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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30 views

Problem with $x^0$ where $x \in(-∞, ∞)$

I understand that when $x \in(-∞, ∞)$, is means that $x^0 = 1$. I also understand why this is. But isn't the following still true that when $x \in(-∞, ∞)$, it means that $x^y = \sqrt [1/y] {x}$? And ...
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0answers
14 views

“root” of a right-continuous function

Suppose $f:[0,1] \longrightarrow [-1,1]$ is a right-continuous function such that $f(0) < 0$, $f(1) > 0$, and $f$ only changes sign once in the interval $[0,1]$. Suppose we define the "root" of ...
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0answers
16 views

Interval of Polynomial Root Finding

Let's say we have a polynomial of a given degree. You don't have any tools to figure out the amount of roots in this polynomial. All you know is the function and you cannot graph it. How would you ...
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1answer
26 views

Prove or disprove this relation between one root of the quadratic and the cubic equation of a certain form, and linear recurrences.

It is well known that the n-anacci (higher degree Fibonacci, that is Tribonacci and so on) numbers can be computed in closed form from roots of polynomials in the way Eric Weisstein at Mathworld ...
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2answers
31 views

Roots of quadratic equation

If the roots of $ax^2+bx+c$ are $\alpha$ and $\beta$, express $\frac1\alpha-\frac1\beta$ in terms of $a$, $b$ and $c$. I know how to express $\alpha+\beta$ or $\alpha\beta$ which is usually enough, ...
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1answer
31 views

Show that $z^n+nz-1$ has $n$ zeros in $D(0,R)$

Let $n\geq 3$. Show that the polynomial $z^n+nz-1$ has $n$ zeros in $D(0,R)$, where $$R=1+\left(\frac{2}{n-1}\right)^{1/2}.$$ I was hoping to use Induction and Rouche's Theorem. For the base case ...
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1answer
59 views

Proving that if the sequence $X_n$ converges to $x$, then ${X_n}^a$, where $a$ is a positive rational, converges to $x^a$.

I've been stuck on this problem for a while. I splitted a into $p/q$, so it would be $({X_n}^p)^{1/q}$, and I got the convergence of ${X_n}^p$ to be $x^p$ since it is just induction using the product ...
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1answer
28 views

Find the fixed points of a function $f(x) := exp(x - 2)$ using a recursive algorithm

I need to find the fixed points (i.e. when $f(x) = x$) of the following function $f(x) := exp(x - 2)$. I understood that the fixed points should be the intersecation points between $f(x)$ and a ...
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1answer
30 views

Effect on existing roots of polynomial when adding small higher-order term

How do existing roots of a polynomial change when adding higher-order term with a small coefficient? Given a sufficiently small coefficient of the new higher-order term, the existing roots shouldn't ...
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3answers
52 views

Find the real root $\alpha$ of the cubic equation $z^3-2z^2-3z+10=0$

Find the real root $\alpha$ of the cubic equation, $$z^3-2z^2-3z+10=0$$ The exam paper is giving just 2 marks for this and the mark scheme isn't very helpful. My idea is that you can use some of this ...
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1answer
27 views

Finding a root by bisection method in Excel

Working on a maths assignment and we're trying to use Excel for a bisection method. $$\frac12 e^{x/2}+\frac{1}{2x}-\frac32=0$$ Here is a pic, I can't get the formula to work with the exponent. This ...
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1answer
45 views

The roots of the cubic equation $z^3-2z^2+pz+10=0$ are $\alpha$, $\beta$ and $\gamma$. Show that $\alpha^2+\beta^2+\gamma^2=p+13$

$$z^3-2z^2+pz+10=0$$ $$ax^3+bx^2+cx+d=0$$ $$\Rightarrow\,\,\,\,\,\,\,\,\,a=1,\,\,\,\,\,\,\,\, b=-2,\,\,\,\,\,\,\,\, c=p,\,\,\,\,\,\,\,\, d=10$$ ...
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3answers
40 views

Prove $x^7+3x^5+1$ has exactly one real root using Bolzano's theorem and the MVT.

Prove $f(x)=x^7+3x^5+1$ has exactly one real root using Bolzano's theorem and the MVT. What I did: $f(-1)=-3$ $f(0)=1$ As $f$ is continuous, there exists a $c \in (-1,0) /f(c)=0$ Then computed ...
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3answers
119 views

Solve $ \left(\sqrt[3]{4-\sqrt{15}}\right)^x+\left(\sqrt[3]{4+\sqrt{15}}\right)^x=8 $ [closed]

I don't know what can I substitute for $x$ so that equation becomes satisfied. Any assistance will be greatly valued. Thanks!
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0answers
60 views

Solving quartic equation? (Cardano/Ferrari)

today I've written a little Code-Snippet that is based upon the steps that are mentionned in this wikipedia-Article to solve a general quartic polynom. Here's my matlab-implementation: ...
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1answer
37 views

general theorem on roots of a polynomial needed to show it's identically zero.

Polynomial degree k, one variable, if it's zero at k+1 values, then it's identically zero. Can someone point me to a proof of this? I know derivatives at points can count as these roots (if k-degree ...
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0answers
10 views

Root finding of a Hermite interpolating polynomial

Consider a Hermite interpolation problem. I have an approach for obtaining the roots of interpolating polynomial. I would like to present an example for this approach. Can you suggest me an applicable ...
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2answers
36 views

Entire functions of order 0

Sorry, this may be a stupid question, but I am just beginning to learn about this and cannot find the answer anywhere I have looked so far. Clearly if we have any polynomial $P(z)$, then it is easy to ...
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1answer
40 views

I have to show $p=p(x-\lambda)$ if and only if they have the same zeros in $F$

Suppose $F$ is a field, $|F|\geq n \geq 2$. Given $\lambda \in F$ I know $p,p(x-\lambda)\in F[x]$ are irreducible monic polynomials. I have to show $p=p(x-\lambda)$ if and only if they have the same ...
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1answer
56 views

Help required! Polynomials

Let $D(p) = p^{20} - p^{18} - p^{16} - \dots - p^2 - 2$ Prove that the sum of fourth powers of all the real roots of $D(p) = 8.$ Please help.
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1answer
41 views

What are 'Regular Products'?

When looking at the functional equation for the Riemann zeta function, I came across the statement: For $s$ an even positive integer, the product $\sin{(\frac{\pi s}{2})}\Gamma({1-s})$ is regular. ...
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1answer
54 views

How to remove duplicate roots from a polynomial?

Given a polynomial equation (with real coefficients of any degree with any number of repeating roots), let say $x^5 + 6x^4 - 18x^3 - 10x^2 + 45x - 24 = 0$, ... (A) it can be written as $(x-1)^2 ...
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0answers
17 views

Root locus vs Matlab's root locus function

Today during a lecture out teacher was demonstrating Matlab's rlocus function, but before that he decided to do it on chalkboard. Below we have a simple transfer ...
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1answer
24 views

Algebraic vs. analytic definition of the multiplicity of a polynomial's root

Let $f(x) = a(x - c_1)^{d_1}(x - c_2)^{d_2} \dots (x - c_n)^{d_n}$ be a polynomial over the complex numbers ($n, d_i \in \{1, 2, \dots\}$, $a \in \mathbb{C}\setminus \{0\}$), where the roots $c_1, ...
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2answers
32 views

Solving recurrences whose characteristic equations have complex roots

In my Discrete Mathematics lecture notes, there is a section regarding solutions for linear recurrences whose characteristic polynomials have complex roots. There is a particular statement which I am ...
0
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1answer
19 views

Finding roots of complex polynomial with conjugates

I am having problem with the following question... I know that I should use De Moivre's formula somewhere... but can't quite get to it $$ (-15w + 34\bar{w})^4 = -1 $$ will be happy to get some help, ...
2
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2answers
51 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 ...
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4answers
159 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 ...
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1answer
39 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 ...
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1answer
23 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 ...
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0answers
72 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 ...
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1answer
31 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 ...
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1answer
26 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 ...
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1answer
38 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 $f$ ...
2
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5answers
151 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
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2answers
58 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
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2answers
30 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 ($-M$) ...
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1answer
42 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} & ...
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1answer
32 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 ...
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1answer
59 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
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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 ...
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2answers
73 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 ...
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3answers
37 views

If the square of a number is added to the number increased by 4, the result is 60. [closed]

So, if $n$ represents the number, which equation can be used to find $n$???
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1answer
40 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. ...
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1answer
52 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 ...
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0answers
17 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
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1answer
49 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
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1answer
33 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
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0answers
27 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
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1answer
16 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) ...