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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0
votes
2answers
162 views

Graeffe's root finding method

What are the practical applications of Graeffe's root finding method?I searched a lot but couldn't find.I found that it is used in aerodynamics and electric circuit analysis.But don't know much about ...
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
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 ...
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 ...
-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!
-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 $f(\xi)=0$
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.
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 ...
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 ...
5
votes
4answers
103 views

What is the minimum value of $abc$

If the roots of the equation $$ax^2-bx+c=0$$ lie in the interval $(0,1)$, find the minimum possible value of $abc$. Edit: I forgot to mention in the question that $a$, $b$, and $c$ are natural ...
-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
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.
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 ...
3
votes
2answers
144 views

Behavior of $f(z)=\int_0^1\mathrm{e}^{\alpha t^2}\sin(tz)\,dt$ when $\alpha <0$

Define $$f(z)=\int_0^1\mathrm{e}^{\alpha t^2}\sin(tz)\,dt,$$ where $\alpha \in \mathbb{R}$. If $\alpha >0$ then $f(z)$ has infinitely many real zeros and at most a finite number of complex ...
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 ...
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 ...
0
votes
1answer
61 views

Rate of convergence of an iterative root finding method similar to Newton-Raphson

We are defining an algorithm as follows: Let $f(x)$ be a function with a root in $[a,b]$. We define a series $\{x_k\}_{k=1}^{\infty}$ as follows: $x_{k+1}=x_k-f(x_k)\frac{b-a}{f(b)-f(a)}$. ...
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 ...
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 ...
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 ...
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 ...
0
votes
2answers
99 views

How to solve the cubic equation $ x^3+3x -2 = 0$ without using matrices?

I am trying to solve $ x^3+3x -2 = 0$ Using the remainder theroem but none of the factors of the constant make the equation equal to $0$. Is there any way I can get the answers without using matrices? ...
3
votes
5answers
1k views

How to solve the cubic equation $x^3-12x+16=0$ [closed]

Please help me for solving this equation $x^3-12x+16=0$
0
votes
2answers
51 views

Number of real roots of $2^x = 1-x^2$ for $x\in (0,1)$

How can I found no. of real roots of $2^x = 1-x^2$ in $x\in (0,1)$ I did not found a method by which i can draw graph of two curve in the interval $x\in (0,1)$ please help me , Thanks Sorry ...
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 $$3^x8^{\frac{x}{x+1}}=36.$$
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
38 views

The dependence of the number of solutions of the equation $x^3-3x=a$ on the parameter $a$

Find the dependence n (a) the number of solutions the equation given a parameter 1) $x^3-3x=a$ 2) $e^2x=ax$ 3) $x^ax=e (x>0)$ For example what I did for (1) is $x(x^2-3)=a$ $x=a , x^2-3=a$ ...
4
votes
3answers
178 views

If $x\in\mathbb R$, solve $4x^2-40\lfloor x\rfloor+51=0$.

If $x\in\mathbb R$, solve $$4x^2-40\lfloor x\rfloor+51=0$$ where $\lfloor x\rfloor$ denotes the integer part of the number. $\lfloor x\rfloor\le x$ and $\lfloor x\rfloor=x-\{x\}$, where $\{x\}$ ...
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 ...
7
votes
3answers
209 views

All roots of the quartic equation $a x^4 + b x^3 + x^2 + x + 1 = 0$ cannot be real

Problem Prove that all roots of $a x^4 + b x^3 + x^2 + x + 1 = 0$ cannot be real. Here $a,b \in \mathbb R$, and $a \neq 0$. Source This is one of the previous year problem of Regional ...
1
vote
2answers
83 views

How to find the zeros of $f(x) = 2x(5-x)$

How do I get the zeros, if $f(x) = 2x(5-x)$. I have told by my classmate that in order to get the zeros of $f(x) = 2x(5-x)$, I need to distribute $2x$ to $(5-x)$. So I distribute it to make it ...
1
vote
2answers
111 views

Solve the equation $x^{2n} + 1 = 0.$ Use these solutions to find a factorization of $x^{2n} + 1$ with real coefficients.

I am asked to solve the equation $x^{2n} + 1 = 0,$ and to use these solutions to find a factorization of $x^{2n} + 1$ with real coefficients. I am given the hint that pairing factors arising from ...
-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 ...
0
votes
1answer
65 views

Prove that the roots of $2x^3 - x + 5 = 0$ are irrational

We want to prove that for the equation $2x^3 -x + 5 = 0$, any root must be irrational. How can this be done? Seems like plugging in $x = a/b$ doesn't really help at 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
6answers
109 views

How to find all solutions of $4^x-3^x=1$?

I have problem with equation: $4^x-3^x=1$. So at once we can notice that $x=1$ is a solution to our equation. But is it the only solution to this problem? How to show that there aren't any other ...
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
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?
1
vote
1answer
147 views

no. of real roots of exponential equation in three questions

How Can i calculate no. of real roots of exponential equation in three questions (1) $2^x = 1+x^2$ (2) $2^x+3^x+4^x = x^2$ (3) $3^x+4^x+5^x = 1+x^2$ My Try:: (1) Let $f(x) = 1+x^2-2^x$ now ...
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
4answers
133 views

Solve the equation $x^x=10^9$.

The main question was to solve $x\log_{10}{x}=9$. I reduced it to this equation. This is $x$ Degree equation. How to solve this? I know this can be solved by newton's method. But I am not getting how ...
6
votes
4answers
127 views

Solve $x^{3}-3x=\sqrt{x+2}$

Solve for real $x$ $$x^{3}-3x=\sqrt{x+2}$$ By inspection, $x=2$ is a root of this equation. So, I squared both sides and divided the six degree polynomial obtained by $x-2$. Then I got a ...