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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Distinct roots of $z^n-z$

How would we prove that for any positive integer $n$ the complex roots of $z^n-z$ are all distinct? In the case that $n=1,2,3$ I have factored it directly but how can we do it in general?
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0answers
27 views

Roots of serie $\sum_{i=1}^N \frac{a_i + b_i \log(c_i - x)}{c_i - x}$

I encountered the following serie during an optimization problem: $\sum_{i=1}^N \frac{a_i + b_i \log(c_i - x)}{c_i - x}$ I need to find the roots of that serie, but I cannot find the way to find a ...
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1answer
31 views

Prove that the line integral on $\beta$ of $f'(z)/f(z) = (A-B)/2 \pi i$ using Rouche's Theorem

Suppose that $\alpha$ is a regular closed contour. $f$, our function, lacks zeros and poles on $\beta$ and if A=the number of zeros of f inside $\beta$ (a zero of order n is counted n times) and B= ...
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1answer
27 views

If $p>0$ demonstrate that the $1/2\pi i$ the line integral of $z^p f'(z)/f(z)$ is $\sum (z_k)^p$

This is basically a deviation of Rouche's Theorem from what I can tell. My first instinct was to do this via induction in which we know that $p=0$ we would have Rouche's theorem. But it gets ...
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1answer
37 views

Approximating non-rational roots by a rational roots for a quadratic equation

Let $a,b,c$ be integers and suppose the equation $f(x)=ax^2+bx+c=0$ has an irrational root $r$. Let $u=\frac p q$ be any rational number such that $|u-r|<1$. Prove that $\frac 1 {q^2} \leq |f(u)| ...
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1answer
48 views

Laguerre theorem

I'm looking for a proof of the theorem 7, page 6, of this document : http://www.nipne.ro/rjp/2013_58_9-10/1428_1435.pdf Theorem 7 (E. Laguerre) Let $f \in \mathbb{R}[x]$ be a polynomial of degree ...
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0answers
22 views

Homework help on eigenvalues function minimization

so I actually have two separate questions which are homework bonuses for my numerical methods course. Unfortunately, because of the time of the semester, our TAs are not available so I don't have many ...
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1answer
73 views

Find the number of roots of a polynomial using Rouche's Theorem

Use Rouche's theorem to find the number of roots of the polynomial $z^5+3z^2+1$ in the anulus $1<|z|<2$. I am looking for a solution to this problem. My thoughts: This is a topic that ...
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1answer
25 views

Let $p$ be a prime number $\ge2$ and $u = \cos\left(\frac 2p\pi\right)+i\sin\left(\frac2p\pi\right) \in \mathbb C$. Prove that …

Let $p$ be a prime number $\ge2$ and $u = \cos\left(\frac 2p\pi\right)+i\sin\left(\frac2p\pi\right) \in \mathbb C$. Prove that $u$ is root of $f(x)=x^{p-1}+x^{p}+...+x+1$. I know that $f(x)$ is ...
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0answers
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Show that the polynomial $x^8 -x^7 +x^2 -x +15$ has no real root. [duplicate]

Please check if my method is correct. Solution : Let $$f(x) = x^8 - x^7 + x^2 -x +15 $$ Now, let $g(x)= x^8 -x^7=x^7(x - 1)$ and $h(x)= x^2 -x=x(x-1)$. Thus, $$f(x)=g(x) + h(x) + 15$$ On analyzing ...
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5answers
427 views

Show that some of the root of the polynomial is not real.

\begin{equation*} p(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_3x^3+x^2+x+1. \end{equation*} All the coefficients are real. Show that some of the roots are not real. I don't have any idea how to do this, I ...
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0answers
22 views

Number of ordered pairs of $(p,q)$ [duplicate]

Let $x^2-px+q=0$ and $x^2-qx+p=0$ have unequal integral roots where $p$ and $q$ are natural numbers. Find the number of ordered pairs $(p,q)$ for which this is possible. I do understand that roots ...
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4answers
65 views

Find the roots of the polynomial? (Cardano's Method)

$y^3-\frac7{12}y-\frac7{216}$ This is part of Cardano's method, so I've gotten my first root to be: $y_1=\sqrt[3]{\frac7{432}+i\sqrt{\frac{49}{6912}}}+\sqrt[3]{\frac7{432}-i\sqrt{\frac{49}{6912}}}$ ...
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2answers
67 views

Showing that an equation has a root in an interval

Show that the equation $x^4 - 7x^3 + 1 = 0$ has a root in the interval $[0,1]$. How would I go about working this out in steps?
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1answer
73 views

Prove $p(x)>0$ for $x>b$

This is a question from a past paper which I have no solution to. Let $p(x)=x^n + a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}, n\geq 1$ be a polynomial of dgree n and let $b=1+|a_{1}|+\cdots ...
4
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1answer
51 views

$\int\limits_{0}^{32/9}\sqrt{1+\frac{9x}{4}}dx$

Question : Solve $\int\limits_{0}^{32/9}\sqrt{1+\frac{9x}{4}}dx$ My Try: Let u = $1+\frac{9x}{4}$ Then, $$du = \frac{9x}{4}dx$$ $$dx = \frac{4du}{9}$$ Substituting the above in the main ...
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2answers
68 views

Prove that $f(x)=x$ can have at most one solution if $f'(x)\ne1$

Prove that $f(x)=x$ can have at most one solution if $f'(x)\ne1$ What I did : Use $g(x) = f(x)-x$, then $g'(x) = f'(x)-1\ne0$ I suspect I have to use Rolle's theorem now, But I am having difficulty ...
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2answers
843 views

Is Vieta the only way out?

Let $a,b,c$ are the three roots of the equation $x^3-x-1=0$. Then find the equation whose roots are $\frac{1+a}{1-a}$,$\frac{1+b}{1-b}$,$\frac{1+c}{1-c}$. The only solution I could think of is by ...
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1answer
59 views

Location of the roots of $f'$ (Laguerre's theorem)

Let $f \in \mathbb{R}[X]$ be a polynomial of degree $n$ having $n$ distinct roots $a_1,...,a_n$. Let $b_1<...<b_{n-1}$ be the roots of its derivative $f'$ (note that $b_i \in ]a_{i}, a_{i+1}[$ ...
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0answers
29 views

Positive Zeroes within a Polynomial

Question: Let $a,b>0.$ Can the polynomial $$x^{10} − x^7 + 2x^5 + ax^3 − bx + 1$$ have exactly three (counting multiplicity) positive zeroes? Can it have three simple positive zeroes together with ...
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1answer
42 views

Confusion about exponents like ${x^m}^{(1/n)}$.

I've been reading this post. It says that $\sqrt[m]{x^n} = x^{n\frac 1m}=x^{\frac mn}=x$ if $m=n$. Let's take $x=-2$, and $m=n=2$. Now we have, $\sqrt[2]{(-2)^2}=\sqrt[2]{4}=2$ But according to that ...
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2answers
63 views

Extreme point of quadratic equation

For the below question read here: Write a function quadratic that returns the interval of all values $f(t)$ such that $t$ is in the argument interval $x$ and $f(t)$ is a quadratic function: ...
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1answer
38 views

Exist another method to solve the problem?

We have $x_1,\:x_2,\:x_3\:\in \:\mathbb{C},\:\:f=x^3+x^2+mx+m,\:m\in \mathbb{R}$. We need to find $m\in\mathbb{R}$ such that $|x_1|=|x_2|=|x_3|$. Here is what I tried: $f=x^3+x^2+mx+m=(x^2+m)(x+1)$, ...
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1answer
49 views

How to evaluate real root of a polynomial equation? [closed]

If $\alpha$ is a real root of the polynomial equation $$300x^{299}+299x^4+343x^3+23x+300=0$$ Then how to find out the value of $[\alpha]\space $ where, '$[ \space]$' denotes greatest integer? I have ...
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3answers
73 views

Zeroes of sin(x)

Consider the function f = $\sin(x)$ defined as $$ \sin(x) = \frac{e^{ix}- e^{-ix}}{2i} $$ How to prove that the only zeroes of this function lie on the line $i = 0$ in the complex plane and ...
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0answers
18 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
23 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
33 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
38 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
34 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
64 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
34 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
51 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
61 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
128 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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2answers
833 views

Only 12 polynomials exist with given properties

Prove that there are only 12 polynomials that have all real roots and whose coefficients are $-1$ or $1$. Zero coefficients are not allowed, and constant polynomials do not count. Two of them ...
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1answer
49 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
49 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
152 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
90 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
41 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
13 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 ...
0
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2answers
47 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
42 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
57 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.
1
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1answer
45 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. ...
3
votes
1answer
73 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
33 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 ...
1
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1answer
29 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, ...
0
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2answers
41 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 ...