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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Proof of Ramanujan's famous cubic identity

Ramanujan found that given a polynomial $y=x^3+ax^2+bx+x$, one can find $\sqrt[3]{u+x_1}+\sqrt[3]{u+x_2}+\sqrt[3]{u+x_3}=\sqrt[3]{3\sqrt[6]{d}+w}$ where $$d=\frac {4(a^2-3b)^3-(2a^3-9ab+27c)^2}{27}$$$$...
2
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2answers
93 views

Does this alternating sum of roots converge to $\sqrt2$?

This problem arose from what I'm hesitant to call an investigation into a certain type of "quadrature". Starting with the unit disk, I partition it into $p$ pieces by cutting the disk with vertical ...
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1answer
30 views

Real roots for exponential-polynomial equations

I am trying to find the number of real roots of an equation such as $k_1 x e^x-e^{k_2 x}-k_3x+k_4=0.$ Setting the first derivative equals to zero is analytically unsolvable, unfortunately. Do you ...
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1answer
42 views

Are minimizing a function and root finding the same?

What is the relationship between minimizing a function and finding a root of an equation? Are the the same? I know in both problem we have similar algorithms, such as gradient decent, or newton's ...
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1answer
24 views

Brumer quintic polynomials - is there a general formula for 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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1answer
21 views

Iteration: Approximation and Errors, finding all possible iterative arrangements

I am looking at a relatively simple problem to reiterate: $x^4=e^x$ I've found 5 different possible forms 1: $x_{r+1}=\frac{e^x}{x^3}$ 2: $x_{r+1}=(\frac{e^x}{x^2})^{0.5}$ 3: $x_{r+1}=(\frac{e^x}...
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2answers
85 views

'Strange' trigonometric roots of $x^5-4x^4+2x^3+5x^2-2x-1$ - could someone explain?

This quintic equation has $5$ real roots: $$x^5-4x^4+2x^3+5x^2-2x-1=0 \tag{1}$$ The roots are, from left to right: $$x_1=\frac{\cos \frac{19}{22} \pi}{\cos \frac{1}{22} \pi}$$ $$x_2=\frac{\cos \...
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2answers
38 views

Symmetric roots of polynomial

Let $\alpha_1, \alpha_2, \alpha_3$ be the roots of the polynomial $x^3 - x^2 + 2x - 3$ $\in \mathbb{C}[x]$. Calculate $\alpha_1^3 + \alpha_2^3 + \alpha_3^3$. What to do here exactly? I already ...
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0answers
83 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-...
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1answer
34 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}...
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2answers
53 views

How to find the roots of this 4th order polynomial?

Can someone explain how to factor/find roots to this 4th order polynomial: $$ s^4 + 14s^3 +45s^2 +650s + 1800 = 0 $$ It's such a nightmare. I've been stuck for hours, any help would be appreciated :)...
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3answers
26 views

Find the algebric form of the zeros(roots) of the following polynomial: $\left(\:z^2+iz+2\right)\left(z^3-8i\right)$

Good morning to everyone. I don't know how to find the zeros(roots) of the following polynomial function: $$\left(\:z^2+iz+2\right)\left(z^3-8i\right)$$. What I've tried: The zero(root) of the second ...
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3answers
112 views

Show the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle

So I need a little help with the following: Considering separately the cases of real and complex roots show that the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle (i.e....
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2answers
280 views

Why Rational Root Theorem only works with integers

Why does the rational root theorem only work when the polynomial has integer coefficients?
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0answers
23 views

Factoring polynomials when roots are external to the ring

I shall avoid maths script since I'm typing on a mobile, anyway I think I can do without. I have a question about factoring polynomials over a ring. Let's call R the ring in question. It is clear to ...
2
votes
2answers
39 views

Which is bigger and how to check?

I have $\dfrac{20\sqrt3-23}3$ and $\dfrac{\sqrt6+12}3$ but I don't know how to check which is bigger and which is smaller. Can someone help me?
2
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5answers
57 views

Solving $2x^4+x^3-11x^2+x+2 = 0$ [duplicate]

I am having no idea how I can solve this problem. I need help! Here's the problem $2x^4+x^3-11x^2+x+2 = 0$ I am learning Quadratic Expressions and this is what I need to solve, and I can't ...
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0answers
32 views

Understanding a calculation deduced for the function $\pi^{-s/2}\Gamma(s/2)\zeta(s)$

With my current knowledges I don't know if this is a bad question, but since I am interesting in this kind of calculations I want to ask you, if I was wrong or if if my statement is obvious. From ...
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1answer
97 views

The other $47$ roots of the minimal polynomial for $\cos 1 ^\circ$

The minimal polynomial for $x=\cos 1 ^\circ=\cos \frac{\pi}{180}$ is: $$281474976710656 x^{48}-3377699720527872 x^{46}+18999560927969280 x^{44}- \\ -66568831992070144 x^{42}+162828875980603392 x^{40}-...
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2answers
108 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(\...
3
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2answers
91 views

If $x\in\mathbb R$, solve $x^{\lfloor x\rfloor}=\frac{9}{2}$, where $\lfloor x\rfloor$ is the integer part of $x$.

If $x\in\mathbb R$, solve $x^{\lfloor x\rfloor}=\frac{9}{2}$, where $\lfloor x\rfloor$ is the integer part of $x$. Of course, $x=\lfloor x\rfloor+\{x\}$, where $\{x\}$ is the fractional part of $x$. ...
3
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5answers
86 views

The roots of the equation $x^2 - 6x + 7 = 0$ are $α$ and $β$. Find the equation with roots $α + 1/β$ and $β + 1/α$.

Quadratic equation question, as specified in the title. The roots of the equation $x^2 - 6x + 7 = 0$ are $α$ and $β$. Find the equation with roots $α + \frac{1}{β}$ and $β + \frac{1}{α}$. I ...
3
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1answer
67 views

Find real parts of the complex roots of this $9^{th}$ order polynomial in explicit form

I have a following polynomial. (See WolframAlpha ): $$x^9-6x^8+14x^7-16x^6+36x^5-56x^4+ 24x^3-320x+\frac{640}{9}=0 \tag{1}$$ Wolfram says that $(1)$ has three real roots and three pairs of complex ...
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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 it not have 3 real roots and 1 complex?
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3answers
54 views

Let $f = 2x^4 + 2(a - 1)x^3 + (a^2 + 3)x^2 + bx + c.$ ,Find out $a, b, c ∈ R$ and its roots knowing that all roots are real.

Let $f = 2x^4 + 2(a - 1)x^3 + (a^2 + 3)x^2 + bx + c.$ Find out $a, b, c ∈ R$ and its roots knowing that all roots are real. The first thing that came into my mind was to use vieta's ...
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1answer
60 views

Zeros of $f_n(z)=1+\frac{1}{z}+\frac{1}{2!z^2}+…+\frac{1}{n!z^n}$ are in $B_{\varepsilon}(0)$

I want to prove that for every $\varepsilon >0$ there is a $N\in\mathbb{N}$ so that for every $n\ge N$ all zeros of $$f_n(z)=1+\frac{1}{z}+\frac{1}{2!z^2}+...+\frac{1}{n!z^n}$$ are in $B_{\...
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0answers
35 views

Existence of solution of equation involving normal distribution

I've tried to show that the following equation has a solution: \begin{equation*} g(x)=\left[1-\left(2\int _{\mu}^{x}f(y)dy\right)^2\right]-8xf(x)\int _{\mu}^{x}f(y)dy=0, \end{equation*} where $f(x)$ ...
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3answers
73 views

prove that the $5$th root of $r$ is irrational if $r$ is irrational [closed]

I am trying to learn mathematics for computer science in own efforts. I got this task to prove that $\sqrt[5]{r}$ is irrational, given that $r$ is irrational. Normally if I had to prove that $\sqrt{2}...
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2answers
43 views

Logistic regression for football results - Estimating coefficient through maximum likelihood

Consider two football teams $V$ and $L$ with strengths $W_V$ and $W_L$, respectively. Let's assume that the draw probability $\mathbb{P}(Draw)$ is known. Then this model is supposed to give estimates ...
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1answer
25 views

Singularity of $f(z)=f(z+a)=f(z+b),\ a,b\in\mathbb C$ does not exist

Let $f(z)=f(z+a)=f(z+b),\ a,b\in\mathbb C$ be a not constant meromorphic function, which is periodic and let $a,b$ be lineare independent. Show that $f$ has no zeros or singularities on the boundary ...
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2answers
54 views

Question regarding roots of a cubic polynomial

If $\alpha$, $\beta$ and $\gamma$ are the roots of a cubic equation with $$\alpha + \beta + \gamma = 1$$ $$\alpha^2 + \beta^2 + \gamma^2 = 2$$ $$\alpha^3 + \beta^3 + \gamma^3 = 3$$ Then find the ...
2
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1answer
39 views

Prove that $f(x+z)$ has $4$ roots $\pm \alpha$ and $\pm \beta$

Let $a$ be a real parameter such that $$f_a(x)= x^4-6x^3+11ax^2-3(2a^2+3a-3)x+1$$ has has four distinct complex roots, that form a parallelogram when plotted on the Argand diagram. Prove That $...
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1answer
55 views

Are there more quadratics with real roots or more with complex roots? Or the same?

Consider all quadratic equations with real coefficients: $$y=ax^2+bx+c \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,, a,b,c \in \mathbb{R}, \, a≠0 $$ I was wondering whether if more of them have real roots, more ...
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1answer
41 views

non-complex cubic roots formula?

Suppose we have a cubic equation $$ ax^3 +bx^2 +cx +d =0 $$ for which we know that all three distinct roots are real. Do we have a formula for them that does not involve complex roots of unity? The ...
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2answers
17 views

Complex Conjugate roots with non real coefficients

I understand that a polynomial with real coefficients must have complex conjugate roots (if complex roots exist) Is it possible for a polynomial with non-real coefficients to have complex conjugate ...
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2answers
413 views

Numerical root finding of function with unknown parameters

I have a multivariate function of which I want to find one of (or all) its roots. However, besides the variables, it also depends on a bunch of parameters. Now I only want to find roots which are ...
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0answers
48 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 ...
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1answer
55 views

How to compute equation with exponents?

I want to find $a$, where that term satisfies this equation: $$a + a(1-a) + a(1-a)^2 + \cdots + a(1-a)^{15} = 0.5$$ I could write that as a sum from 0 to 15, but still it is unclear to me how should ...
3
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3answers
105 views

Need help solving $x^4-3x^3-11x^2+3x+10=0$

Solve $x^4-3x^3-11x^2+3x+10=0$ I have tried to solve this equation using 'general formula from roots' from https://en.wikipedia.org/wiki/Quartic_function. $$ax^4+bx^3+cx^2+dx+e=0$$ $$x_{1,2}=-\frac ...
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0answers
32 views

Specific fucnction has 11 different zeros

Let $f : \mathbb{C} \to \mathbb{C}$ be given by $$ f(z) = z^{11} + 4 e^{z + 1} - 2 $$ Show that $f$ has 11 different zeros in the annulus $\{z \in \mathbb{C} : 1 < |z| < 3\}$. This is an old ...
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1answer
21 views

Solving a characteristic Polynomial of the Hilbert Matrix

I need to find the eigenvalues of the following characteristic polynomial but I can't seem to successfully find the roots of the equation: $P[λ]$ = $λ^5$ - $563/315λ^4$ + $0.3476λ^3$ - $0.0038λ^2$ ...
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7answers
12k views

How to prove that a polynomial of degree $n$ has at most $n$ roots?

How can I prove, that a polynomial function $$f(x) = \sum_{0\le k \le n}a_k x^k\qquad n\in\mathbb N,\ a_k\in\mathbb C$$ is zero for at most $n$ different values of $x$? (Except $n=0$ where $f(x)$ is $...
4
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2answers
88 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 $-2\cos\left(\frac {4\pi}{9}\right)$ ...
0
votes
2answers
42 views

Function satisfying inequality has no root

Let $f$ be an entire function such that, for all $z \in \mathbb{C}$ with $|z| > 1$, $$ |f'(z)| < \frac{|f(z)|}{|z|^2} < 1 $$ Show that there is no $a \in \mathbb{C}$ such that $f(a) = 0$. ...
0
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1answer
49 views

Show that … Has No Real Roots

When $f(x) = 3x-4$ and $g(x) = \frac{5}{3-x}$, Question 1: Find the value of x for which fg(x) = 5 Question 2: Show that the equation $f^{-1}(x) = g^{-1}(x)$ has no real roots. I understand that ...
3
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0answers
40 views

How to solve a quintic polynomial equation?

I know that not all quintics are solvable. But how do I identify the class of solvable ones?
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2answers
46 views

How can I imagine double/repeated root of a quadratic equation?

A quadratic equation such as $(x-2)^2=0$ has a repeated root $(2,2)$. A lot of things in math (equations, matrixes and so) can be nicely drawn on a $2D, 3D$ etc plane (with $x$-$y$ axis). I mean, I ...
13
votes
3answers
642 views

How would you find the exact roots of this equation?

My friend asked me what the roots of $y=x^3+x^2-2x-1$ was. I didn't really know and when I graphed it, it had no integer solutions. So I asked him what the answer was, and he said that the $3$ roots ...
1
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2answers
63 views

Finding an interval in which all the real roots of a polynomial lie

I'm making a program which uses simple bisection method to find the roots of a polynomial. For me to implement this method, I need a rough interval where it can be said with absolute certainty that ...