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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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
112 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
222 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
203 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
193 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
299 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
716 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
50 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
197 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
181 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
102 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
94 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
353 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
250 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
273 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
69 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
699 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 ...
2
votes
1answer
110 views

How to find all zeros of a polynomial

Let $$f(x) = x^4 - 3x^3 + 2x^2 - 7x - 11. $$ I want to find the roots of $f(x)$. I know that there are $3$ or $1$ positive roots, $1$ negative root, and $2$ or $0$ imaginary roots but I can't figure ...
4
votes
1answer
192 views

Solving a transcendental equation consisting of a quadratic part and a part involving inverse Lambert W functions

Question statement I would like to solve the following equation in the two variables $x$ and $y$: \begin{gather} 0 = x^2 - a y^2 + i b [x y - W^{-1}(x)W^{-1}(y)] , \end{gather} where $a$ and $b$ are ...
2
votes
4answers
117 views

Finding the root of a degree $5$ polynomial

$\textbf{Question}$: which of the following $\textbf{cannot}$ be a root of a polynomial in $x$ of the form $9x^5+ax^3+b$, where $a$ and $b$ are integers? A) $-9$ B) $-5$ C) $\dfrac{1}{4}$ D) ...
0
votes
1answer
75 views

Write the 2nd degree equation which have the following roots

$y_1$=${(x_1+x_2\varepsilon+x_3\varepsilon^2)}^3$ $y_2$=${(x_1+x_2\varepsilon^2+x_3\varepsilon)}^3$ where $x_1,x_2,x_3$ roots for the $x^3+ax^2+bx+c=0$ and $\varepsilon$ = ...
2
votes
1answer
74 views

Solution to set of three equations

I have the following three equations: $$\cos\theta \left(\cos\psi - k_3\sin\psi\right) = k_1$$ $$\sin\phi\sin\theta\cos\psi - \cos\phi\sin\psi - k_3\left(\cos\phi\cos\psi + ...
2
votes
2answers
63 views

Finding Root of an Equation with Variables Dependent on Each other

Sorry for the title. I'm sure there is better terminology. I'd be interested to here what that terminology is haha. Here is my problem: ...
2
votes
4answers
161 views

Finding root of equation

This question was asked in one of the enterance test of mathematics in India which is For the equation $1+2x+x^{3}+4x^{5}=0$, which of the following is true? (A) It does not possess any real root ...
16
votes
2answers
608 views

Adriaan van Roomen's 45th degree equation in 1593

Adriaan van Roomen proposed a 45th degree equation in 1593(see this book, picture reference as follows): $$ \begin{gathered} f(x) = x^{45} - 45x^{43} + 945x^{41} - 12300x^{39} + 111150x^{37} - ...
2
votes
0answers
67 views

Overdetermined system - showing that there are no roots that satisfy the set of equations

We consider an overdetermined set of equations, consisting of two equations for one complex variable $x$. I want to show that there are no roots for $x$ in the complex unit disc but without the ...
6
votes
3answers
138 views

Approximating the roots of $\epsilon^{2}x^{3}+x+1$

I saw the following in my lecture notes, and I am having difficulties verifying the steps taken. The question is: Assuming $0<\epsilon\ll1$ find all the roots of the polynomial ...
1
vote
0answers
49 views

Rouche's theorem: $g(z)$ has no roots in $L$, $|g(z)| > |f(z)|$ for the contour $\partial L$. Does $f(z)$ have no roots in $L$?

Let $f(z)$ and $g(z)$ be analytic functions. Let $L$ be the complex unit disc and its contour is $\partial L$, the complex unit circle $|z| = 1$. If $g(z)$ has no roots in $L$, e.g. $g(z) = z + 2$, ...
2
votes
2answers
56 views

Determining the sign of a polynomial given its factorization

Is there a quick way of determining where a polynomial is positive/negative without actually plugging values? Say you have a polynomial $$1) f(x)=(x+a)(x+b)$$ or $$2) f(x)=(x-a)(-x+b)(x-c)$$ ...
3
votes
6answers
520 views

Solving $\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$

Where do I start to solve a equation for x like the one below? $$\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$$ After squaring it, it's too complicated; but there's nothing to factor or to ...
13
votes
3answers
144 views

$\sum_i x_i^n = 0$ for all $n$ implies $x_i = 0$

Here is a statement that seems prima facie obvious, but when I try to prove it, I am lost. Let $x_1 , x_2 \dots x_k$ be complex numbers satisfying: $$x_1 + x_2 \dots + x_k = 0$$ $$x_1^2 + x_2^2 ...