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 (radicals) and (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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0
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0answers
19 views

solving equation with variable on both sides

Is there a way to solve the following equation for $v_n^2$? I am working through a problem and I feel like there should be a way to solve it, but I am not sure how to do it: $$v_n^2-(1+\eta)=-i\...
-1
votes
1answer
22 views

Show that any polynomial of odd degree 2n+1: $f(x)=\sum_{k=0}^{2n+1} a_kx^k $, $a_{2n+1}\neq0$ has at least one real root.

Show that any polynomial of odd degree 2n+1: $$f(x)=\sum_{k=0}^{2n+1} a_kx^k $$ $a_{2n+1}\neq0$ has at least one real root. I would like to prove this using IVT, how would I go about starting ...
0
votes
1answer
35 views

solving a pair of simultaneous equations

I have a rather messy pair of simultaneous equations, which I need to solve for x: $\left(x+2n-1\over2\right)^2+\left(\sqrt{1-\left(x^2-2\over2\right)^2}+\sqrt{1-\left(-x^2+x+2n+1\over2\right)^2}\...
5
votes
2answers
698 views

Solving a 6th degree polynomial equation

I have a polynomial equation that arose from a problem I was solving. The equation is as follows: $$-x^6+x^5+2x^4-2x^3+x^2+2x-1=0 .$$ I need to find $x$, and specifically there should be a real ...
1
vote
0answers
32 views

Quadrant in which the zeros of a polynomial lies

Consider a polynomial $$p(z) = z^6 + 9z^4 + z^3 + 2z + 4 $$ I need to find which quadrant of the complex plane contains how many zeros that lie in unit circle. Also, I need to find which quadrant ...
0
votes
0answers
22 views

Newton Method Variant with convergence of order 3

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be twice continuously differentiable for all $x$ in the neighborhood $U(\xi)=\{x\in\mathbb{R}:|x-\xi|<r\}$ of a simple zero $\xi$ of $f$ such that $f(\xi)=0$,...
1
vote
2answers
25 views

Roots of a perturbed equation

I'm looking to show that the equation $$\displaystyle \psi(\delta) := e^{\alpha g(\delta)} - \delta$$ has a real root for $\alpha$ sufficiently small that converges to $\delta = 1$ as $\alpha \...
2
votes
1answer
79 views

A simple cubic equation problem:

Consider the cubic equation $$az^3-bz^2+\bar{b}z-\bar{a}=0$$ where $a$ and $b$ are non-zero complex numbers. Suppose $z_1, z_2$ and $z_3$ are the roots. Question: Which $a$ and $b$ gives $|z_1|=|...
0
votes
1answer
23 views

Zero functions on open interval

Are there non-constant differentiable functions that are zero on an open interval of real line? I've tried using the product integral: $$ f(x) = \exp(\int_0^1 \log(x-u) \mathrm{d}u ) = \frac{x^x (x-1)^...
1
vote
0answers
9 views

Finding the roots with the largest magnitude

Given a non-constant polynomial $p\in\mathbb{Z}[x]=\alpha\prod_{k=1}^nx-\alpha_k$ how can I find the roots $R=\{\beta_1,\ldots,\beta_t\}\subseteq\{\alpha_1,\ldots,\alpha_n\}\subseteq\mathbb{C}$ with $|...
0
votes
0answers
13 views

Is there an algebraic solution for this rootfinding problem?

I would like to solve for the roots of $f(x)=a_0 + a_1x^\gamma + a_2x^{\gamma+1}$, where $a_0,a_1 \in \mathbb{R}$ and $\gamma \in \mathbb{R}_{\geq 0}$ are arbitrary coefficients. This is possible ...
1
vote
2answers
45 views

$e^z=3z^5$ - Rouche's theorem

Question : Show that the equation $e^z=3z^5$ possesses five distinct real roots. In using the Rouche's theorem with the function $f(z)=-e^z+3z^5$ and $g(z)=-3z^5$, I succeeded to prove the first ...
0
votes
0answers
20 views

Computationally check for roots/positiveness of a big polynomial in a given interval

For a proof, I need to check that given a little interval $(0, 0.28)$ some concrete polynomials $\in \mathbb{Q}[w]$ (polynomials in one variable ranging over the real numbers, with degrees around 50) ...
0
votes
1answer
27 views

Construct a degree $n$ polynomial with roots $a_1, a_2, a_3, \ldots, a_n$

We have the numbers: $a_1, a_2, a_3, \ldots, a_n$ Show that there is a polynomial $P(x)$ of degree $n$ such that $a_1, a_2, a_3, \ldots, a_n$ are roots of $P(x)$
0
votes
0answers
14 views

Equation involving Bessel and Struve functions

I need to solve the equation $Z(\gamma) = r$ of the function $$Z(\gamma) = 1 - \frac{2}{\gamma} \left(J_1(\gamma) - i H_1(\gamma)\right),$$ where $J_1$ is the Bessel function and $H_1$ the Struve ...
0
votes
1answer
31 views

Roots of a fourth order polynomial [duplicate]

I am looking for the roots of $x^4=-1$, I have written $ -1 $ using Euler as $e^{j180}$. Therefore, $x=\pm e^{j45}$. But the fourth order equation should have two other roots, how can I get them?
3
votes
2answers
38 views

Number of real roots of $f ' ( x )$

Let $$f(x)=(x-a)(x-b)^3(x-c)^5(x-d)^7 $$ where $a,b,c,d$ are real numbers with $a < b < c < d$ . Thus $ f ( x )$ has $16$ real roots counting multiplicities and among them $4$ are ...
2
votes
1answer
79 views

How many roots does the polynomial $p(z) = z^8 + 3z^7 + 6z^2 + 1$ have inside the annulus $1 < |z| < 2$?

How many roots does the polynomial $p(z) = z^8 + 3z^7 + 6z^2 + 1$ have inside the annulus $1 < |z| < 2$? I know I can use Rouche's Theorem. I'm just not sure how. It states that $|f(z) − g(z)| &...
0
votes
1answer
24 views

Estimation for points in a neighbourhood of a root of a polynomial

Let $p(x)$ be a polynomial with complex coefficients and $p(\tilde x)=0$. Choose $\delta>0$ small enough, such that $\tilde x$ is the only root of $p$ in $B_\delta(\tilde x)$. I want to show that ...
0
votes
0answers
25 views

Finding root from implicit definition

I have the following implicit equation which defines $x$, $x$ is such that: $ (x-y) - \frac{x-(2y - f(y))}{2} \Phi( \frac{(2y - f(y)) - x}{\sigma}) - \frac{x-f(y)}{2} \Phi(\frac{f(y) - x}{\sigma}) + ...
3
votes
2answers
56 views

Finding Roots of tenth degree polynomial

I know that there are no explicit formulas to find roots for polynomials of degree higher than $4$. I have to find all the roots of the polynomial $ f(z) = 1+z^2+z^4+z^6+z^8+z^{10}$ I found two ...
0
votes
1answer
55 views

Prove that the roots are equal

Suppose that all roots of the polynomial equation $x^4 - 4x^3 + ax^2 +bx + 1 = 0$ are positive real numbers. Show that all roots of the polynomial are equal. I am not getting any idea as to how to ...
0
votes
2answers
29 views

find number of real roots

The number of real roots of the equation $$2\cos((x^2+x)/6) = 2^x + 2^{−x}$$ My approach: If I put $x=0$ in both side then $LHS=RHS$ so one real solution is zero but I'm not able to find if it has ...
2
votes
1answer
70 views

zeros of two functions are alternate

Let $a,b,c,d$ be real numbers. Show that the zeros of the functions $f(x)=a\cos x+b\sin x$ and $g(x)=c\cos x+d\sin x$ are distinct and alternate whenever $ad-bc\neq 0$. Suppose $x_0\in \mathbb{R}$ ...
3
votes
2answers
88 views

Solve the equation $x^3-6x-6=0$

Evaluate the roots of $$x^3-6x-6=0$$ I solved it using Cardano's method, but I'm looking for other elementary approaches through substitutions and properties of polynomials. Thanks.
2
votes
2answers
197 views

Calculus approach to solve this Quadratic equation problem

Both roots of the equation $$(x-b) (x-c) +(x-a) (x-c) +(x-a) (x-b) = 0$$ are always positive , negative or real. Prove your result. By solving this equation I got $3x^2 - 2(a+b+c)x +ab + bc + ca = ...
9
votes
2answers
103 views

Fundamental Theorem of Algebra for highschool

My teacher has told me about the Fundamental Theorem of Algebra, but I can't seem to find any proofs on it which I can understand. For something so important I'm hoping to find a proof that a ...
1
vote
3answers
108 views

How to solve $x^3 = 1$?

My intuitive side tells me to take the cube root of both the sides and get the answer $1$. However, I realize that it might be a problem for I'll lose solutions as given here: Is it the case that ...
3
votes
2answers
32 views

Find the cubic polynomial given linear reminders after division by quadratic polynomials?

A cubic polynomial gives remainders $(13x-2)$ and $(-1-7x)$ when divide by $x^2-x-3$ and $x^2-2x+5$ respectively. Find the polynomial. I have written this as: $P(x)=(x^2-x-3)Q(x)+(13x-2)$ $P(x)=(...
0
votes
1answer
15 views

Solve using auxiliary variable

solved using auxiliary variable (so they ask) I can not build the auxiliary variable for this problem, if they ask log in base 10 $$10^{\log ( \log x )}-10^{\log (16/\log x)}=6$$
0
votes
1answer
102 views

Approximate roots of nonlinear equation (non-integer polynomial)

In case of pulsating bubble arising from underwater explosion, bubble radius satisfies the following equation. $x^3\dot{x}^{2} + x^3 + \frac{k}{x^{3(\gamma-1)}} = 1$ The minimum and maximum bubble ...
2
votes
0answers
42 views

Find how many solutions of the equation $z^6+6z+10=0$ are in each quadrant. [duplicate]

Find how many solutions of the equation $z^6+6z+10=0$ are in each quadrant. This polynomial has six solutions by TFTA. I just don't know how to show what they are and where they lie. Any solutions or ...
2
votes
2answers
44 views

Roots of a Quartic (Vieta's Formulas)

Question: The quartic polynomial $x^4 −8x^3 + 19x^2 +kx+ 2$ has four distinct real roots denoted $a, b, c,d$ in order from smallest to largest. If $a + d = b + c$ then (a) Show that $a + ...
1
vote
1answer
38 views

Maximum/minimum of a special function

I was given a function $f(x)=\mbox{Li}_{-n}(x)$, where Li is the polylogarithm of order $-n$ ($n>0\in\mathbb{N}$) and $x\in(-\infty,0)$. The function in this domain is bounded and has some extremes....
0
votes
1answer
50 views

What are the roots of quintics?

I've been teaching myself a bit of Galois theory and from what I understand, arithmetic operations ranging from addition to taking roots are not enough to express all of the roots of a general ...
5
votes
0answers
76 views

The probability that a random (real) cubic has three real roots

We can formalize the notion of the probability that a randomly selected quadratic real polynomial has real roots as follows: Suppose $R > 0$, and suppose the random variables $a, b, c$ are (...
0
votes
1answer
39 views

Solving a “simple” quadratic/quartic equation

Despite having solved quadratic quations for years I can't seem to be able to get the same result than maple on this one (not as simplified as Maple's), so I wonder if someone could not explain: I'm ...
27
votes
6answers
2k views

Function with no roots

Given a non-constant function $f(x)$, is it possible for it to have no zeroes (neither real nor complex)? Say for example, $f(x)=\cos x-2$, does a complex solution exist for this because for real $x$,...
4
votes
2answers
113 views

The Gamma function has no zeros

How can I prove the Gamma function has no zeros in its holomorphy domain $\Bbb C\setminus\Bbb Z_{\le0}$ using only its integral definition $\Gamma(z)=\int_0^{+\infty}t^{z-1}e^{-t}\,dt$ valid when $\Re ...
0
votes
0answers
35 views

Suppose $m/l$ is a root of $f$, where $m$ and $l$ are relatively prime integers. Show that $m$ divides $a_0$ and $l$ divides $a_n$.

Suppose $f(x) = a_nx^n + \dots + a_1x + a_0$ is a polynomial with integer coefficients, and suppose $m/l$ is a root of $f$, where $m$ and $l$ are relatively prime integers. Show that $m$ divides $a_0$ ...
2
votes
2answers
39 views

Please express the first 3 7-adic digits of a root of $x^3-1$ in $\mathbb{Z}_7$ other than 1.

Please express the first 3 p-adic digits of a root of $x^3-1$ in $\mathbb{Z}_7$ other than 1. Does this just mean find the 7-adic expansion of 2 or 4? Wouldn't their expansions just be 2 and 4?
1
vote
2answers
34 views

if $f(x)=x^2$ and $g(x)=x\sin x+\cos x$ then number of points where $f(x)=g(x)$?

The question is if $f(x)=x^2$ and $g(x)=x\sin x+\cos x$ then number of points where $f(x)=g(x)$? My approach:- $$f(x)=g(x)\implies x^2=x\sin x+\cos x\implies x^2-x\sin x-\cos x=0$$ Let $$h(x)=x^2-x\...
2
votes
1answer
30 views

Show that this sum of polynomials has no zeros with positive real part

Let $0 < \lambda_1 \leq \ldots \leq \lambda_n $ and $k_1, \ldots, k_n> 0$. Let further $$ \begin{align} P(x)&:=\prod_{i=1}^n (x+\lambda_i) = (x+\lambda_1)\cdot \ldots \cdot (x+\lambda_n) \\...
4
votes
2answers
50 views

The Set $x:\left |x+\frac{1}{x}\right|>6=?$

The question is that ,the Set $x:\left |x+\frac{1}{x}\right|>6$ equals what intervals of $x$? My approach:- I tried to solve the inequality and get interval for $x$'s value as follows:- $$\left|\...
4
votes
4answers
76 views

Show that over $\mathbb{Z}_7$, $x^5-1$ has no roots other than 1.

Show that over $\mathbb{Z}_7$, $x^5-1$ has no roots other than 1. I know I can iterate through each member of $\mathbb{Z}_7$ and show that each one, when raised to the 5th power, is not equal to 1 ...
-2
votes
1answer
46 views

Find the roots of the equation - $z^2 +12jz+64 = 0$

Just needing a little guidance. This is what I've done so far and I'm not sure if I'm doing it right. Using quadratic formula: $$z^2+12jz+64=0$$ $$ z= \frac{-12j ±\sqrt{(12j^2-4\times1\times64)}}{2\...
-5
votes
1answer
59 views

Proof that $0^0 \neq 1$ [closed]

Suppose that $t = \sqrt{t}^{\sqrt{t}}$, then, it follows that; $$ t^{\sqrt{t}} = \sqrt{t}^{t} \\ \frac{1}{2}t\ln{\left(t\right)} = \sqrt{t}\ln{\left(t\right)} \\ \ln{\left(t\right)}\left[\frac{1}{2}t ...
2
votes
1answer
39 views

A non-constant polynomial with odd-integer co-efficients and of even degree , has no rational root?

Let $f(x)$ be a non-constant polynomial in $\mathbb Z[x]$ with odd-integer co-efficients and even degree ; then is it true that $f$ has no rational root ?
0
votes
1answer
25 views

Construct a Sinusoidal Equation for an Irregular Period

I would like to be able to construct a sinusoidal function of limited domain given a set of real roots, assuming that the function is graphically centered on $y=0$. I expect that this would ...
0
votes
1answer
53 views

How to show that $a^3+b^3+c^3+d^3\geq abc+abd+acd+bcd$ if $a,b,c,d>0$

How can I prove that if $a,b,c,d>0$ then $$a^3+b^3+c^3+d^3\geq abc+abd+acd+bcd?$$ I think there is some simple proof but I can't remember... is this a special case of some general inequality? ...