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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35 views

Roots of the Taylor approximation of the exponential

While answering another question, I looked at the roots of the $n^{th}$ degree Taylor approximation of the exponential. $$e^x\approx E_n(x):=\sum_{k=0}^n\frac{x^k}{k!}.$$ Apparently, these root are ...
2
votes
3answers
81 views

The number of distinct real roots of a polynomial of degree 4

Suppose I have a equation of a degree of 4 and I don't know a proper method of solving this type of equation (like completing the square is a proper method to solve the quadratic equation) so how or ...
0
votes
2answers
28 views

Roots of a Non-Monic Cubic Polynomial

Find all roots of $f(x)=231x^3+68x^2-9x-2$ I cannot use the cubic formula or Viete's theorem here because the polynomial is not monic. The only other way I can think of doing this is by the rational ...
3
votes
1answer
79 views

Find real roots of the equation

Find all real solutions to $$\dfrac{\sqrt{x+1}}{2+\sqrt{2-x}} - \dfrac{\sqrt{x^2-x+2}}{2+\sqrt{-x^2+x+1}} = x^3-x^2-x+1$$ This question is very similar to one of my previous problem, ...
2
votes
1answer
52 views

Solving a mixed radical and quadratic equation

Solve for $x \in \mathbb{R}$ $$4x^2(x+2) +3(2x^2-4x-3)\sqrt{4x+3} +6x = 0$$ I tried taking square by isolating the radical, but the resultant equation couldn't be solved. Any help ...
10
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1answer
152 views

Finding all real roots of the equation $(x+1) \sqrt{x+2} + (x+6)\sqrt{x+7} = x^2+7x+12$

Find all real roots of the equation $$(x+1) \sqrt{x+2} + (x+6)\sqrt{x+7} = x^2+7x+12$$ I tried squaring the equation, but the degree of the equation became too high and unmanageable. I ...
5
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1answer
108 views

Solve $ 1 + \dfrac{\sqrt{x+3}}{1+\sqrt{1-x}} = x + \dfrac{\sqrt{2x+2}}{1+\sqrt{2-2x}} $

Solve for $x \in \mathbb{R}$ $$ 1 + \dfrac{\sqrt{x+3}}{1+\sqrt{1-x}} = x + \dfrac{\sqrt{2x+2}}{1+\sqrt{2-2x}} $$ I tried some substitutions and squaring but that didn't help. I also tried ...
5
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1answer
108 views

Solving a radical equation for real $x$

Solve for $x \in \mathbb{R}$ $$\dfrac{\sqrt{x^2-x+2}}{1+\sqrt{-x^2+x+2}} - \dfrac{\sqrt{x^2+x}}{1+\sqrt{-x^2-x+4}} = x^2-1$$ I tried squaring the equation but it became a sixteen degree ...
2
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0answers
23 views

Finding all complex roots of an equation with exponentials.

I know that $$ (-1)^x + 2^x - 2 x - 1 = 0 $$ has a single real root $(x =3)$ and an infinite number of complex roots whose real part appears often negative. Don't the complex roots also have their ...
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2answers
56 views

How to find the roots of this 4th order polynomial?

Can someone explain how to factor/find roots to this 4th order polynomial: $$ s^4 + 14s^3 +45s^2 +650s + 1800 = 0 $$ It's such a nightmare. I've been stuck for hours, any help would be appreciated :)...
6
votes
1answer
102 views

Is there any nice explanation of why the complex exponential function has no roots in the complex plane? [duplicate]

Here I am not looking for an explanation that uses basic properties that complex exponential function has, such as $e^{z+w}=e^ze^w$ or $e^0=1$ or any other, if this fact can be explained by using ...
0
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0answers
15 views

Basic Linear Algebra/Root finding question

What is the general method for solving this problem? $\theta_n.1_T'.z_T=0_n$ where $\theta_n$ is a n x 1 vector of parameters that are free to vary, $1_T'$ is a 1 x T vector of ones, $z_T$ is a T x ...
1
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0answers
25 views

Computing $n^{\text{th}}$ root of a positive integer to arbitrary precision using integer arithmetic

There are various questions on this forum that appear similar, but my question pertains to writing code that can compute the $n^{\text{th}}$ root of a number $a$ correct to $p$ decimal places, where ...
1
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1answer
47 views

Let $α$ and $β$ be the roots of equation $px^2+qx+r=0,p≠0$

Let $α$ and $β$ be the roots of equation $px^2+qx+r=0,p≠0$, If $p,q,r$ are in A.P and $\dfrac{1}{α}+\dfrac{1}{β}=4$, then the value of $|α−β|$ is $:$ $\dfrac{\sqrt{61}}{9} $ $\dfrac{2\sqrt{17}}{9}$ ...
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0answers
23 views

Conjugate roots of a polynomial

If $\sqrt 2 - i$ is a root of $x^5-x^4-2x^3+mx^2+9x+m-11=0$, $m \in \Bbb Q$ find m and the other roots. My question is what other roots can i deduce from what is given? Is $\sqrt 2 + i$ the only one ...
0
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0answers
20 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
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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
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1answer
38 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
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2answers
722 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 ...
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0answers
34 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
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0answers
24 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
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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
81 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
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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
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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
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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
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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 ...
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0answers
22 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
28 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 $P(x)$
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0answers
15 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
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2answers
41 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
81 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
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0answers
26 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
57 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
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1answer
56 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
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2answers
32 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
71 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
92 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
198 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
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2answers
105 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
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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
33 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
21 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
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1answer
106 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
48 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
55 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 ...