb{R}$ and $\lambda ,{\lambda _0} \in \mathbb{C}$ and $r\in(0,1)$. $w(x) = {\alpha _m}{x^m} + \cdots + {\alpha _1}{x^1} + {\alpha _0}$. $f(\lambda)$ is function such that $f(\...
4
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1answer
57 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. ...
3
votes
0answers
80 views

Is this equation $(n+1)~x^{2n+1}-n~x^{2n}-n=0$ solvable in radicals for some $n \geq 2$?

Consider this polynomial equation: $$(n+1)~x^{2n+1}-n~x^{2n}-n=0,~~~~n \geq 2,~~~n \in \mathbb{N}$$ It's related to another question of mine, but I don't think the context matters here. I'm ...
1
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2answers
66 views

Can a polynomial of $n$ degree have $n+1$ distinct real roots?

Question : Let $f(x) = \sum^n_{k=0}c_kx^k$ be a polynomial function then prove that if $f(x) = 0$ for $n+1$ distinct real values, then every coefficient $c_k$ in $f(x)$ is $0$ , thus $f(x) = 0$ for ...
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0answers
13 views

Connecting First Passage Time to Power Spectrum

Let $f$ be a real function. Is there a connection between The first positive abscissa for which its autocorrelation function is equal to zero (which I call the first passage time, fpt) The largest ...
-1
votes
1answer
30 views

If $w$ is an imaginary cube root of unity, then the polynomial whose roots are $2w+3w^2$ and $2w^2 + 3w$ is?

What polynomial with complex coefficients has the following as its roots? $2w+3w^2$ and $2w^2 + 3w$ I have tried doing this all the ways I know of, still can't get my pen over it... Can you guys ...
4
votes
2answers
101 views

How would you find the roots of $x^3-3x-1 = 0$

I'm not too sure how to tackle this problem. Supposedly, the roots of the equation are $2\cos\left(\frac {\pi}{9}\right),-2\cos\left(\frac {2\pi}{9}\right)$ and $-2\cos\left(\frac {4\pi}{9}\right)$ ...
1
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3answers
64 views

Is there a difference between $\sqrt{x+2}+x=0$ and $x^2-x-2=0$

Is there a difference between $\sqrt{x+2}+x=0$ and $x^2-x-2=0$ Solutions are $x=2$ or $x=-1$. But $x=2$ does not satisfy $\sqrt{x+2}+x=0$.Because $\sqrt{4}+2 \neq0$ So does it mean that they are ...
6
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2answers
137 views

Roots of $y=x^3+x^2-6x-7$

I'm wondering if there is a mathematical way of finding the roots of $y=x^3+x^2-6x-7$? Supposedly, the roots are $2\cos\left(\frac {4\pi}{19}\right)+2\cos\left(\frac {6\pi}{19}\right)+2\cos\left(\...
0
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2answers
50 views

Finding a value based on the roots of an equation

So I saw this question recently: Known $a^2+b^2+6a-12b+45=0$. Find $\dfrac{b-a}{b+a}$. I tried to factorize it but I don't really know how. Can someone help me with this?
10
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2answers
138 views

Only five solvable quintic equations of the form $x^5+ax^2+b=0$? What are their solutions?

According to Wikipedia there is only five solvable quintic equations of the form $x^5+ax^2+b=0,~~a,b \in \mathbb{Q}$ (up to a scaling constant $s$). $$x^5-2s^3x^2-\frac{s^5}{5}=0 $$ $$ x^5-100s^3x^2-...
6
votes
3answers
640 views

Finding the roots (contest math)

So the problem is : $x^4-4x^3-x^2-8x+4=0$, find all solutions A tip that I have gotten, is to divide both sides by $x^2$. I've tried so, but I do not manage to see any further. Do anyone know how ...
2
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3answers
176 views

Hi! Just wondering if any one can help me out with this roots question? [closed]

(i). Factorise $z^2 - 5z + 6$ and hence, solve the equation $ z^2 - 5z + 6 = 0$ (ii). Show that $z^2 - 5z + 6$ is a factor of $z^3 + (-4 + i)z^2 + (1 - 5i)z + 6(1 + i)$. (iii). Find the three roots ...
3
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3answers
119 views

Brumer quintic polynomials - is there a general formula for the roots?

There exist a family of quintic polynomials, called Brumer's polynomials (or Kondo-Brumer), which have the form: $$x^5+(a-3)x^4+(-a+b+3)x^3+(a^2-a-1-2b)x^2+bx+a,~~~a,b \in \mathbb{Q}$$ According to ...
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2answers
62 views

A solvable quintic with the root $x=(\sqrt[5]{p}+\sqrt[5]{q})^5$ - what are the other roots?

I derived a two parameter quintic equation with the root: $$x=(\sqrt[5]{p}+\sqrt[5]{q})^5,~~~~~p,q \in \mathbb{Q}$$ $$\color{blue}{x^5}-5(p+q)\color{blue}{x^4}+5(2p^2-121pq+2q^2)\color{blue}{x^3}...
3
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0answers
28 views

Finding all real roots of an equation

I am looking for a computational method to find real roots of a function. The function looks like this: $$F(x) = \sum_{i=0}^n \frac{k_i}{\sqrt {(x - x_i)^2 + c_i^2}}.$$ I would like to use something ...
1
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1answer
35 views

$F(x,t)=a_n(t)x^n+ \ldots +a_1(t)x+a_0(t)$. Show that $F(\cdot , t_0)$ has exactly one zero using the Implicit Function Theorem

$F(x,t)=a_n(t)x^n+ \ldots +a_1(t)x+a_0(t)$ is a through $t$ parametrized family of polynominals. $a_i : I \to \Bbb R \:\:\:\mathrm{ are }\: \mathcal C^k$- functions with $k \ge 1$. Let $x_0$ be a zero ...
0
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1answer
36 views

Find m so that the equation has integer solutions

We are given the following equation: $(m+1)x^2-(2m+1)x-2m=0$, where $m\neq-1$. We have to find all integers $m$ so that the equation above has integer solutions. I know that $m=0$ and $m=-2$ satisfy ...
1
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1answer
66 views

Simplifying $\frac{4\sqrt{7}}{3}\cos{\left(\frac{1}{3}\arccos{\frac{1}{\sqrt{28}}}\right)}+\frac{1}{3}$

I was finding the roots of the polynomial $y=x^3-x^2-9x+1$. And I got one of the roots of the polynomial to be $$\dfrac{4\sqrt{7}}{3}\cos{\left(\dfrac{1}{3}\arccos{\dfrac{1}{\sqrt{28}}}\right)}+\dfrac{...
3
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0answers
131 views

What methods are known to visualize patterns in the set of real roots of quadratic equations?

I came across a previous awesome question about the visualization of the distribution of polynomial roots and tried to do a simpler version applied to the set of real roots of quadratic equations $ax^...
0
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1answer
533 views

Inverse Quadratic Interpolation and the secant method

I am currently completing a maths project that aims to approximate the roots of functions using MATLAB. The two root finding methods that I have used are inverse quadratic interpolation and the ...
0
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1answer
31 views

non-standard exponential-squared fog attenuation [closed]

I inherited a formula that I'm hoping to simplify. $d = \frac{\sqrt{-\log_2(t)}}{f\sqrt{\ln(2)}}$ Any ideas? Thanks, Jason EDIT (for context): This formula determines the exponent for exponential-...
0
votes
3answers
34 views

Localization of roots of complex quadratic equations

Let $a,b,c\in\mathbb C\setminus\{0\}$ be complex numbers such that $$b^2-4ac \neq 0.$$ We consider the equation $$ax^2+bx+c=0.$$ I am interested in general statements about the roots of this equation ...
4
votes
1answer
162 views

Find the roots of $e^x+e^{1/x} + a = 0$

Find the roots of this equation $e^x + e^{1/x} + a = 0$ where $a \in \Bbb R$ Is there any nice formula for this type of equation?
1
vote
1answer
66 views

Trigonometric Roots of a Polynomial

After wondering on this question, I wondered how would you be able to find the roots of a polynomial, in the form $y=x^3+ax^2+bx+c$ if they are the sums of cosines? I'm wondering if it can, too, be ...
0
votes
3answers
53 views

Have $n$ real root then show that ${(n-1)\left(c_{n-1}\right)^2} \geq 2 n c_{n-2} c_n$

What should I do here? I don't even know where to start from. Please help me by giving me a hint. If $$x^{n } - c_{1} x^{n-1}+c_2 x^{n-2} -c_3 x^{n-3}+\cdots+(-1)^{n-1} c_{n-1} x+(-1)^n c_n=0(c_1,...
0
votes
4answers
48 views

Under what conditions will $x^2+bx+c=0$ have both roots real and positive?

Obviously, $x=\frac{-b\pm \sqrt {b^2-4c}}{2}$ and for real roots we must have $b^2-4c\geq 0$. But for what values of $a,b,c$ will the quadratic have both roots positive?