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 (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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12
votes
2answers
124 views

Fully factored integer polynomials with constant differences

Given a degree $d$, it is possible to construct a pair $(F,\delta),$ where $F$ is a polynomial in $\mathbb{Z}[X]$ and $\delta$ a non-zero integer, such that $F(X)$ and $F(X)+\delta$ both split into ...
1
vote
2answers
126 views

verifying a polynomial is positive on the half-line

Math people: I am running experiments that produce polynomials $P(z)$ that, in every experiment I have run, are always positive on the half-line $\{z \geq 1\}$. I want to prove analytically that the ...
3
votes
1answer
256 views

how to find the roots of a cubic equation?

Given a formula $$x^3+ax^2+bx+c=0$$ how can I get the value of x without having an $i$ in my roots? Because Cardano's formula does have imaginary numbers if the discriminant is less than zero. My ...
7
votes
4answers
575 views

How many zeros does $z^{4}+z^{3}+4z^{2}+2z+3$ have in the first quadrant?

Let $f(z) = z^{4}+z^{3}+4z^{2}+2z+3$. I know that $f$ has no real roots and no purely imaginary roots. The number of zeros of $f(z)$ in the first quadrant is $\frac{1}{2\pi ...
0
votes
1answer
66 views

Find the value of $\sqrt{(b-a-4)^2}- \sqrt{(a-b+1)^2}$ if a>0 and b<0

Find the value of $\sqrt{(b-a-4)^2}- \sqrt{(a-b+1)^2}$ if $a>0$ and $b<0$. How do i find the value? This doesn't make any sense.
2
votes
0answers
60 views

Roots of a polynomial plus a logistic equation

I would like to know if there are any methods to find the roots (analytically) of complex valued equations of the following form: $$ f(z)=P(z)+\frac{e^{-z}}{(1+e^{-z})^2} $$ where $P(z)$ is a ...
1
vote
2answers
707 views

How to factorize $x^3 - 7x + 6$?

How do you factorize this polynomial: $\mathbf{x^3 - 7x + 6}$ Some online solver doesn't even work saying: using GCF method doesn't work, but sites like Mathway.com gave me the answer, is there a ...
1
vote
1answer
54 views

What's a good reference for the continuity of the number of zeros?

What is a good reference for the following statement, or something that easily implies it? For all sequences $\:\langle\:f_0,f_1,f_2,f_3,...\rangle\:$ of (complex) analytic functions $\:\:f_n : ...
4
votes
2answers
98 views

What this sine function equation means?

Apostol's book "Calculus" asks to prove that $$\sin\frac{\pi }{6}=\frac{1}{2}$$ using the fact that $$\sin 3x=3\sin x-4\sin^3 x$$ and $$\sin \frac{\pi}{2}=1$$ So, we take $x=\frac{\pi}{6}$ and ...
1
vote
2answers
118 views

Prove that if a polynomial $P$ has no roots in the upper half plane, then so does $P'$

Prove that if a polynomial $P$ has no roots in the upper half plane, then so does $P'$ This is a part of an exam preparation and I would appreciate a hint. My approach was to use Rouche's theorem but ...
1
vote
3answers
388 views

If $(2x^2-3x+1)(2x^2+5x+1)=9x^2$,then prove that the equation has real roots.

If $(2x^2-3x+1)(2x^2+5x+1)=9x^2$,then prove that the equation has real roots. MY attempt: We can open and get a bi quadratic but that is two difficult to show that it has real roots.THere must be an ...
6
votes
2answers
501 views

Show that a polynomial has at least 1 real root

I have the polynomial $P(x)=x^{2}+2013x+1$ and a number $n\in\mathbb{N}$. Now I have to show that the polynomial $P(P(...P(x)...)$ $(n$ times$)$ has at least one real root. How can I do this?
4
votes
1answer
35 views

How does the set of algebraic numbers compare to the set of possible fixed points for polynomials (with integer coefficients but not y=x)?

I was thinking of a way to map any polynomial $P$ with at least one real root onto some polynomial $Q$, s.t. the real roots of $P$ are exactly the real fixed points of $Q$, (There could be many, so we ...
5
votes
2answers
707 views

$n,a\in \mathbb Z,n\geq1,$ prove that $x^3+x+1\nmid x^n+a$

$n,a\in \mathbb Z,n\geq1,$ prove that $x^3+x+1\nmid x^n+a.$ In other word, they have no common roots. My idea: Let $x_1,x_2,x_3$ be the roots of $x^3+x+1=0,$ we need to prove that $\dfrac{x_1}{x_2}$ ...
12
votes
2answers
436 views

Number of real positive roots of a polynomial?

Consider the polynomial $$f(x)=x((1+x^n)^n+a^n)-a(1+x^n)^n,$$ where $n\geq 2$ is a positive integer and $a$ is a positive real number. I'm interesting in deducing the number of positive real roots ...
2
votes
3answers
130 views

How to prove that a given polynomial $P(x)$ has no interger roots.

How to prove that a given polynomial $P(x)$ has no integer roots.
5
votes
3answers
786 views

How to solve problems involving roots. $\sqrt{(x+3)-4\sqrt{x-1}} + \sqrt{(x+8)-6\sqrt{x-1}} =1$

How to solve problems involving roots. If we square them they may go to fourth degree.There must be some technique to solve this. $$\sqrt{(x+3)-4\sqrt{x-1}} + \sqrt{(x+8)-6\sqrt{x-1}} =1$$
0
votes
3answers
93 views

Find the eigenvalues of the matrices.

The characteristic equations for the two matrices are: $x^3-8x-7=0$ and $x^3-6x^2+11x-6=0$ I know that in order to find the eigenvalues, I need to factor these two equations out. I'm just having a ...
0
votes
2answers
67 views

Root of an exponential equation

Let $0 \le a \le 1$ and $-\infty < b < \infty$. I am looking for a solution of the exponential equation. $$ a^x + abx = 0. $$ I guess closed form expression of the root in terms of $a$ and $b$ ...
1
vote
2answers
377 views

what is the maximum number of roots of quadratic function with 3 variables?

Given the general quadratic form with $3$ variables $(x,y,z):ax^2 + by^2 +cz^2 + dxy + eyz +fzx + gx + hy + iz = 0$ satisfies $x^2 + y^2 + z^2 = 1$ I would like to ask what is the maximum number of ...
1
vote
1answer
121 views

Analog to bisection: Converging on complex roots of a polynomial

I am working on a Perl module that, among other features, will solve all the zeroes of a polynomial. Thus far, I am doing OK for $2$, $3$, $4$th degree using quadratic, Cardano's and Ferarri's ...
3
votes
1answer
81 views

Proof needed for the Obreschkoff-Hermite-Kakeya Theorem

I am having trouble proving the Obreschkoff-Hermite-Kakeya Theorem: Theorem (OBK). Let $P$ and $Q$ be two non-constant real valued polynomials with no common zeros. $P$ and $Q$ have only real zeros ...
2
votes
2answers
175 views

Prove that $f$ has finite number of roots

Let $f:[0,1]\to \mathbb{R}$ be a differentiable function. If there do not exist any $x\in[0,1]$ such that $f(x)=f'(x)=0$, prove that $f$ has only finite number of zeros in $[0,1]$. I'm not ...
2
votes
1answer
56 views

A problem in polynomials [duplicate]

Let c be a fixed number.Show that a root of the equation x(x+1)(x+2)...(x+2009)=c can have multiplicity at most 2.Determine the number of values of c for which the equation has a root of ...
8
votes
0answers
154 views

Let $x_n$ be the (unique) root of $\Delta f_n(x)=0$. Then $\Delta x_n\to 1$

Note that by Cesaro's Theorem, we have as a consequence $$\frac{x_n}n\to 1$$ Consider $$r_n(x)=e^{-x}-\sum_{k=0}^n (-1)^k\frac{x^k}{k!}$$ and $$f_n(x)=(-1)^{n+1}e^{-x}r_n(x)$$ One can argue by ...
3
votes
3answers
117 views

Behavior of the Nth root of N?

Taking the Nth root of some real number $N$ (ie: $R(N) = N^{1/N}$), generally $R(X) > R(Y)$ when $X < Y$. This obviously isn't the case though when $ X< Y < 3$. Put another way, starting ...
5
votes
0answers
119 views

Generating Functions, Recursive Polynomials

At the CMFT international conference in Turkey (2009), the following open problem was given: Show that $$p_n(x):=\sum_{k=0}^n \frac{(n-k)^k}{k!}x^{n-k}$$ has only real simple zeros for every $n$. ...
4
votes
1answer
226 views

Expressing the solutions of the equation $ \tan(x) = x $ in closed form.

I know that the equation $ \tan(x) = x $ can be solved using numerical methods, but I’m looking for a closed form of the solutions. In my opinion, having only numerical solutions means that we don’t ...
7
votes
3answers
202 views

Proving that $\sum\limits_{n = 0}^{2013} a_n z^n \neq 0$ if $a_0 > a_1 > \dots > a_{2013} > 0$ and $|z| \leq 1$

I'm going to teach a preparation course for the complex analysis qualifying exam from my university (which basically consists of me doing some problems from past exams) and I'm trying to solve some ...
4
votes
1answer
65 views

Polynomials and Trig

Question: The equation $x^{2}-x+1=0$ has roots $\alpha$ and $\beta$. Show that $\alpha ^{n}+\beta ^{n}=2\cos\frac{n\pi }{3}$ for $n=1, 2, 3...$ Attempt: $x^{2}=x-1 \Rightarrow ...
2
votes
1answer
135 views

Expressing polynomial roots expression in terms of coefficients

This is my first question on MSE. Apologies in advance for any textual or LaTeX errors. I'm stuck with this problem: Given $x^3 - bx^2 + cx - d = 0$ has roots $\alpha$, $\beta$, $\gamma$, find ...
1
vote
1answer
131 views

How do I solve $\; 3^{2x+1}-10\cdot 3^x+3=0 \quad?$

Solve the following equation for $x$ : $ \quad3^{2x+1}-10\cdot 3^x+3=0 $ I am baffled to solve this equation. With graphing I have found the answers to be x=1 and x=-1. I would like to know how ...
3
votes
1answer
67 views

root of an equation

I have the following equation: $$\sum_{k=0}^n \frac{a_k}{a_k+x}=1$$ where all the $a_k$'s are positive real numbers. For $n=2$ the roots are $x={}_{-}^+\sqrt{a_1a_2}$, but for $n\geq 3$ the ...
5
votes
0answers
114 views

Prove that $F_{2n}(x)=0$ has exactly one root in the interval $x\in(0,1),$ and this root $\to 0$ when $n \to \infty.$

Define $f_1(x)=x,f_2(x)=x^x,\dots f_{n+1}(x)=x^{f_n(x)}~(n\geq 1).$ Let $F_n(x)=f_n^{'}(x).$ Hence $F_1(x)=1, F_2(x)=x^x(1+\log(x))\dots.$ Prove that $F_{2n}(x)=0$ has exactly one root in the ...
10
votes
2answers
107 views

Behavior of zeros of $f'$ for complex polynomials $f$ with zeros on the boundary of the unit disc.

Suppose we have $f(z) = (z-r_1)\cdots(z-r_n)$, $|r_j| = 1$. According to the Lucas-Gauss theorem, all of the zeros of $f'$ lie in the convex hull of the $r_j$, but I discovered some behavior I don't ...
7
votes
2answers
207 views

Roots of $8x^3-4x^2-4x+1$

It is known that the roots of polynomial $8x^3-4x^2-4x+1$ are $\cos\frac{\pi}{7}$, $\cos\frac{3\pi}{7}$ and $\cos\frac{5\pi}{7}$. However this is what Wolfram Alpha/Wolfram Mathematica gives: $$x = ...
1
vote
1answer
105 views

Prove that there not real roots with $P(x)=ax^3+bx^2+cx+d, $

let $P(x)=ax^3+bx^2+cx+d,a,b,c,d\in R$, such that $$\min{\{d,b+d\}}>\max{\{|c|,|a+c|\}}$$ show that $P(x)=0$ have no real roots in $[-1,1]$
1
vote
2answers
106 views

How many real roots for $ax^2 + 12x + c = 0$?

If $a$ and $c$ are integers and $2 < a < 8$ and $-1 < c$, how many equations of the form $$ax^2+12x+c=0$$ have real roots?
3
votes
5answers
201 views

Polynomials - Solutions

How I can find the exact solutions of this polynomial? I can not get to the exact roots of the polynomial ... what methods occupy for this "problem"? $$x^3+3x^2-7x+1=0$$ Thanks for your help.
4
votes
2answers
327 views

Relation between root systems and representations of complex semisimple Lie algebras

I'm trying to understand the machinery of root systems for the purpose of classifying complex semisimple Lie algebras. During this process i lost the overview, espacially when it came to highest ...
1
vote
2answers
786 views

Product and Sum of Polynomial Roots

The ratio of the sum of the roots of the equation, $8x^3+px^2-2x+1=0 $ to the product of the roots of the equation $5x^3+7x^3-3x+q=0 $ is $3:2$. What is the value $\frac{p-q}{p+q}$? Well I found out ...
3
votes
2answers
30 views

Finding a function with properties

I am looking for a function $f(x)$ with the following properties: Positive for $x\in(-\infty, 0)$ but tangent to the x-axis at $x=-1$ A root at $x=0$ and negative for $x\in(0, 2)$ A root at $x=2$ ...
2
votes
1answer
57 views

For a fixed and small $\epsilon$, finding the number of real roots of $x^{2}+e^{-\epsilon x}-2+\sin(\epsilon x)$

I saw the following question in an introduction to applied mathematics exam (this is only the first part of the question): Assume $0<\epsilon\ll1$ . Denote $$ f(x,\epsilon):=x^{2}+e^{-\epsilon ...
3
votes
1answer
226 views

Showing that a root $x_0$ of a polynomial is bounded by $|x_0|<(n+1)\cdot c_{\rm max}/c_1$

I have doubts about the following problem (Problem 3.21 from Sipser's "Introduction to the Theory of Computation"): Let $c_1 x^n + c_2 x^{n-1} + \cdots + c_n x + c_{n+1}$ be a polynomial with a ...
0
votes
1answer
184 views

The function defined by $f(x)=\sin\pi x$ has zeros at every integer. Show that when $-1<a<0$ and $2<b<3$ , the bisection method converges to

The function defined by $f(x)=\sin\pi x$ has zeros at every integer. Show that when $-1<a<0$ and $2<b<3$ , the bisection method converges to (a) $0$, if $a+b<2$ (b) $2$, if $a+b>2$ ...
0
votes
1answer
104 views

Show that $|f(p_n)|<10^{-3}$ whenever $n>1$ but that $|p-p_n|<10^{-3}$ requires that $n>1000$.

Let $f(x)=(x-1)^{10}$. The root of the equation , $p=1$. The approximates of the root, $p_n=1+\frac{1}{n}$ Show that $|f(p_n)|<10^{-3}$ whenever $n>1$ but that $|p-p_n|<10^{-3}$ requires ...
0
votes
1answer
101 views

Difficulty to solve the exercise of Bisection method.

Find an approximation to $ {25}^{\frac{1}{3}}$ correct to within $10^{-4}$ using the Bisection algorithm. How to solve it? Where are the function and interval here?
0
votes
3answers
355 views

I am not understanding what has asked to compute of the following exercise.

let $f(x)=(x+2)(x+1)x(x-1)^3(x-2)$. To which zero of $f$ does the Bisection method converges when applied on the interval $[-3,2.5]$ Have i asked to find the root of $f(x)$ ?
0
votes
1answer
75 views

Determine the number of iteration to find solutions accurate to within $10^{-2}$ for $f(x)=x^3-7x^2+14x-6=0$ on $[a,b]=[1,3.2]$

i got the number of iteration,$n$, to achieve the accuracy, $\epsilon=10^{-2}$ is $n=5.5\approx 6$ But in answer script, $n=8$. My procedure is $ \frac{(b-a)}{2^n}<\epsilon$ ...
2
votes
1answer
113 views

Correct answer of the following math related to Bisection Method.

Use the Bisection method to find $p_3$ for $$f(x)=\sqrt x-\cos(x)$$ on $[0,1]$ I have got the answer $p_3=0.875$ But in answer script , $p_3=0.625$ Which one is correct? let $[a,b]=[0,1]$ ...