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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2
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1answer
34 views

Formula for the midpoint in the hyperbolic geometry

I have two questions. First, is there a relatively simple formula for the midpoint of two points $a_1$ and $a_2$ in the disk with respect to the hyperbolic geometry? That is, the point on the ...
1
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3answers
51 views

property of roots

I need to prove $\gamma_n$ is an integer for any value of $n$. $\gamma_n$ is defined as $= \alpha^n + \beta^n$. $\alpha$ and $\beta$ are roots of equation $x^2 + mx - 1 = 0$, $m$ is an odd integer. I ...
0
votes
8answers
164 views

The number of real roots of $x^5 + 2x^3 + x^2 + 2 = 0 $ is

The number of real roots of $x^5 + 2x^3 + x^2 + 2 = 0 $ is A. 0; B. 3; C. 5; D. 1. I don't know how to solve this.
-1
votes
1answer
62 views

How to find the polynomial corresponding to positive roots of $f(x)$?

Let $$P_0(x)=1$$ $$P_1(x)=x$$ $$P_n(x)=\frac{(x+\sqrt{x^2-4})^{n+1}-(x-\sqrt{x^2-4})^{n+1}}{2^{n+1}\sqrt{x^2-4}}; n\ge2$$ and $$f(x)=x P_{2n+1}(x)− P_{2}(x).P_{2n-2}(x)$$ It is given that roots ...
3
votes
2answers
56 views

Understanding quadratic root

Quadratic root is defined as $\sqrt{ x^2} = |x|$. Easy to remember, but seems to lack logic. And this topic is about you proving me wrong. 1) This definition of a square root is not universal and is ...
2
votes
0answers
29 views

Need reference for fact about roots of characteristic polynomials of recurrences

Many famous sequences $\{a_n\}$ satisfy recurrence relations. For example, the Fibonacci numbers $\{0,1,1,2,3,5,\ldots\}$ and Lucas numbers $\{2,1,3,4,7,11,\ldots\}$ both satisfy $$ a_n = a_{n-1} + ...
2
votes
2answers
59 views

Is it true that for any $x>2$ and $k,j \in N$ with $k>j$, $p_k(x)$ is larger than $p_j(x)$?

Let $p_k(x)=(x+\sqrt{x^2-4})^k-(x-\sqrt{x^2-4})^k$. Is it true that for any $x>2$ and $k,j \in N$ $k>j$, $p_k(x)$ is larger than $p_j(x)$?
2
votes
2answers
56 views

Is this polynomial irreducible? $f(x)=x^3-117x+53$

I am trying to find if $f(x)=x^3-117x+53$ is irreducible or reducible. I tried Eisenstein's criterion but this fails as $53$ does not divide $117$ After some thinking, if f(x) has any roots then ...
1
vote
1answer
51 views

Quadratic Equation; Roots' Magnitude Less than 1

What are the conditions on $a$ and $b$ so that the roots (real or complex) of the equation have magnitude $< 1$. $$λ^2 − (a − b + 1)λ + a = 0$$ On a separate note, if you could explain (NOT ...
0
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1answer
25 views

Is there a closed form expression for the intersection point of a linear function and a exponential function?

$$ a x = be^{cx} - 1 $$ where $a,b,c$ are constant and $a > 0, 0 < b < 1$ and $c>0$. Is there a closed form expression for this function?
10
votes
1answer
67 views

Closed form solution for the zeros of an infinite sum

Does there exist a closed form expression for the zeros of the following equation? $$\sum\limits_{n=1}^\infty\frac{1}{n^4 - x^2} = 0 \text{ where } x \in \rm \mathbb R$$ Could you suggest a ...
0
votes
1answer
58 views

Infinite sum solution

I was wondering if there is a closed form expression for the zeros of the following equation: $$\sum\limits_{n=1}^\infty\frac{1}{n^4 + x^2} \text{ where } x \in \rm I\!R$$ If it not exists, could ...
3
votes
2answers
55 views

Prove that largest root of $Q_k(x)$ is greater than that of $Q_j(x)$ for $k>j$.

Consider the recurrence relation: $P_0=1$ $P_1=x$ $P_n(x)=xP_{n-1}-P_{n-2}$; 1). What is closed form of $P_n$? 2). Let $k,j.\in\{1,2,...,n+1\}$ and $Q_k(x)=xP_{2n+1}-P_{k-1}.P_{2n+1-k}$. Then ...
0
votes
0answers
31 views

Comparison between roots of $p$ and $q$

Let $p(x)$ and $q(x)$ are two monic polynomials with non negative roots such that $p(x)=f(x)-g(x)$ and $q(x)=f(x)-h(x)$. If we know all the roots of $f,g,h$, can we say anything about largest positive ...
0
votes
1answer
33 views

Does this Question Have Enough Information to Answer?

I need to determine the equation of the function in the graph below. Attached is a graph with x-intercepts at (-4, 0) and (3,0) and another point given at (2,10). I know its not quadratic. Are ...
0
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0answers
33 views

Implementation of Poincaré–Miranda theorem

To test whether a continuous function has one simple root in a given interval $[x0, x1]$ is relatively easy: according to Intermediate value theorem when the sign of function value at $x0$ is opposite ...
0
votes
1answer
29 views

How to find roots provided by integer values on a function?

If we have a function (defined on the real or complex numbers) is there a technique that we can use to find roots provided by integer arguments? For example, Some f(x) have some roots like x=17.45, ...
0
votes
1answer
33 views

Find the exact solution of f(x)=0 where f(x)=3^(3x+1)-7*5^(2*x)

Find the exact solution of $$f(x)=0$$ where $$f(x) = 3^{3x+1}-7\cdot 5^{2x}$$ In my Numerical Analysis class, we have used methods such as Newton's method, Secant method, and bisection method to ...
2
votes
2answers
50 views

Zeros of order $m$

Suppose that $f$ has a root $\alpha$ of order $m$. Prove that $\alpha$ is a root of order $2m-1$ for $$ \theta(x)=f(x+f(x))-f(x). $$ My attempt: We can take $f(x)=(x-\alpha)^{m}q(x)$ where ...
1
vote
0answers
41 views

Removing exponent from equation

I'm trying to solve the following equation numerically: This is problematic, because the term $ t^{\beta_i}_{j} $ becomes extremely large ($> 10000^{300}$), and unrepresentable with typical ...
1
vote
0answers
37 views

Are there iterative formulas to find zeta zeros?

I am wondering whether one could find Riemann zeta zeros iteratively by using relationships such as this one: $$\rho _1=\lim_{s\to 1} \, \frac{\zeta (s) \zeta \left(s \cdot \rho _1\right)}{\zeta ...
2
votes
0answers
33 views

Characterizing infinitely oscillating functions on $L_2[0,1]$

Descarte's Rule of Signs tells me, as a function of the (sign pattern of the) vector $\mathbf{a}\in\mathbb{R}^{n+1}$, a bound on the number of roots of the polynomial $p(k)=\sum_{k=0}^n a_k x^k$. Is ...
1
vote
0answers
31 views

Characterization of roots for a cubic equation

I have been trying to characterize the roots of the equation $$ f(x) = x \left(1-x^2 \right) + \left(1+x^2 \right) \frac{\sum w_i y_i }{n} - x \frac{\sum w_i^2 + \sum y_i^2}{n} = 0 $$ where ...
3
votes
1answer
45 views

Determine all the roots of the equation given by $z^2(1-z^2)=16.$

For my third year Complex variable course, the question is Determine all the roots of the equation given by $$z^2(1-z^2)=16.$$ My attempt: Let $z^2 = x$ $x(1-x) = 16$ $x-x^2 = 16$ $x^2-x-16 ...
1
vote
1answer
29 views

Asymptotic expansion of roots of function

Find expansions for all roots of the equations below as epsilon → 0 with two nonzero terms in each expansion I don't see how drawing the graph will help. Also how do I go about balancing the sizes ...
0
votes
1answer
16 views

Bessel Function and roots

How would you prove that if $p_k$ and $p_{k+1}$ are two consecutive zeros of the Bessel function $J^{(0)}(t)$ then there exist a $t_1$ such that $p_k\leq t_1 \leq p_{k+1}$ and $J^{(1)}(t_1) = 0$?. Is ...
0
votes
1answer
18 views

Using either x-roots or logarithms to simplify equation

I have the equation: $.5 = 1-(1-s^r)^b$ And want to solve for $s$. Would the correct solution involve (1) b-roots and r-roots or (2) logarithms? Here's approach (1): Here's approach (2): ...
0
votes
0answers
30 views

Root-finding algorithm for shifted Signum function

So a simplified version of my problem is essentially this : I am working with a discrete function of the form $f(x)=\text{sgn}(x-i)$ for a natural number $i$ (where $\text{sgn}(x)$ is $1$ for $x\geq0$ ...
1
vote
1answer
49 views

Prove that if p is a prime, then each element of $\Bbb{Z}_p$ is a root of $x^p-x$.

Question is given in the title. I have tried picking an arbitrary element $\overline{a}$ and then using the division algorithm to try to show that my remainder is zero when dividing $x^p-x$ with ...
3
votes
2answers
51 views

I think I've found all roots to $f_k(x)=\sum_{j=1}^k x^j-x^{-j}$ for any $k$ - how to prove it?

Conjecture: The set of unique roots of $$f_k(x)=\sum_{j=1}^k x^j-x^{-j} \;,\;\; x \not=0$$ is given by $e^{i \pi \phi_k}$, where $$\frac{1}{2}\phi_k=\{0, \frac{1}{2}, ...
0
votes
2answers
25 views

Find lambdas so that two polynomials share a root and find such root

Find $ \lambda \in \mathbb{C} $ such that the following polynomials share a root, and find that root. $$ \lambda x^3-x^2 - x - (\lambda + 1) \quad and \quad \lambda x^2 - x - (\lambda +1) $$ I ...
0
votes
1answer
37 views

Newton's method for function $f(x)=1- e^{x}$

I am trying to find a solution for Newton's method for this function, $f(x)=1- e^{x}$, and figuring out if it converges for every initial starting point. Now, from my understanding, Newton's method ...
0
votes
1answer
38 views

Polynomial with odd number of real roots

I have been trying to characterize the number of roots on $\theta$ for the following polynomial $$ \sum_{i=1}^n \frac{\theta- x_i}{1+ \left(x_i - \theta \right)^2} = 0$$ If we were to put ...
0
votes
5answers
46 views

What are the roots of this function with absolute values?

The function is the following $$f(x)= (\lvert x\rvert + x^2)e^{-x}$$ I don't understand how to deal with the $\lvert x\rvert$
1
vote
1answer
38 views

How to find two square roots whose difference is greater than one.

How do you find the greatest $n$ such that the difference of its square root from some other integer is greater than or equal to one? For example : $$2011^{1/2} - n^{1/2} \ge1$$ What should be the ...
1
vote
5answers
56 views

$ p \in Q[x] $ has as a root a fifth primitive root of unity, then every fifth primitive root of unity is a root of $p$.

I'm extremely stuck. Can't figure it. The conjugate is easy: let $w$ be a primitive root of unity, then $w^{-1}$ will also be a root, that's easy. But I'm missing $w^2$ and $w^3$. Why would they be ...
2
votes
1answer
35 views

Polynomials and their (real) roots

The following exercise can be found in the book Some exercises in pure mathematics with expository comments by authors J. D. Weston and H. J. Godwin (it is the exercise $166$ at page $43$). Three ...
1
vote
1answer
40 views

Find the three positive values of p for which the equation $px^2-4x+1=0$ will have rational roots

Question: Find the three positive values of p for which the equation $$px^2-4x+1=0$$ will have rational roots. My attempt (Algebraically): Usually if it has to have rational ...
0
votes
1answer
21 views

Muller's method Three Initial values

I am trying to sole these two questions. I know how to do the Muller's method. How can I find the "Three initial values(x1, x2, x3)" for question 3? My guess of initial value for question 3 is that ...
1
vote
1answer
74 views

How to solve cubic equations with given coefficients?

I have a large data set that requires a cubic equation to be solved for each point. There are too many points to use Goal Seek (numerical Excel method) on them all. For example: $$y=7\cdot ...
1
vote
3answers
57 views

approximate the root of perturbed polynomial

Approximate the root of $$f(x)=(x-1)(x-1)(x-3)(x-4)-10^{-6}x^6$$ near $r=4$. Do I have to use iterative method of finding the root, such as Bisection, Secant, etc? Is there other way?
1
vote
0answers
39 views

Analytic formulas for recursive sequences of polynomials

I have a sequence of polynomials $\{ f_{m} \}$ with $f_{m} \in \mathbb{Z}[x]$ that satisfies an order 2 linear recurrence relation. In particular, I have a polynomial $g \in \mathbb{Z}[x]$ such that ...
2
votes
0answers
46 views

Asymptotic behavior of zeros of a function

Let $f(x,m)=(2m-1)\Gamma(m)\,x^{-m}$ where $x>0$ and $\Gamma(z)$ denotes the Gamma function. Let $g(x,m)=f(x,m)+f(x,-m)$. I'm interested in the solution $m=m(x)>0$ of the equation $g(x,m)=0$ ...
3
votes
2answers
64 views

How does this factoring work?

$$ (z^2 - 2i ) = (z -1 -i)(z + 1 +i) $$ I see if you multiply out the right-hand side, you obtain the left-hand side, but how does one know to factor like that or this? $$ (z^2 − 3iz − 3 + i) = (z − ...
0
votes
0answers
23 views

Question on polynomials

What I know is that if the above said equation has three equal roots that meant that the above equation can be written as (x-q)^3*(x-w)*(x-e)... Now do we have to compare this equation with the ...
0
votes
0answers
24 views

What are the necessary and sufficient conditions to guaranty this property for sequences?

In this page we know that if a function $f:ℝ→ℝ$ or a sequence has infinitely many zeros, then it is not necessary to changes its sign infinitely many times. My question is: What are the necessary ...
2
votes
2answers
54 views

Prove that a polynomial of odd degree in $ \Bbb R[x]\\$ with no multiples roots must have an odd number of real roots.

The coefficients of the polynomial are in the ring of real numbers. Prove that a polynomial of odd degree in $ \Bbb R[x]\\$ with no multiple roots must have an off number of real roots. I hate to ask ...
1
vote
2answers
59 views

Existence of real roots for vector quadratic equations

I have a vector quadratic equation of the form $\boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{x} + \boldsymbol{x}^{T} \boldsymbol{b} + c = 0$ where $\boldsymbol{A}$ is symmetric and for my particular ...
0
votes
2answers
45 views

Finding all the roots of a function - Numerical methods

I'm trying to find all the roots of a function f(x) without algebraically solving the function. What I have done so far: I attempted to use a numerical method, more specifically the secant method to ...
0
votes
4answers
145 views

Are there real solutions to $x^y = y^x = 3$ where $y \neq x$?

I need to solve the following equation for (x,y) $$x^y = y^x = 3$$ Everytime I run a numerical method for this problem, I get $$ (x,y) = (1.82546...,1.82546..) $$ I expect there to be a solution ...