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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2
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2answers
56 views

Roots of a Polynomial Minus It's Constant Term

Suppose we have a sequence of integers $a_1,\dots,a_n$. Is there any way to determine the roots of the polynomial $$P(x) = (x+a_1)\dots(x+a_n) - a_1\dots a_n$$ Clearly $P(0) = 0$, but can anything ...
1
vote
4answers
166 views

Find the square root of $404.11$.

Find the square root of $404.11$ without using calculator accurate upto $2$ decimal places . It is clear that $20<\sqrt{404.11}<21$ so it will be $20.ab$ without trial and error what ...
1
vote
1answer
47 views

The number of zeros of a polynomial that almost changes signs

Let $p$ be a polynomial, and let $x_0, x_1, \dots, x_n$ be distinct numbers in the interval $[-1, 1]$, listed in increasing order, for which the following holds: $$ (-1)^ip(x_i) \geq 0,\hspace{1cm}i ...
1
vote
1answer
24 views

Is Gershgorin bound of roots sharp?

Gershgorin circle theorem tells that the eigenvalues of a matrix $A$ lie in the union of the associated Gershgorin circles. $A=\begin{pmatrix} 0 & 0 & \dots & 0 & -a_0 \\ 1 & 0 ...
2
votes
0answers
266 views

nth-root of continued fraction with Raney transducers

There are some algorithms for doing basic arithmetic by using regular continued fraction expansions. These algorithms are mainly due to Gosper (1972) and Raney (1973). These two approaches use ...
0
votes
1answer
35 views

Is there closed form solution for this infinite polynomial or high-order polymonial?

The equation is as follows \begin{align} \sum_{N=1}^{\infty}P(N)x^N=Z, \end{align} where $P(N)$'s are real number satisfying $0\leq P(N)\leq 1$. Another equation is \begin{align} \sum_{N=1}^{\bar ...
0
votes
1answer
29 views

Roots of trigonometric equation

In the following trigonometric equation $$1 + \alpha^2 \cos^2 (n \theta) = 0$$ The complex solutions are $$\cos (n \theta) = \pm i/\alpha$$ So I thought that the correspondant angles were $$n ...
1
vote
1answer
55 views

Numerically find the minimum distance between given point and the curve of given function.

Let $f\colon\Bbb{R}^n\to\Bbb{R}$ and $\mathbf{x}_0\in\Bbb{R}^n$. How could I (numerically) find the minimum Euclidean distance between the curve $f(\mathbf{x})=0$ and $\mathbf{x}_0$, granted that $f$ ...
3
votes
5answers
175 views

Prove that equation $x^6+x^5-x^4-x^3+x^2+x-1=0$ has two real roots

Prove that equation $$x^6+x^5-x^4-x^3+x^2+x-1=0$$ has two real roots and $$x^6-x^5+x^4+x^3-x^2-x+1=0$$ has two real roots I think that: ...
1
vote
2answers
74 views

Given some zeroes of a real polynomial of a given degree, how can one find the remaining zeroes?

Here is what the problem says: If $2$, $-\sqrt{5}$, and $3+i$ are three zeroes of a $5$th degree polynomial function with real coefficients, find the other zeroes of multiplicity $1$. I don't ...
3
votes
2answers
34 views

Elementary bound theorem of a monic real polynomial

An elementary bound theorem on the roots of a real monic polynomial states that $$M := \operatorname{max} (1, |a_0| + \cdots + |a_{n-1}|) := \operatorname{max} (1, B)$$ is an upper and lower ($-M$) ...
1
vote
1answer
49 views

Can a quartic equation be reduced to a cubic/quadratic knowing that two roots are real?

I have a quartic equation that is the determinant of a 4-by-4 matrix that looks like: $det(M-\lambda I) = det \left( \matrix{m_{11}-\lambda & m_{12} & m_{13} & 0 \\ m_{21} & ...
1
vote
1answer
55 views

How to upper-bound the smallest positive root of a polynomial?

Is there any algorithm for (upper-)bounding the smallest positive root of a polynomial of an arbitrary degree if it exists, or detecting that it does not exist otherwise? Edit: I'm looking for a ...
-2
votes
1answer
75 views

Root question help needed [closed]

$$\sqrt{3+\sqrt{3+\sqrt{3+x}}}=x$$ Question is: How to find x? Could you help me? Thanks in advance
0
votes
2answers
41 views

Conditions needed for a unique root to also be a “clear-cut” root

Suppose $f:[0,1] \longrightarrow [-1,1]$ is a continuous function that has a unique root $r_{0} \in (0,1)$. I want $r_{0}$ to be a ``clear-cut root" (not sure what to call it) in the following ...
1
vote
2answers
85 views

Solution of recursive polynomial functions

Is there anything that can be said about the roots of the polynomial $f_n(x)$ if $f_n(x) = xf_{n-1}(x) + f_{n-2}(x)$ where these are polynomials of degree $n, n-1,$ and $n-2$, respectively? My goal is ...
0
votes
1answer
54 views

How to use newton's method on a function of multiple variables?

I have a function $f \colon R^3 \to R$. I want to find $x$, $y$, $z$ such that $f(x,y,z)=0$. I'm using the method from here: http://en.wikipedia.org/wiki/Quasi-Newton_method#Search_for_zeroes. ...
2
votes
1answer
71 views

How do I use Rouche's theorem here?

Suppose I had the polynomial $f(z) = z^5+3z+1$ and I want to find the number of complex roots in the first quadrant. How would I use Rouche's theorem? or is there a simpler way. I was thinking of ...
0
votes
0answers
18 views

Reducibility and Roots of Functions in Various Fields

I was just wondering what it means to be "reducible/irreducible" as well as having "(no) roots" for a given polynomial in a given field. For example, consider polynomial $p(x)$ being an element of ...
2
votes
1answer
50 views

If $z$ is an $n$th root of unity, prove that $1/z$ is an $n$th root of unity

I'm not sure if how I'm going to prove this to be correct: Since $z$ is an $n$th root of unity, it means $z^n = 1$ For $1/z$ to be an nth root of unity, lets take it to the power of $n$, ...
0
votes
1answer
34 views

Field of characteristic $0$ and simple roots

Suppose $g(x)=f(x)m(x)^{-1}$ where $f(x)\in F[x]$ and $F$ has characteristic $0$ and $m(x)=(f,f')$. Then show that all roots of $g(x)$ are simple. I assume g(x) has multiple roots that is ...
3
votes
0answers
30 views

Squares of finite fields (mod p*q)

Lets say we have $\mathbb{Z}_p$, where p is prime. For each element(x) we have two squares(y) so that $y^2=x$ ie if $p=7$ for $x=4$ we have $y_1=2,y_2=7-2=5,y=\pm2 $ ok, lets have ...
0
votes
1answer
21 views

find coefficients of a polynomial given k roots

Lets say that I have k roots for a polynomial and I am trying to find the coefficients of the terms in the polynomial. (x - r1)(x - r2)(x - r3) ... (x - rk) ...
2
votes
0answers
29 views

Roots of polynomial equations

Is there anything that can be said about the roots of the polynomial $f_n(x)$ if $f_n(x) = xf_{n-1}(x) + f_{n-2}(x)$ where these are polynomials of degree $n, n-1,$ and $n-2$, respectively. In ...
1
vote
2answers
29 views

Show that $\frac{1}{x}-\sin(x)$ has exactly one root in the interval $(0,\frac{\pi}{2}]$

I have problems showing, that this function has exact one root in the interval $\left(0,\frac{\pi}{2}\right]$: $$f(x):=\frac{1}{x}-\sin(x)$$ My idea was to use the Intermediate value theorem, but ...
0
votes
3answers
62 views

Solving a radical equation for real roots

I'm attempting to solve the derivative of my function $f(x)$ for real roots. $$ \\ \begin{align*} \\ f(x) &= 3x^2 + 3\arcsin{x} \\ f^{\prime}(x) &= 6x + \dfrac{3}{\sqrt{1-x^2}} \\ \\ 0 &= ...
1
vote
1answer
64 views

Müller's Method

I have these question and I cannot solve it. Can somebody help me? Use Müller’s method to determine the roots of $$ f(x)=2x^5−2x^4+6x^3−6x^2+8x−8 $$ Choose $x_2=0.8 $, $x_0=0.808$ ...
-1
votes
3answers
34 views

Find the speed of a jet given the time of travel back and forth

The problem: A jet flew from Tokyo to Bangkok, a distance of 4800km. On the return trip, the speed was decreased by 200 km/h. If the difference in the times of the flights was 2 hours, what ...
2
votes
1answer
158 views

Why is this answer wrong? (quadratic functions)

Question: Determine the quadratic function that has the given roots (x-intercepts) and passes though the given point. $x=2+\sqrt{5}, x=2 - \sqrt{5}$. The given point is $(2,10)$. This is my ...
0
votes
0answers
30 views

How do I solve this quadratics problem? [duplicate]

The problem: A jet flew from Tokyo to Bangkok, a distance of 4800km. On the return trip, the speed was decreased by 200 km/h. If the difference in the times of the flights was 2 hours, what ...
0
votes
1answer
42 views

Which are the conditions for a biquadratic equation to have 4 different roots?

Which are the conditions for a biquadratic equation to have 4 different roots in R? I think D>0, If we have $$t=x^2$$ then t>0. Is there any other condition?
1
vote
2answers
43 views

Prove that the rounding error can contaminate half the digits of computed root

I am trying to resolve the following problem: If $b^2 \approx 4ac $ the rounding error can contaminate half the digits of the root computed with the formula: $\dfrac {-b \pm \sqrt {b^2 - 4ac}} ...
1
vote
1answer
23 views

Discriminant with non-Real result

I have the following equation: $ ax^2 + (a+1)x - a = 0 $ Where $a$ is not $0$ When calculating the discriminant $\Delta$ i get a non-real result. But what does it mean? I know that a negative ...
0
votes
1answer
45 views

Polynomial in $Q[x]$ with 2 complex roots

I need to show that for any $n$, I need to show that there is an irreducible polynomial in $Q[x]$ of degree $n$ having exactly $n-2$ real roots. As a hint I have that if $f(x) \in \mathbb{R}[x]$ is a ...
2
votes
1answer
93 views

Irreducible polynomial in $\mathbb Q[x]$ of degree $n$ having exactly $n-2$ real roots.

I need to show that for any $n$ there is an irreducible polynomial in $\mathbb Q[x]$ of degree $n$ having exactly $n-2$ real roots. I know that from a previous exercise that if $f(x) \in ...
1
vote
2answers
54 views

Polynomial $f(x)$ over $\mathbb{R}$ has $k$ distinct roots, then $f(x) + a$ too

I am trying to learn Galois theory by myself. When reading a section for applications to polynomials, I got stuck in the following exercise: If $f(x) \in \mathbb{R}[x]$ is any polynomial having ...
3
votes
0answers
26 views

Magnitudes of roots of random polynomials

I'm looking at the roots of random polynomials with integer coefficients, and constant term=leading term = 1. Using the Mathematica code ...
0
votes
1answer
59 views

Solving for $x$, ${\sqrt{7x-5}} - {\sqrt{2x}} = {\sqrt{15 - 7x}}$

could I please have some help solving this equation for $x$ ? ${\sqrt{7x-5}} - {\sqrt{2x}} = {\sqrt{15 - 7x}}$ Thank you
0
votes
0answers
29 views

Iterpolating to find the zeros of a complex function

I have an $N\times M$ grid of complex points sampled from some unknown complex function. I would like to interpolate and find the zeros of that function. I believe that this function can be well ...
1
vote
1answer
28 views

If $\tan\alpha,\, \tan\beta,\,\tan\gamma$ are roots of $au^ 3 +(2a-x)u+ y=0$

If $\tan\alpha,\, \tan\beta,\,\tan\gamma$ are roots of $au^ 3 +(2a-x)u+ y=0$ for fixed $x$ and $y$ and $\tan\alpha + \tan\beta = h$ Find $ah^3 +(2a-x)h$. Options are A) $y$ B) $-y$ C) $2a- x$ D) ...
0
votes
0answers
35 views

Analytical solution to equation $ \arctan(x)-k \arctan(x/k)=c$

For the equation:$$\arctan(x)-k \arctan(x/k)=c$$ which is part of a gasdynamics function called Prandtl–Meyer function, it is not difficult to find the solution numerically, however, I'm wondering, ...
0
votes
0answers
17 views

bounds on the non real zeros of a polynomial $P_n(z)$

There are several bounds on the complex zeros (including real zeros and nonreal zeros) in the literature. There are also several bounds on the positive/negative zeros in the literature. Are there ...
0
votes
1answer
65 views

Number of solutions to an equation $6\ln(x^2+1)-e^x = 0$

I need to determine how many solutions the equation $6\ln(x^2+1)-e^x = 0$ has. I wanted to find monotonic intervals of this function and check function's values at local extrema so I calculate ...
1
vote
0answers
42 views

Irrational roots of polynomials with integer coefficients

Consider an equation of the form $$ p(x) = 0, $$ where $p(x) = \sum_{i=0}^{n} a_i x^i$ and all $a_i \in \mathbb{Z}$. Is there an algorithm which can tell us that this equation has a root of the form ...
0
votes
3answers
46 views

How to find the range of zeroes in a polynomial

I am writing a java program that finds all of the zeroes of a polynomial by bisection. The first step, clearly, is to iterate through integers in a certain range looking for sign changes. I could ...
0
votes
0answers
50 views

How to get the proper fixed point iteration function?

When we find the approximated root of a function $f(x)$ in an interval $[a,b]$ from the fixed point iteration method, we derive a new function $g(x)$ which has a fixed point as a root of $f(x)$. Is ...
0
votes
1answer
24 views

Showing that at least one of the equations has two real roots

Let's suppose that $b_1, b_2, c_1, c_2$ are real numbers. We know that $b_1b_2=2(c_1+c_2)$. The task is to prove that at least one of the equations $x^2+b_1x+c_1=0$, $x^2+b_2x+c_2=0$ has two real ...
0
votes
1answer
17 views

How do we know that $\tan z$ or $\cos z$ don't have singularities off of the real axis?

Since $\sin z$ is bounded, $\tan z = \frac{\sin z}{ \cos z}$ has singularities when $\cos z = 0$. We know $\cos z = 0$ for $z = \frac{\pi}{2} + n\pi$ for $n \in \mathbb{Z}$, but could it not also be ...
4
votes
2answers
46 views

Show that $x+e^{-Bx^2}\mbox{cos}(x)$ has only one root over all reals ($B>0$).

Let $B>0$ and define for all $x\in \mathbb{R}$ the function $f(x)=x+e^{-Bx^2}\mbox{cos}(x)$. Prove that $f$ has exactly one root over $\mathbb{R}$. My original attempt was to show that the ...
7
votes
1answer
112 views

Can the general septic be solved by infinitely nested radicals?

I. Quintic. The general quintic can be reduced to the form, $$x^5=p+x\tag1$$ $$x = \sqrt[5]{p+x}$$ Hence by an iterative process, $$x =\sqrt[5]{p+\sqrt[5]{p+\sqrt[5]{p+\sqrt[5]{p+x\dots}}}}$$ ...