Tagged Questions

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.

learn more… | top users | synonyms (1)

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
378 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
485 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
420 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
779 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
92 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
66 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
349 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
117 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
79 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
170 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
150 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
115 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
115 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
223 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
200 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
64 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
132 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
123 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
112 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
104 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
206 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
102 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
105 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
195 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
307 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
736 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
54 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
213 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
182 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
103 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
99 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
354 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
73 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
110 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 $[a,b]=[0,1]$ ...
0
votes
1answer
77 views

Roots of $x^{2}+e^{0.1x}-1$

I saw an exercise that asks to prove that $f(x):=x^{2}+e^{0.1x}-1$ have a root $r<0$. The solution stated that $f''(x)=2+(0.1)^{2}e^{0.1x}>0$ hence there is a maximum of two roots, since $0$ is ...
1
vote
0answers
78 views

prove that polynomial has root of unity

Prove that $ f=x^n\pm x^m\pm1 $ is either irreducible over rationals or has a root which is a of unity. I tried to see what if $x=|r|e^{i\phi}$ but I have no proper result.
3
votes
2answers
74 views

Showing how the roots of this complex polynomial are different.

I want to show that the complex polynomial $p(z) = z^5 + 6z - 1$ has four different roots in the annulus $\{z \in \mathbb{C} : \frac{3}{2} < |z| < 2 \}$. I used Rouché's theorem to proof that ...
3
votes
1answer
127 views

Roots of $z^{2n} + \alpha z^{2n -1} + \beta ^2$

I've been looking at a problem available here. The problem is: Let $n$ be a natural number, and $\alpha$, $\beta$ nonzero reals. Show that the number of roots of $p(z) = z^{2n} + \alpha z^{2n -1} + ...
1
vote
1answer
252 views

Do the false position method really need that there exists only one root inside $[a; b]$?

I'm studying the False Position Method for finding zeroes of real functions and in the book I'm reading the author says that it is required that only one root of $f$ is contained inside the initially ...
4
votes
3answers
274 views

Prove $x^{n}-5x+7=0$ has no rational roots

This question arises in STEP 2011 Paper III, question 2. The paper can be found here. The first part of the question requires us to prove the result that if the polynomial ...
0
votes
1answer
70 views

number of solutions in homogeneous system

What is the maximum possible number of solutions of homogeneous system $N \times N$ ($N$ variables, $N$ equations) of degree $2$, where in each equation we have linear terms in $x_i$ and quadratic ...
4
votes
1answer
730 views

Calculating the Roots of Sine

Aside from the obvious knowledge that the roots of $\sin x$ are all integer multiples of $\pi$, is there a formal, algebraic method to calculate the roots of trigonometric functions similar to the ...