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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5
votes
1answer
120 views

Can the measure of zeroes of a harmonic function be positive?

Let $u$ be a non-constant harmonic function of two variables defined, say, in the unit disk (or on the half plane for example). It is known that $u$ can vanish on some lines, as it discussed in here. ...
2
votes
3answers
46 views

Show that a polynomial $P(x)$ has $r$ as a double root if and only if $P'(r)=0$ and $P(r)=0$

Assuming that $r$ is a double root. Then $$P(x)=(x-r)^2\cdot k(x).$$ We also have the derivative: $$P'(x) = 2(x-r)k(x) + (x-r)^2k'(x).$$ Hence, $$P(r) = (r-r^2)k(r)=0$$ and $$P'(r) = 2(r-r)k(r) + ...
1
vote
1answer
46 views

polynomial over finite field, roots forming additive subgroup

Let $q=2^h$ and $t=2^r$ for some $h\ge r$ and denote by $\mathbb{F}_q$ the finite field of order $q$. (since the previous, simple version was wrong, I'm posting here a new version) Let $f$ be a ...
4
votes
0answers
51 views

How many iterations of the Newton's method are needed to achieve a given precision

There is a formula for bisection method to estimate number of iterations that are needed to achieve a given precision (desired significant figures) in the interval $[a,b]$ $$ n\ge ...
8
votes
3answers
250 views

Properties of Roots of polynomials

Today in highschool we were doing a chapter called "Roots of polynomials" where we learnt something new and interesting which is : $ax^2+bx+c=0$ Has roots $\alpha$ , $\beta$ Then: $$\alpha + ...
1
vote
2answers
51 views

If $\frac{[x_n]^n [1-[x_n]^n]}{(1-x_n) n} = a$, is $[x_n]^n$ increasing for $n\geq 3$?

Let $x_n$ be the solution to $\frac{x^n [1-x^n]}{(1-x) n} = a$, where $x \in [0,1], a \in [0,1]$ and $n \in \mathbf{N}$. I want to prove that $[x_n]^n$ is increasing in $n$ for $n\geq 3$. (From ...
1
vote
3answers
78 views

How to find all solutions of $\tan(x) = 2 + \tan(3x)$ without a calculator?

Find all solutions of the equation $\tan(x) = 2 + \tan(3x)$ where $0<x<2\pi$. By replacing $\tan(3x)$ with $\dfrac{\tan(2x) + \tan(x)}{1-\tan(2x)\tan(x)}$ I've gotten to $\tan^3 (x) - 3 \tan^2 ...
1
vote
2answers
176 views

$x^3+b^2x^2+2x+3=0$,Find several integer values of b such that the equation has roots.

$x^3+b^2x^2+2x+3=0$, Find several integer values of $b$ such that the equation has roots. My solution: I use the rational root theorem. $-3,3$ can be its rational roots. $$P(x)=x^3+b^2x^2+2x+3$$ ...
3
votes
1answer
94 views

How to solve this equation in radicals?

How to solve the equation $x^6-2\varphi^5x^5+2\varphi x+\varphi^6=0$ in radicals? where $\varphi$ is the golden ratio.
-2
votes
1answer
71 views

How to Solve $-3^x+617x+1625=0$

can anyone please help me solve this : $$-3^x+617x+1625=0$$ I can't do it analytically. originally the problem was to find intersection point of $$y=1625+617x$$ and $$y=3^x$$ i did the regular ...
3
votes
1answer
158 views

Cube root of complex number without trigonometric functions

Is there a general equation for a cube root of a complex number that does not exploit De Moivre's Theorem or in any way use trigonometric functions? For example, a square root of a complex number $x$ ...
1
vote
3answers
69 views

Roots of this third degree polynomial

I've got the following polynomial $$ x^3-6x^2-2x+40 $$ and I want to find its roots. The only option I see at the moment is to compute all the divisors of $40$ and their inverse, and manually check if ...
2
votes
1answer
88 views

How to find the polynomial which has the sum of two cube roots as one of its roots?

For example. How do I find the polynomial which has $\sqrt[3]2 + \sqrt[3]3$ as one of its roots? ( Hint: polynomial is $x^9-15x^6-87x^3-125$ )
0
votes
0answers
22 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
votes
1answer
62 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) ...
1
vote
2answers
50 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
votes
2answers
67 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
votes
0answers
40 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{?}= ...
0
votes
1answer
25 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?
3
votes
5answers
93 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 ...
0
votes
0answers
33 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 ...
0
votes
1answer
54 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
1
vote
1answer
62 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 + ...
1
vote
2answers
489 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
30 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 ...
1
vote
0answers
39 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
50 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
votes
1answer
212 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
49 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
39 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
47 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
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
votes
1answer
33 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
47 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
83 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
41 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
20 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
69 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
63 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, I'm not sure.
7
votes
1answer
103 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
44 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
49 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!
1
vote
1answer
30 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
67 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
30 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
vote
1answer
62 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
27 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
85 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
24 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 $t = f(A, \alpha, \omega, \phi)$.