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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4
votes
3answers
253 views

How can I show why this equation has no complex roots?

I've been asked to show why an equation has no complex roots but i'm at a complete loss. The equation is $F_{n+2}=F_n$ Where $F_n=(x-1)(x-2)...(x-n)$ and n is a positive integer. I'd really ...
-3
votes
1answer
22 views

Find two roots for $\cos 5x=a$

For some $a>0$ equation $\cos 5x=a$ has two roots, difference between them is $\large\frac{7\pi}{4}$. Find all roots of this equation!
0
votes
0answers
48 views

Polynomial of Degree 3 Solutions [duplicate]

If $p(x) \in F[x]$ is of degree $3$, and $p(x)=a_0+a_1x+a_2x^2+a_3x^3$, show that $p(x)$ is irreducible over $F$ if there is no element $r\in F$ such that $a_0+a_1r+a_2r^2+a_3r^3 =0$. If $p(x)$ is ...
-1
votes
2answers
65 views

Existence of a root

Let $f:[a,b] \rightarrow \Bbb R$ continuous, such that for every $x$ there is a $y$ such as that $|f(y)|\leq|f(x)|/2$. Show there exists a $\xi$ such that $f(\xi)=0$
0
votes
1answer
34 views

Finding complex roots of integer polynomials

How would one find approximates for complex root of polynomial with integer coefficients,I know for example the Newton's method $$x_n=x_{n-1}-\frac{f(x_{n-1})}{f'(x_{n-1})}$$ Anyway is it possible to ...
2
votes
2answers
39 views

Difference between the complex roots of $f(x)$ and $|f(x)|^2$

I suppose a basic question, but it's causing me more problems than I envisioned! I have some polynomial $f(x)$ for which the roots are complex, $x+iy$. How will these roots change if I now take ...
3
votes
1answer
115 views

The number of solutions of $z^5+2z^3-z^2+z=a$ for $a\in \mathbb{R}$

How we can calculate the number of solutions of $$z^5+2z^3-z^2+z=a\;\;,\;\;a\in \mathbb{R}$$ in the half-plane $\mathfrak {Re}(z)\ge 0$. Any hint would be appreciated.
2
votes
0answers
32 views

What is the (currently) optimal root finding algorithm for multivariate functions? [closed]

Let's say we wish to find the roots of the function: $f(x,y,\cdots) = 0 \;,$ so, for a minimal example: $xy - 1 = 0 \; .$ I know there are different methods to solve this problem for the ...
4
votes
2answers
222 views

Problem getting the real roots of this complex expression

I'm trying to get the real roots of this expression: $$\dfrac{1}{z-i}+\dfrac{2+i}{1+i} = \sqrt{2}$$ Where $i^2=-1$ and $z=x+iy$. I tried to simplify that with Algebra, and then separate the real ...
0
votes
4answers
44 views

One root of the equation $x^2-(r+3)x+(5r-3)=0$ is twice the other root. Find the two possible values of r. [closed]

One root of the equation $x^2-(r+3)x+(5r-3)=0$ is twice the other root. Find the two possible values of $r$. I need help with this question, thank you.
-1
votes
2answers
61 views

Form a quadratic equation whose roots are $\sqrt{3}+2$ and $\sqrt{2}+3$ [closed]

Form a quadratic equation whose roots are $\sqrt{3}+2$ and $\sqrt{2}+3$ I need help with this question, thank you
7
votes
3answers
174 views

Upper and lower bounds for the smallest zero of a function

The function $G_m(x)$ is what I encountered during my search for approximates of Riemann $\zeta$ function: $$f_n(x)=n^2 x\left(2\pi n^2 x-3 \right)\exp\left(-\pi n^2 x\right)\text{, ...
4
votes
2answers
126 views

How to solve $\displaystyle x=\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-x}}}}$ for $x$?

How to solve $\displaystyle x=\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-x}}}}$ for $x$? I tried this way: Let $$f(x)=\sqrt{4+\sqrt{4-x}}$$ So, $x=f^2(x)=f^{2n}(x)$ where $n\in\mathbb{N}$. Then, I tried to ...
0
votes
1answer
39 views

Zeros of quadratic form of vectors

I have a set of vectors defined as $[\mathbf{v}(x)]_n = e^{jn\pi x}; \quad n = 0 ~\text{to}~ (N-1)$ where $\mathbf{v}$ is an $N \times 1$ vector, $j$ is $\sqrt{-1}$, and $-1 \leq x < 1$. For a ...
7
votes
1answer
147 views

Do perfect polynomials of degree $4$ exist?

I asked this question already, but I cannot find it anymore. If it is a duplicate, I will delete it. Is there a polynomial $$p(x)=x^4+ax^3+bx^2+cx+d$$ such that p and all the derivates upto the ...
3
votes
1answer
42 views

Find zero of sum of 4 modified Bessel functions

I am trying to find the (positive) root of the function $f(x) = I_{-3/4}(x) + I_{3/4}(x) - I_{-1/4}(x) - I_{1/4}(x)$ where $I_\alpha(x)$ denotes the modified Bessel function of the first kind. ...
0
votes
0answers
32 views

Counting Zeros of complex functions in the upper half plane

I have a question about counting zeros. Here it goes Given $f(x)= i z^5+z-2010$. Find the number of zeros of $f$ in the upper half plane $\operatorname{Im}(z)>0$. I have tried to use the Argument ...
1
vote
2answers
28 views

Do polynomials $ P(t)$ of an odd degree have at least one real root belong to $(t-a)Q(t)$?

This is a continuation of a question where ker(T) = (t-a)Q(t) = P(t). Show that {P(t) ∈ R[t] | deg(P(t)) = 3} ⊂ $∪_{a∈R}$ker(T). So the mark scheme says that all polynomials in R[t] of an odd ...
0
votes
3answers
73 views

How do I solve the trigonometric equation $\sec^3x - 2 \tan^2 x = 2$? [closed]

A friend asked to me how could she resolve this equation, but I don't know how to resolve it?? Could you help me?. The equation is : $\sec^3x - 2 \tan^2 x = 2$ Note: She told me that I can use ...
0
votes
1answer
19 views

Marking the roots of a quadratic function in Scilab

I have 2D plotted a simple quadratic function in Scilab and now have to mark the roots with an X. Is there any simple way of doing that? I have written a function that calculates the roots and ...
5
votes
1answer
62 views

Entire function with zeros of even multiplicity is the square of another entire function

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an entire function such that the multiplicity of each of its zeros is even. Must there exist an entire $g$ such that $f(z) = g(z)^{2}$? Progress I ...
4
votes
0answers
92 views

Level curves of a polynomial and the zeros of its higher derivatives.

The Gauss--Lucas Theorem states that all zeros of a degree $n$ complex polynomial $p(z)$ are contained in the convex hull of the zeros of $p$. By iteration, this implies that the zeros of ...
4
votes
1answer
96 views

Proof that $\sqrt[m]{a} + \sqrt[n]{b}$ is irrational

Is there a way to prove that $\sqrt[m]{a} + \sqrt[n]{b}$ ($\sqrt[m]{a}$ and $\sqrt[n]{b}$ are irrational); $a, b, m, n \in \mathbb{N}$; $m, n \neq 2$; is irrational without using the theorem mentioned ...
2
votes
2answers
44 views

Number of irrational roots of the equation $3^x8^{\frac{x}{x+1}}=36$

Find the number of irrational solutions of the equation $$3^x8^{\frac{x}{x+1}}=36.$$
1
vote
3answers
65 views

Find polynomial whose root is sum of roots of other polynomials

We have two numbers $\alpha$ and $\beta$. We know that $\alpha$ is root of polynomial $P_n(x)$ of degree $n$ and $\beta$ is root of polynomial $Q_m(x)$ of degree $m$. How do you find polynomial $R_{n ...
0
votes
1answer
61 views

Show that $\sqrt{2}$ is irrational using integer root theorem

Show that $\sqrt{2}$ is irrational using integer root theorem. Let $P(x)=x^2-2$. Since $\sqrt{2}$ is a root of this polynomial, had it been a rational (suppose $\sqrt{2}=\frac{p}{q}$) no, by ...
-1
votes
2answers
25 views

Multiplicity of roots in finite fields of order prime. [closed]

I am having trouble with completing this question from last years exam (part a and d) Let p be a prime, and $f = x^5-1 = (X-1)(X^4+X^3+X^2+X+1) \in \mathbb{F}_p[X]$ Show: (a) if $p\neq 5$ then every ...
4
votes
2answers
81 views

Connection with golden ratio?

Consider the following problem: Let $p\in\mathbb{Z}[x]$ be a polynomial with integer coefficient. Suppose that the leading coefficient is 1, all roots are real and in $(0, 3)$. Find all ...
0
votes
1answer
43 views

How to obtain the number of real valued zeroes of a polynomial?

While I know there's no analytical formula for the roots of a general polynomial of degree five and higher, I wonder whether there is at least something like a parabola's discriminant to determine how ...
0
votes
2answers
51 views

Integer roots to cubic equation

If I have a cubic equation $x^3 + ax^2 + bx + c = 0$, what constraints exist on $a,b,c$ when we have three integer solutions? How do I choose $a,b,c$ to force integer solutions?
2
votes
3answers
41 views

What are the roots of $\sin(ax) + \sin((a + 2)x)$?

I was playing around with $\sin(5x) + \sin(7x)$, wondering where the roots of the function are. I graphed it on wolframalpha and from the list of solutions I guessed that the solutions to $\sin(5x) + ...
1
vote
1answer
52 views

If $f$ is the limit of polynomials with only real zeros, then all zeros of $f$ are real

Problem Let $f$ be a non-constant entire function. Suppose that there is a sequence of polynomials ${P_n(z)}$, $n=1,2,...$ such that $P_n(z)$ converges uniformly to $f$ on every bounded set ...
4
votes
1answer
43 views

Is it possible to calculate the roots of the difference between successive terms of this polynomial series $\rm{P}_n (x)=x\rm{P}_{n-1}-r\rm{P}_{n-2}$

Consider the polynomial series defined by the following recursion formula: $$ \begin{align} &\mathrm{P}_0 = 1 \\ &\mathrm{P}_1 = x-r \\ &\mathrm{P}_n = x\mathrm{P}_{n-1} - ...
2
votes
1answer
43 views

Limit solution to a transcendental equation

Let $n\ge 1$ be a positive integer. The question is to solve the following transcendental equation: \begin{equation} \left(1+q\right)^{2 n} = \frac{\sqrt{\pi}}{2} \frac{1-q}{\sqrt{q}} \sqrt{n} ...
0
votes
2answers
56 views

Simple equation $2^x = 16$ [closed]

Solve the following equation: $$2^x = 16$$ What is $x$? For $x = 4$, how do the $16$ and $2$ relate?
1
vote
0answers
28 views

Roots of this trigonometric polynomial

Let $f:[0,2\pi) \rightarrow \mathbb{R}$ with $f(x):=\sum_{n=0}^{k}a_n \left(1+\cos(x)\right)^n$ for arbitrary $a_n$ with $a_k \neq 0$. My question is: What is the maximum number of zeros that this ...
2
votes
0answers
30 views

Gaps between roots of trigonometric polynomials?

Given a polynomial in $e^{\mathrm{i}k t}$ of the form $$ p(t) = \sum_{-n\leq k\leq n} c_k e^{\mathrm{i}k t} $$ with $\bar c_{-k} = c_k$, is there a good way of characterising how close its roots can ...
1
vote
3answers
50 views

Solving a Perturbed Cubic Equation

Consider a cubic equation $(1 + \epsilon)x^3 - 2ax^2 + (a - 3\epsilon)x + 2\epsilon = 0$ where $\epsilon > 0$ and $a \gg 1$. In the limit of $\epsilon \rightarrow 0$, $x(x^2 - 2ax + a) = 0$ so ...
3
votes
3answers
190 views

Roots of polynomial of degree 6

I'm struggling to find the complex roots of $x^6-9x^3+8 = 0$. I've managed to find the real roots (1 and 2) by letting a variable, say $α = x^3$ and substituting where relevant, leading to a ...
2
votes
1answer
53 views

Checking tolerance of Newton-Raphson method to calculate square root

Finding the square root of $c$ is finding the solution to: $$x^2 - c = 0.0$$ We can use Newton's method to successively approximate the solution. My question is how to check whether we are within ...
1
vote
1answer
72 views

Roots of $\tan x - x$

The function $\tan x - x$ has exactly one root $x_n$ in the interval $(n\pi, (n + \frac{1}{2})\pi)$. Show that $$x_n = n\pi + \frac{\pi}{2} - \frac{1}{n\pi} + r_n$$ where $\lim_{n\rightarrow \infty} n ...
2
votes
4answers
41 views

Number of distinct real roots with $e^{-x}$ in the equation

How to find the number of distinct real roots of the equation $$13x^{13}-e^{-x}-1=0$$ I know that we generally find number of real roots by observing number of sign changes in $f(x)$ and $f(-x)$ but ...
1
vote
1answer
28 views

Sum of Non Real Roots of Bi Quadratic

Consider $$f(x)=8x^4-16x^3+16x^2-8x+k=0$$ where $k \in \mathbb{R}$,then find sum of non Real roots of f(x). My approach: we have $$f'(x)=32x^3-48x^2+32x-8=0$$ Also ...
2
votes
1answer
105 views

Solving equation $a^{-x} + \log x/\log a = 0$

Please can you instruct me how should I start writing an algorithm (pseudo-code, to be implemented) for finding all solutions for the following equation: $a^{-x} + \log x/\log a = 0$ where $a$ ($a$ ...
0
votes
1answer
21 views

Projective roots of a homogeneous polynomial

Suppose that $f(X,Y)\in\mathbb C[X,Y]$ is a homogeneous polynomial of degree $n$, then we can consider it as a function on $\mathbb P^1_\mathbb C$. It has at most $n+1$ projective roots (points of ...
1
vote
0answers
29 views

Quartic Polynomial Manipulation

I have a quartic polynomial in $x$ (too long to write here) $f(x,c_1, c_2, c_2)$ where $c_1, c_2, c_3$ are constants which I have complete freedom over how to fix their values, as long as they are ...
0
votes
2answers
26 views

Understanding Multiplicities

I am having troubles understanding what 'multiplicities' mean. In example what does $-1/3(multiplicity 2)$ translate into?? To clarify this is for finding zero's in a polynomial function Any help ...
5
votes
1answer
80 views

To prove this complex polynomial has all zeros on unit circle

I'm trying to prove a self-inversive polynomial $P(z) = \sum\limits_{n=0}^{N-1}a_nz^n$ has all its roots on the unit circle. The coefficients are such that $ a_n = e^{j(n-\frac{N-1}{2})\pi u_0} - ...
3
votes
1answer
77 views

Find roots of $3z^{100} - e^z$ in the unit disc.

This question was given in an exam in complex analysis: Let $f \left( z \right) = 3z^{100} - e^z$. Find all of $f$'s roots in $D \left ( 0,1 \right)$ and show that they are simple roots. I've seen ...
1
vote
1answer
80 views

The number of zeros in a polynomial

Does anyone know of a relationship between the number of zeros (complex and real) and the degree of a polynomial? Specifically, if a polynomial has a double root, is it compensated with complex ...