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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4
votes
2answers
96 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 \...
0
votes
2answers
41 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 ...
1
vote
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}...
0
votes
3answers
29 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 ...
10
votes
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-...
-1
votes
1answer
26 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}...
0
votes
0answers
25 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 ...
1
vote
1answer
46 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 ...
2
votes
3answers
126 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....
2
votes
2answers
48 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
votes
5answers
60 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 ...
0
votes
0answers
34 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 ...
4
votes
1answer
102 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}-...
6
votes
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(\...
3
votes
5answers
102 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
votes
1answer
70 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 ...
1
vote
3answers
55 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 ...
1
vote
1answer
65 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_{\...
12
votes
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?
1
vote
0answers
36 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 $f(x)$ ...
-4
votes
3answers
74 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}...
1
vote
2answers
46 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 ...
0
votes
1answer
26 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 ...
1
vote
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
votes
1answer
40 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 $...
1
vote
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 ...
0
votes
1answer
43 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 ...
0
votes
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 ...
1
vote
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 ...
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 ...
1
vote
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 ...
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
votes
1answer
50 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 ...
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 $-2\cos\left(\frac {4\pi}{9}\right)$ ...
3
votes
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?
0
votes
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 ...
-1
votes
1answer
57 views
13
votes
3answers
700 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
vote
2answers
67 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 ...
1
vote
3answers
82 views

Why does degree determine the amount of zeros?

We just learned about complex numbers in my math class and I have a question. Why does the degree of a polynomial equal the amount of zeros it has? The degree of $f(x) = x^3 - x^2 + x - 1$ is $3$, ...
0
votes
1answer
30 views

The function $\zeta(\frac{1}{2}+it) \left[ \sqrt{2}\left( \cos(t\log 2)+i\sin(t\log 2) \right) -2 \right]$ has a numerical root

Using the complex exponentiation (this is the MathWolrd's Page) one can deduce for $t>0$ $$2^{\frac{1}{2}+it}=\sqrt{2}e^0(\cos(t\log 2)+i\sin(t\log 2)),$$ since $a=2,b=0,c=\frac{1}{2}, d=t$ and $\...
2
votes
1answer
37 views

Falsi regula using maple

Hi I'm using falsi regula algorithm in maple. For first function it worked fine : restart; epsilon := 1e-3: f := x->x^3+x^2-3: a:= 1: b:=2: step:=infinity: while abs(step) >= ...
0
votes
1answer
32 views

Get a matrix of polynomial coefficients from the roots

I've got the polynomial $P(z) = \Phi_0 - \Phi_1z $ defined by the following matrices of coefficients: $$ \begin{eqnarray} \Phi_0 = \left[ \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0.2 & 1 &...
0
votes
2answers
30 views

How to numerically find zeros of a system of first-order differential equations (Airy function)?

To numerically approximate the Airy function y = Ai(x) which satisfies the equation $$ y'' - xy = 0 $$ I converted this second-order diff. eq. into a pair of first order diff eq. and solved them using ...
2
votes
1answer
9 views

How to approximate the order of a root given the base and the power

I need to narrow the range (minimum/maximum) to search for a root given the base and the power, the values are all integers, the base can be very large relative to an always positive power. Is there a ...
1
vote
3answers
31 views

Help with sum and product of roots.

I'm having trouble with a question from my textbook relating to roots of an equation. This is it: Let a and be roots of the equation: $x^2-x-5=0$ Find the value of $(a^2+4b-1)(b^2+4a-1)$, without ...
5
votes
1answer
91 views

Prove that $f(x) = x^4+ax^3+bx^2+cx+d$ does not has all rational roots

The quartic polynomial $f(x) = x^4 + a x^3 + b x^2 + c x + d$ is such that $ad$ is odd and $bc$ is even. Prove that $f(x)$ does not has all rational roots. My attempt: Clearly, f(x) will have ...
1
vote
1answer
28 views

$/6 = 10 \pmod {7^2}$

I am trying to understand Hensel's lemma[wiki] with Newton iteration. But I dont understand how $r_{k+1}=r_k+t p^k=r_k-\frac{f(r_k)}{f'(r_k)}$ is the same. More detailed in examples: They get $10^2 \...
3
votes
0answers
44 views

Solutions of an equation of degree $n>4$

I know that the Abell-Ruffini theorem prove that we cannot solve a general equation of degree $n>4$ with radicals. But I've read that quintic equations can be solved by means of elliptic modular ...
0
votes
0answers
22 views

Finding the roots of this multivariable polynomial?

My polynomial is this ten term monster $P(x,y,z) = 6561 x^3+486 x^2+12 x+6561 y^3+1944 y^2+192 y+6561 z^3+6318 z^2+2028 z+223$ It's simplest form is ${1 \over 81} \left( (81x+2)^3 + (81y+8)^3 +(81z+...