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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4
votes
2answers
99 views

Real solutions of $x^n + y^n = (x+y)^n$

I have to find all real solutions of the following equation: $x^n + y^n = (x+y)^n$ Clearly for $n = 1$, the equation holds for every $x,y$ real numbers. If $n$ is greater or equal to $2$, we do ...
19
votes
3answers
2k views

Why is it so hard to find the roots of polynomial equations?

The question that follows was inspired by this question: When trying to solve for the roots of a polynomial equation, the quadratic formula is much more simple than the cubic formula and the cubic ...
0
votes
0answers
20 views

Root with bolzano theorem

Given this equation $a\cos{x}+b=x$ with $a,b>0$ how to prove that there is at least one root between $(0,a+b]$ ? For $x=0$ its $a+b$ which is >0 For $x=a+b$ its $a\cos(a+b) ...
1
vote
1answer
66 views

The diophantine equation $y^2=x^3+7$ has no solutions.

In my lecture notes there is the following example: The diophantine equation $y^2=x^3+7$ has no solutions. Proof: If the equation would have a solution, let $(x_0, y_0)$, $y_0^2=x_0^3+7$, then ...
1
vote
1answer
88 views

Number of real roots of $2 \cos\left(\frac{x^2+x}{6}\right)=2^x+2^{-x}$

Find the number of real roots of $ \cos \,\left(\dfrac{x^2+x}{6}\right)= \dfrac{2^x+2^{-x}}{2}$ 1) 0 2) 1 3) 2 4) None of these My guess is to approach it in graphical way. But equation seems ...
2
votes
0answers
23 views

Check my answer - complex analysis, using residue and rouche's theorem

I was asked the following questions and I am unsure of my solutions, any advice would be appreciated, maybe there is a better way of doing this. Question: We are given $f(z)=2z-\sinh (z)$ defined on ...
1
vote
0answers
24 views

Finding roots of a discrete complex valued function [on hold]

I am struggling with a numerical problem. I have a discrete dataset with complex valued numbers which are the function of a real variable. The function is a black box. Is there any way to find the ...
1
vote
1answer
26 views

Two sets of polynomials with distinct roots build the ring of polynomials.

Definitions: $i \in K$ $U_{i}:=\{f\in K[X] |f(i)=0 \}$ $K[X]$ is the ring of polynomials HINTS: K[X] is a vector space Every $U_{i}$ is a vector subspace of $K[X]$ Question: (i) With $s \neq ...
3
votes
4answers
154 views
1
vote
2answers
558 views

Roots of biquadratic equation

This question also was a part of my today's maths olympiad paper: If squares of the roots of $x^4 + bx^2 + cx + d = 0$ are $\alpha, \beta, \gamma, \delta$ then prove that: $64\alpha\beta\gamma\delta ...
0
votes
1answer
52 views

Closed form of $\cot x=x$

I plotted the graphs of $y=\cot x$ and $y=x$. Its clear that they have infinite intersections. I tried to solve for the first root but it doesn't seem to be any known number to me. Even Wolfram Alpha ...
1
vote
1answer
44 views

Roots less than 1 if at least one coefficient is greater than one

I have this doubt. If you have this equation with $\alpha_i \in \mathbb R$ $$P(z)=1-\alpha_{1}z-\alpha_{2}z^{2}- \cdots - \alpha_{p}z^{p}=0$$ I believe that if there exist an $\alpha$ greater or equal ...
0
votes
0answers
25 views

What is a one-parameter Newton's method?

The Newton's method that I know is defined as follows: $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ However, I've recently encountered a paper that talks about a one-parameter family of Newton's method ...
-1
votes
0answers
21 views

Solving a nonlinear equation of two unknowns (a matrix and a vector) using fsolve in Matlab?

I am solving a nonlinear equation in Matlab using fsolve. The equation is $$W(BKhQ)e^{W(BKhQ)+Ah}-BKh=0,$$ where $W$ is the Lambert $W$ function. Also $A$, $B$, ...
2
votes
1answer
42 views

Inverting complex cosine

I have been working out problem 3a in chapter 1 section 3 in Basic Complex Analysis by Marsden. He asks to solve $$ \cos z=\frac{3}{4}+\frac{i}{4} $$ After putting cosine in its exponential form and ...
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
18 views

Sixth root of -64 using Euler's formula and De Moivre's theorem

I am attempting to solve: $$(-64)^{\frac{1}{6}}$$ Using the relation: $$a+bi=re^{i(\tan^{-1}(\frac{b}{a})+2\pi n)}$$ And then applying De Moivre's theorem: ...
0
votes
0answers
28 views

root of $a-b{{e}^{cx}}-{{e}^{\left( c+d \right)x}}=0$

I am trying to find the root(s) of this equation, basically write variable x in terms of parameters a, b, c, and d. not sure how to proceed. Thanks! $$a-b{{e}^{cx}}-{{e}^{\left( c+d \right)x}}=0$$
0
votes
0answers
34 views

Different ways to prove Fundamental Theorem of Algebra

This is just a curosity .I know some proofs of the fact that Every non constant polynomial with complex coefficient has a complex root via using Liouville's theorem in Complex Analysis.Proof goes as ...
3
votes
4answers
95 views

Is the zero polynomial the only polynomial that vanishes at every point of $\mathbb C$?

The zero polynomial has the property that every value it takes on $\mathbb C$ is zero. Is the converse true, or are there other polynomials $f$ such that $ f(x)=0$, for all $x \in \mathbb{C}$?
2
votes
3answers
83 views

How many $n$th roots does $0$ have?

Do we say that $0$ has $n$ $n$th roots, all nondistinct, or only one? I don't think it makes any difference, but I'm curious what the convention is.
-1
votes
1answer
35 views

Find the algebraic set $V(S)$

How can we find the algebraic set $$V(x^2+y^2-1)$$ ? $$V(S)=\{(a_1, a_2, \dots , a_n ) \in K^n |f_a(a_1, a_2, \dots , a_n )=0, \forall a \in A\}$$ where $$S=\{f_a \in K[x_1, x_2 , \dots , x_n] | a \in ...
2
votes
1answer
18 views

Find the maximum number of a continuous function

Lets define a function $z:\mathbb{R}^\mathbb{R}\to\mathcal P(\mathbb R)$ that gives you the set of zeros of any $\mathbb R ^\mathbb R$ function. Now, we define a set $S=\{z(f):f\in\mathbb R ^\mathbb ...
1
vote
2answers
29 views

Problem with the rational root theorem

Consider this polynomial: $f(x)=(2x+5)(x-3)(x+8/3)=0$. Then $f(x)=2x^3+...+(-40)$ Here is a list of all factors of $40$ and $2$: $40$: $±1$, $±2$, $±4$, $±5$, $±8$, $±10$, $±20$ $2$: $±2$, $±1$ ...
1
vote
1answer
24 views

Fourth Order Homogeneous Ordinary Differential Equation With Double Complex Conjugate Roots (2.10-14)

This is actually a problem in algebra as shall be seen. I need to find the general solution for the following differential equation: $$y''''+8y''+16y=0$$ The characteristic equation for this is: ...
1
vote
1answer
43 views

Use Newton's method to find root for the following equations

I have to use Newton's method to find the roots with accuracy $10^{-5}$ of the following equation : $e^{x} + 2^{-x} +2\cos x -6 =0$ in the interval $(1,2)$ So $f'(x)= e^x - [2^{-x}]*[\log(2)] ...
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.
5
votes
0answers
289 views

How prove this $f^{(n)}(x)=0$ has at least $n-1$ distinct roots

Let $f\in C^{(n)}\left(]-1,1[\right)$ and $\displaystyle\sup\limits_{-1<x<1}|f(x)|\le 1$. Let $m_{k}(I)=\inf\limits_{x\in I}|f^{(k)}(x)|$, where $I$ is an interval contained in $]-1,1[$. ...
16
votes
2answers
1k views

Showing that the roots of a polynomial with descending positive coefficients lie in the unit disc.

Let $P(z)=a_nz^n+\cdots+a_0$ be a polynomial whose coefficients satisfy $$0<a_0<a_1<\cdots<a_n.$$ I want to show that the roots of $P$ live in unit disc. The obvious idea is to use ...
1
vote
1answer
34 views

Roots of polynomial equation $2a x^\gamma + ax^{\gamma - 1} - 2 = 0$

I would like to find roots of the following polynomial equation $$2a x^\gamma + ax^{\gamma - 1} - 2 = 0$$ where $a,\gamma>0$ (we might also assume that $\gamma \in\mathbb{N}$ if needed). Playing a ...
0
votes
1answer
226 views

Number theory - Primitive root of $338$ [closed]

Im having problem $338$ root. I know it has a root because $13^2\times2=338$ but what is the correct way to find it??
1
vote
1answer
48 views

Finding roots of a complex polynomial in $\{\operatorname{Re}(z) < 0 \} $

How many roots does $P(z)=2z^4+z^3-5z^2+z+2$ have in $\{ \operatorname{Re}(z)<0 \}$? I was told that I should compute $P(it)$ for $t \in \Bbb R$ which is: $P(it)=2t^4 +5t^2 +2 +it(1-t^2) $. ...
0
votes
2answers
60 views

If square root is the inverse function of $5^2$ what is the inverse function of $5^1$

I am not great at maths or anything, but just had a general question: If square root is the opposite of $5^2$, what is the opposite of $5^1$, $5^3$, $5^4$? Is there an opposite? How would I work it ...
3
votes
1answer
83 views

Geometry: How to find cube root, fourth root, fifth root… and so on?

As we know that square root of a number $n$ can be found by using a compass and a straight edge, given the line of length $n$. What I want to know is how to find cube root, fourth root, fifth root or ...
1
vote
1answer
58 views

Multivariate polynomials at bounded evens

Univariate polynomials Given $n$, is there a degree $cn^{c'}$ polynomial $p(x)\in\Bbb R[x]$ and a degree $dn^{d'}$ polynomial $q(x)\in\Bbb R[x]$ with fixed $c,c',d,d'>0$ such that $$m\in\Bbb ...
0
votes
1answer
21 views

Indefinite integral fractional roots

I know the result of these indefinite integrals, but I don´t understand how the calculaton gets there: $$\int \frac{1}{\sqrt{x}}dx = 2 \sqrt{x}$$ $$\int \frac{1}{\sqrt[3]{x}}dx = ...
1
vote
1answer
340 views

Solution of Bessel equation

Prove that for a Bessel function in its normal form that is: $$u'' + \left(1 + \frac{1-(4*p^2)}{4x^2}\right)u=0$$ if $p > \frac12$ then every interval of length $\pi$ contains at most one zero of ...
1
vote
1answer
32 views

Degree of min distance function between two algebraic curves

Suppose I have two algebraic curves $C_1$ and $C_2$ in the plane. I would like to find the minimum distance between the two curves. If the two curves have degrees $n_1$ and $n_2$, what is ...
0
votes
2answers
479 views

How many iterations of the bisection method are needed to achieve full machine precision

Suppose that an equation is known to have a root on the interval $(0,1)$. How many iterations of the bisection method are needed to achieve full machine precision in the approximation to the location ...
1
vote
0answers
22 views

Integer roots of polynomial

I have a polynomial with integer coefficients on the form $$ p_{l,N}(x) = x^{2l+1} - 2\sum_{m=0}^{l}\binom{2l+1}{2m}x^{2m}\sum_{j=1}^N j^{2l+1-2m} $$ and I am interested in finding the possible ...
0
votes
3answers
80 views

Is it possible to find the complex roots of $x^3 + 2 x^2 - 3 = 0$

Sorry to ask so many roots questions in such short bursts, but I want to know if it is humanly possible to compute the complex root of $$x^3 + 2 x^2 - 3 = 0$$ through algebraic manipulation? ...
0
votes
5answers
80 views

How to compute the roots of $x^3 - x^2 - 4x + 4$ = $0$

I am wondering whether there is a simple way to find the roots of $x^3 - x^2 - 4x + 4$ = $0$ by algebraic manipulation I will accept if this is not a trivial equation to compute the roots of Any ...
2
votes
0answers
34 views

Number of integral solutions to a polynomial

Given a polynomial of $n$th order, represented by $$f(x)=a_{0}x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-2}x^{2}+a_{n-1}x+a_{n}=0$$ Is it possible to find the number of integral solutions/roots to ...
1
vote
1answer
34 views

reconstructing a quadratic equation from roots

I have this quadratic equation $$2x^2+x-3=0$$ that I wish to reconstruct from its roots. $$D=b^2-4ac=25$$ $$x_1=\frac {-b\pm \sqrt D} {2a} = 1 \text{ and } \frac {-2} 3$$ Now, I've always learned that ...
1
vote
1answer
37 views

How to get the polynomial which roots are almost “equal and opposite of sign ?”

Respected All I got stuck in it and need your help. We know that if $\alpha_1, \cdots, \alpha_5$ be the roots of $p(x):=x^5+ax^4+bx^3+cx^2+dx+e=0$ then the equation which roots are opposite in sign ...
1
vote
1answer
44 views

find a quadratic polynomial p ( x ) and a number n such that p ( x ) and a number $n \pmod n $ has at least 2015 roots?

I understand what the question is asking for, but I don't know how to prove my answer. Let's say I took an equation of the form: $x^2+ 6x+ 8 \equiv0 \pmod {15}$. The first four roots are ...
0
votes
1answer
43 views

Secant method and false position method exercise

We have $f(x)=x^2-6$. I have to find $p_3$ if $p_0 = 3$ and $p_1 = 2$ by using a) Secant method b) False position method So for the first one I have $p_2=p_0- \dfrac ...
1
vote
4answers
70 views

how to factor this cubic polynomial

Let $f(t)=36t^3-19t+5$ be a cubic polynomial. How we can factor $f$ to its roots? Mathematica says that $f(t)=(-1+2 t) (-1+3 t) (5+6 t)$. How?
-2
votes
2answers
41 views

Nature of The Roots of The Quadratic Equation $(a-1)x^2+(4a-2)x+4a+1=0$ [closed]

For which values of the real parameter $a$ are the roots of the quadratic equation: $$ (a-1)x^2+(4a-2)x+4a+1=0 $$ a) Real b) Positive
1
vote
1answer
21 views

Slight problem with solving a trigonometric equation.

I had to prove the identity $4\cos^3x-3\cos x=\cos 3x$ and then use it to solve the equation $(4\cos^2x-3)(4\cos^23x-3)(4\cos^29x-3)=1$. After proving the identity I proceeded to simplify the ...