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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15 views

Find roots of $\sum_i \alpha_i\,\cos(\beta_i\,t)$

I'd like to solve analytically the following equation, where $\alpha_i$ and $\beta_i$ have known values in $\mathbb{R}$: \begin{equation} \sum_{i\leqslant N} \alpha_i\,\cos(\beta_i\,t)=0 ...
5
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1answer
58 views

Determinant of a matrice $a_{ij}=e^{a_ib_j}$

1) Let $a_1<\dots<a_n$ real numbers and $\lambda_1,\dots,\lambda_n\in\mathbb{R}\backslash\{0\}$ Let $f(x)=\lambda_1e^{a_1x}+\dots+\lambda_ne^{a_nx}$ Show that $f$ has at most $n-1$ zeroes 2) ...
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2answers
49 views

Finding more than one root using Newton's Method

The problem is stated as follows: Find the two roots of $x^{4}-8x^{2}-x+16 \:\:in \: [1,3].$ What is a good first guess / a good way to make a first guess when more than one root is involved, if one ...
3
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2answers
64 views

Zeros of a function of degree 4

I'm trying to show that the following function has no zeros $$ 60x^4-44x^3-25x^2-44x+60=0. $$ I already tried using Eisenstein's criterium, but since the first and the last coefficient are both $60$, ...
3
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0answers
37 views

Decide if radical expression equals a given rational number

Given a radical expression an a rational number. Is there an algorithm to decide if the expression equals the number? Example: $(\sqrt{\sqrt[5]{74} - \sqrt[14]{78}})^{356}+\sqrt[6]{63} \overset{?}= ...
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1answer
24 views

Solution of $p(z)=0$ with $z\in\mathbb C$ and $a_k\in\mathbb R$ for all $k$

Suppose $p(z)=a_0+...+a_nz^n$ with $a_k\in\mathbb R$ for all $k$. How can I prove that if $p(z)=0$ then $p(\bar z)=0$? I know it's true, but how can I prove it?
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5answers
79 views

How to solve $z^6+i=0$

I'm trying to solve $z^6+i=0$. I would have say that it's equivalent to $$z^6=-i\iff |z|^6e^{i6\arg(z)}=e^{i\frac{3\pi}{2}}\iff|z|^6=e^{i\left(\frac{3\pi}{2}-6\arg(z)\right)}$$ But I'm not able to ...
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0answers
30 views

Using Sturm sequences to locate the roots of a polynomial

So I've been doing the sequences and I understand the method of constructing a Sturm sequence but there is few things I don't get. Firstly, how does division using the remainder of division of fuction ...
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1answer
52 views

an exercise about finite extension field and root of a polynomial

Suppose $K|F$ is a field extension of degree $n$ and $f(X)\in F[X]$ is an irreducible polynomial of degree $m\ge 2$ and $(m,n)=1$. Prove that $f(X)$ has no root in $K$. thanks
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1answer
60 views

Confused by the solution of $x^3+bx^2+cx+d=0$

From $x^3 + bx^2 + cx + d = 0$, we have $(x-x_1)(x-x_2)(x-x_3)=0$ for some roots $x_1$, $x_2$ and $x_3$. Expanding this second expression gives us $$x^3 + \left(x_1+x_2+x_3\right)x^2 + \left(x_1x_2 + ...
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2answers
478 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 ...
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0answers
22 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 ...
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0answers
35 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
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2answers
48 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 ...
13
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1answer
206 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
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0answers
47 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 ...
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0answers
37 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
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2answers
44 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
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0answers
70 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
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1answer
29 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 + ...
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1answer
44 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
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2answers
82 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
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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
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1answer
33 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
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1answer
19 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
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0answers
66 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 ...
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0answers
47 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
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1answer
86 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 ...
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2answers
42 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
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1answer
43 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!
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1answer
27 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
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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
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1answer
29 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
1
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1answer
48 views

How to use Newton's method to find the roots of an oscillating polynomial? [closed]

Use Newton’s method to find the roots of $32x^6 − 48x^4 + 18x^2 − 1 = 0$ accurate to within $10^{-5}$. Newton's method requires the derivative of this function, which is easy to find. Problem is, ...
0
votes
1answer
26 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
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1answer
81 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
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2answers
20 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
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1answer
25 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
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0answers
69 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
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1answer
86 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 ...
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1answer
20 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
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1answer
48 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
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1answer
40 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
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2answers
33 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 ...
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0answers
27 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
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1answer
42 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
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0answers
36 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 ...
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2answers
45 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 ...
3
votes
2answers
124 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
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0answers
45 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 ...