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)

1
vote
2answers
423 views

What does a complex root signify?

What does it tell me when I find that a polynomial has complex roots, except for the obvious fact that it crosses zero for these values? To me it seems that the existance of complex roots must have ...
0
votes
0answers
15 views

Roots of the Lagrange polynomials

This question follows my previous one Coefficients of Lagrange polynomials. Notations : $ n\in\mathbb{N}^*$ $[|1,n|]=\{1,2,\dots,n\}$ $A=(a_1,\dots,a_n)\in\mathbb{K}[X]^n$ all different numbers ...
0
votes
0answers
20 views

Finding the roots of a polinomial function obtained by a Binomial c.d.f.

I came across with the following question and I am also attempting to solve it. Let $B(K/2;K,1-x)$ be the Binomial c.d.f. with $K$ trials having at least $K/2$ success with each trial having success ...
1
vote
0answers
23 views

Location of Roots Symmetric Polynomial

I'm trying to prove (or disprove) that the roots of an even degree real symmetric coefficient polynomial are all on the unit circle. If it is not true, I will then try to find the conditions such that ...
0
votes
2answers
41 views

Roots of $x^{4} -28 x^{2}+49$ with Horner

I am studying Horner's algorithm and I got a problem I can't solve. The polynomal is $x^{4} -28 x^{2}+49$. After trying $\pm 1, \pm 7, \pm49$ with Horner I couldn't find any solution. Wolfram alpha ...
12
votes
1answer
153 views

Fitting a closed curve on the roots of ${x \choose k}-c$

Let $${n \choose k} = \frac1{(n+1) \operatorname{B}(n-k+1, k+1)}.$$ be the generalized binomial coefficient, and here $\operatorname{B}$ is the Beta function. Let $f_{k,c}(x)$ be the following ...
2
votes
0answers
41 views

Numerically solve for maximum root

I am looking for an efficient algorithm that can numerically solve a piecewise function for its maximum zero root. The piecewise function will normally take the form of the plots below where by below ...
0
votes
0answers
29 views

Show that the $n^{th}$ eigenfunction has $n-1$ zeros

Just a quick question. If I have an eigenfunction of a Sturm-Liouville form problem: $\phi_n = \sum_{n=1} A_n \sin\left[\frac{n\pi}{\log{b}}\log{x}\right]$, with $x$ between $1$ and $b$ - why is ...
0
votes
2answers
41 views

A Polynomial that Passes through the following four points?

I'm trying to do this for practice but I'm just going nowhere with it, I'd love to see some work and answers on it. Thanks :) Find a polynomial that passes through the points (-2,-1), (-1,7), ...
6
votes
0answers
61 views

Implications from $f(z)\in\mathbb{R} \Longleftrightarrow z\in \mathbb{R}$ [duplicate]

Let $f:D(0,1)\longrightarrow \mathbb{C}$ be a holomorphic function such that $f(z)\in\mathbb{R} \Longleftrightarrow z\in \mathbb{R}$. How to prove that $f$ has at most one zero on the disk. By ...
3
votes
1answer
23 views

Solving equations: reasoning doesn't work backwards?

In doing my (high school) math homework, I came to an issue that doesn't make sense to me. Given an equation $0 = a_1 + a_2x + a_3x^2 + \dots$, we can multiply both sides by $x$ to obtain $0 = a_1x + ...
2
votes
1answer
38 views

A positive polynomial is the sum of two squares in $\mathbb{R}[X]$ [duplicate]

Let $P\in\mathbb{R}[X]$ be a positive polynomial. I want to show that there exists $A,B\in\mathbb{R}[X]$ so that $P=A^2+B^2$ $\displaystyle ...
3
votes
2answers
72 views

Prove that $\prod_{k=0}^{n-1}(z-\mathrm{e}^{2k\pi i/n})=z^n-1$

Prove that $$ \prod_{k=0}^{n-1}(z-\mathrm{e}^{2k\pi i/n})=z^n-1. $$ In my some problem I have used $$ \prod_{k=0}^{7}(z-\mathrm{e}^{2k\pi i/8})=z^8-1. $$ I have verified this. So I think in general ...
0
votes
2answers
29 views

Stuck finding the zeros of a polynomial (complex and real)

Stuck finding the zeros of this polynomial (complex and real): $$x^4+2x^2+1$$ I am not sure how I would factor this. The constant value is really throwing me off. I just need a hint on how to get ...
1
vote
1answer
14 views

Horizontal Cylinder Gas Problem

We have a perfect cylinder with a diameter of 3 ft that lies horizontal. The gas gauge is broken so we are forced to use a dipstick to determine how much gas in our tank. In this problem we are ...
1
vote
1answer
17 views

Root of function involving trig and exponential

Would anyone know an analytical solution to finding the root of $$ f(x) = \sin(x^2) - e^x $$ in $[-1,1]$? I'm writing a simple root finding program and thought I'd try this as a test case, but ...
4
votes
0answers
63 views

Writing the roots of a polynomial with varying coefficients as continuous functions?

Consider the monic polynomial $$p_{\zeta}(z) = z^n + a_{n-1}(\zeta)z^{n-1} + \dots + a_0(\zeta), $$ where the $a_{i}$'s are continuous functions defined over $\mathbb{C}$. As is well known, the ...
0
votes
0answers
40 views

derivative and roots of polynomials

Given a polynomial $g(x)=\frac{f(x)}{(x-x_1)(x-x_2)$ can it be proven that the roots of $g'(x)=0$ would lie in the interval [x_1,x_2]? Real/Complex, im not sure
6
votes
1answer
74 views

Coincidence? : $d(ax^2+bx+c)/dx=\pm \sqrt{\Delta}$

As the title says, is it just a coincidence that $d(ax^2+bx+c)/dx=\pm \sqrt{\Delta}$? (where $\Delta=b^2-4ac$, i.e. discriminant of the quadratic). We can get this easily from rearranging the ...
0
votes
2answers
35 views

Prove the number of solutions a function has?

What methods/theorems are commonly used when trying to prove that a function has exactly one root within a given interval $(a,b)$, or that it has no roots? I have the function ...
0
votes
1answer
38 views

solve x for equation: square root (b^2 + 2ax) = x+a

At an exercise I get the following equation: solve x for equation: $$\sqrt{ b^{2} + 2ax} = x+a$$ My steps would be: $b^2 + 2ax = x^2 + a^2$ so $b^2 + 2a - a^2 = x$ But this is completely wrong!
-4
votes
4answers
37 views

Solve the equation $\sqrt{x^2+a^2} = 3a$ for $x$ [closed]

Solve for $x$ in the equation: $\sqrt{x^2+a^2} = 3a$
1
vote
1answer
23 views

Sum of $n^{\text{th}}$ powers of roots of quadratic

How would I go about finding an expression (preferably closed form) for the sum of $\alpha^n+\beta^n$ in terms of $\alpha + \beta$ and $\alpha\beta$ (where $\alpha$ and $\beta$ are the roots of a ...
3
votes
1answer
66 views

Why do we say that $\sqrt{-0} = -0$?

According to wikipedia's page on signed zeroes, we agree that $\sqrt{-0} = -0$. I would always have guessed that it would be $0i$, as $(0i)^2 = 0^2*i^2 = 0 * (-1) = -0$. I know that my own ...
1
vote
1answer
28 views

The root of $x^2+[1]=[0]$ in $\mathbb{Z}_p$

In $\mathbb{Z}_p$, where $p$ is a prime, how many roots of $x^2+[1]=[0]$? It is equivalent to show $[x^2]=[p-1]$,when p=3,there is non. When p=5, $x=2$,does there exist any rule of it
0
votes
1answer
25 views

Solution of $A = e^{\alpha t}\cos(\omega t + \phi)$

I would like to find the real roots of the function $$i(t) = \frac{\hat{V}}{R}\left(\frac{\omega^2}{(\alpha^2 + \omega^2)} \cos\left(\omega t + \tan^{-1}\left(\frac{\alpha}{\omega}\right)\right) + ...
1
vote
1answer
54 views

Roots of $f(x)=a_0+a_1\cos x+a_2\cos 2x+\dots+a_n\cos nx$

If $a_i$'s are nonzero real numbers such that $a_n > {\sum^{n-1}_{i=0}}|a_i|$ prove that the number of roots of $f(x)=a_0+a_1\cos x + a_2\cos 2x+\dots+a_n\cos nx$ is at least 2n.
2
votes
2answers
16 views

Roots of $i(t) = Ae^{\alpha t}cos(\omega t + \phi)$

I would like to find the roots of the function $i(t) = Ae^{\alpha t}\cos(\omega t + \phi)$ in the form $t = f(A, \alpha, \omega, \phi)$.
2
votes
1answer
19 views

Roots of a complex polynomial with leading coefficient larger than absolute sum of rest

Suppose I have an $N^{\text{th}}$ degree polynomial $P_N(z)=\sum_{i=0}^N a_i z^i$ where $\{a_i\}_{i=0}^N$ are complex numbers such that $|a_N|> \sum_{i=0}^{N-1}|a_i|$, can I claim that all its ...
0
votes
0answers
61 views

Real roots of a quintic polynomial with constraints

This is a question on the edge of math and programming. I pondered about the best way to state the problem: should I provide context, or get straight to the point of the question? Given various ...
2
votes
1answer
76 views

About the identity $e^{i\pi}=-1$

I have a question about the famous identity of Euler $e^{i\pi}=-1$. I opened the other day this question about the number of roots of a complex number with irrational exponent. Under this light and ...
0
votes
1answer
19 views

Lebesgue Measure of the set of roots of a complex exponential equation

In the following equation $\{\beta_i\}_{i=1}^N$ and $\{\alpha_i\}_{i=1}^N$ are non-zero complex numbers: $\sum_{i=1}^N \beta_i e^{\alpha_i t} = 0$. I would like to know if the Lebesgue measure of the ...
0
votes
1answer
47 views

Sign of Laguerre root finding iteration

I'm trying to understand the method by Laguerre for polynomial root finding. However, I have some difficulties to understand one sentence of the book Applied Computational Complex Analysis (vol. 1) by ...
4
votes
1answer
38 views

Prove that this Newton sum value is unique

$$\begin{align}a+b+c+d&=1\\ a^2+b^2+c^2+d^2&=2\\ a^3+b^3+c^3+d^3&=3\\ a^4+b^4+c^4+d^4&=4\\ a^5+b^5+c^5+d^5&- ?\end{align}$$ The usual method I see for solving this kind of ...
0
votes
2answers
27 views

Finding a cubic equation from the relation between the roots

I'm trying to solve this problem: $ x^3 - x^2 - 3x -10 = 0$ has roots α,β,γ. Let u = −α+β+γ. Show that u+2α=1, and hence find a cubic equation having roots −α+β+γ, α−β+γ, α+β−γ. I was able to ...
0
votes
0answers
26 views

Find all functions: $f:C\rightarrow C$

Find all functions $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))$ Extension: Find all $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))(x-f(2\pi))(x-f(-2\pi))\cdots ...
0
votes
1answer
41 views

Writing roots of f(x) as f(a) for some a

I was solving a problem when this random thought popped into my head. Suppose you have a function, say $f(x)=x^2-1=(x-1)(x+1)$. The roots of this function are $-1$ and $-1$. We can write these roots ...
0
votes
0answers
29 views

Rayleigh quotient iteration and root finding

I'm trying to find the roots of a polynomial by finding the eigenvalues of its companion matrix. I understand that it is possible to use QR algorithm as the matrix happens to be in Hessenberg form ...
1
vote
2answers
37 views

Relation between the roots and the coefficients of a polynomial

I have studied that: For the polynomial $ax^3+bx^2+cx+d=0$, with roots $\alpha, \beta, \gamma$: We have: $$\begin{align} & \alpha + \beta + \gamma = -\frac ba \\ & \alpha\beta + \beta\gamma ...
2
votes
2answers
92 views

Method of dominant balance and perturbation

Approximate the solutions of $$\epsilon x^4 + (x-1)^3=0$$ I can't perform a singular perturbation because if I let $\epsilon=0$ then I lose a root. My professor suggests The Method of Dominant ...
1
vote
0answers
41 views

Solve $z+\sin{z}=i$

How can I find how many solutions following equation have? $$z+\sin{z}=i$$ I can make substitution $z=it$ and get $$t+\sinh{t}=1$$ which has one real solution $t\approx0.4900730685$ thus ...
2
votes
2answers
34 views

Finding the Roots of Cubic (Boas 2.14.25)

I'm working my way through Mathematical Methods in the Physical Sciences and came across the following problem: Use a computer to find the three solutions of the equation $x^3−3x−1=0$. Find away ...
1
vote
1answer
37 views

Numerical root finding for 5th degree polyomial

I have the equation $y^5 -ay -b=0$. I need to get a solution whether numerical or analytical. I heard $5$th order polynomials are not solvable analytically, so how can I get the root numerically. ...
-1
votes
2answers
55 views

Cube root equations 1

$$E_{1} : \sqrt[3]{1+z}-\sqrt[3]{1-z}=\sqrt[6]{1-z^{2}} $$ Let $a=\sqrt[3]{1+z}$ and $b=\sqrt[3]{1-z}$ $E_1$ is equivalent to $E_2$ : $$ E_2:\ ...
0
votes
1answer
32 views

Polynomial with one rational root or one imaginary root

In my textbook there is an example where we have to find all the roots of $2x^3-5x^2+4x-1$. After applying the Rational Root Theorem we can conclude that $1$ and $1/2$ are two solutions to this ...
7
votes
0answers
127 views

Is it true that $\gamma_{\lfloor\log\Gamma(x)\rfloor}\sim 2\pi x$?

I realise that Gram points can approximate the imaginary part on the $x$th zeta zero $(\gamma_x)$ accurately, and indeed, Guilherme França, André LeClair give another formula, namely ...
1
vote
0answers
52 views

How to prove the uniqueness of a specific root?

Let us define: $$F(x):=\int_0^Tf(t)\cos(x\,t)dt-\frac{\sin(T_0\,x)}{T_0\,x}$$ where: 1). $0<T_0<T \in\mathbb{R}^+$ are both positive real constants, and 2). $0\leqslant f(x)\in C^{\infty}$ ...
1
vote
1answer
33 views

Roots of a polynomial with one parameter

Let $P$ be the polynomial defined by $P (Z) = Z^{4}-aZ^{2} +1$ Calculate $P\left(\sqrt{\dfrac{a}{2}}\right)$ and deduce the number of real roots of $P$ as the case $a = 2$, $a >2$ or $a ...
1
vote
2answers
69 views

How many roots has the equation?

Let $f(x)=x^3-3x+1$. How many roots has the equation: $f(f(x))=0$? I tried to solve it graphically and found 7 roots. If there exists analytical solution?
1
vote
1answer
39 views

Asymptotic behavior of the solution to an equation

Let $c\in(0,1)$ be a constant and let $k$ be a positive odd integer, and let $a(k)$ denote the value of $a$ that satisfies the equation $$(1-a)^kk\sqrt{a}=c$$. As $k\rightarrow\infty$, what can we ...