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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2
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2answers
44 views

Is it possible to find an integer solution $r≥4$ to an equation?

Is it possible to find an integer solution $r≥4$ to this equation? $$11r²³-7r²¹+11r¹⁸-7r¹⁶-2r¹²+11r¹¹- 7r⁹-2r⁷-2 =0$$ I try some special values of $r$ but without any sucess.
0
votes
2answers
74 views

Is there any geometric interpretation or significance of the complex roots of a derivative?

I was doing some reading online when I stumbled here and learned about this geometric way of viewing the complex roots of a function. It got me thinking; the zeros of the derivative of a function $f$ ...
2
votes
1answer
59 views

How small would $|x_0 - a|$ be in order for $f(x)$ to converge to a for Newton's Method

I found that $f(x) = \cos(x) + \sin(50x)^2$ has a root $a = \pi/2$. Whenever we take our initial value $x_0$ close to a we get convergence, if we are far away from a we do not get convergence to our ...
1
vote
1answer
29 views

Proof of the proposition $V(S)=V(\langle S \rangle )$

In my lecture notes we have the following: Proposition: $$V(S)=V(\langle S \rangle )$$ Proof: $$\langle S \rangle=\left \{\sum_{i=1}^m g_i f_i | f_i \in S, g_i \in R=K[x_1, x_2, \dots , ...
2
votes
4answers
60 views

Given that the equation, $(k-1)x^2-2(k-1)x-(3k+1)=0$ has real roots, show that $k^2-k≥0$

I can get to $k^2-k≥0$ but only when I make $b^2$ negative. The problem is why would I make $b^2$ negative other than the fact that $b$ is negative in the original equation? The problem with this is ...
1
vote
2answers
51 views

Given some of the roots of the function $f(x) = x^3+bx^2+cx+d$, how do I find the coefficients of that function?

Two of the roots of $f(x) = x^3+bx^2+cx+d$ are $3$ and $2+i$. How do I find b+c+d? The answer choices are -7, -5, 6, 9, and 25.
2
votes
1answer
48 views

Possible integer roots of polynomial with real coefficents

If $p\in\mathbb{Q}[X]$, then the rational root theorem gives us possible integer roots of $p$. If $p\in\mathbb{R}[X]$, the theorem cannot be applied. Nevertheless, triangular inequality gives us lower ...
1
vote
1answer
30 views

$K-$rational solution of the equation - Is $\mathbb{Q} \leq \mathbb{Q}_p$?

Let $P(x, y) \in \mathbb{Q}[x, y]$. We consider the equation $P(x, y)=0$. If $a, b \in \mathbb{Q}$ such that $P(a, b)=0$ then $(a, b) \in \mathbb{Q}^2$, is called a rational solution. If $K$ a ...
4
votes
2answers
139 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 ...
1
vote
0answers
36 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) ...
4
votes
1answer
90 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
1answer
101 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 ...
1
vote
1answer
35 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 ...
1
vote
1answer
39 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 ...
2
votes
2answers
60 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
1answer
65 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 ...
-1
votes
1answer
70 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 ...
2
votes
1answer
328 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
33 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! $$a-b{{e}^{cx}}-{{e}^{\left( c+d \right)x}}=0$$
1
vote
0answers
124 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
116 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
85 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.
2
votes
1answer
32 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
36 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
967 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
61 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)] ...
1
vote
1answer
35 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 ...
1
vote
1answer
56 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
77 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 ...
4
votes
1answer
345 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 ...
0
votes
1answer
29 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
48 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
3answers
86 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
91 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 ...
1
vote
0answers
48 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 ...
3
votes
0answers
67 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
141 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
89 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
104 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 ...
1
vote
1answer
156 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 ...
-2
votes
2answers
67 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
30 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 ...
1
vote
2answers
204 views

Find root of equation using bisection method?

Question : Find an approximate value of $\sqrt[3]{25}$ using Bisection Method. Since it doesnt state the accuracy in the question,how many iterations am I going to do to get that approximate value? ...
1
vote
1answer
58 views

What is meant by positive root of $x^3-x^3-1$?

I am a bit confused. I think there must be a mistake. In a text I read: The entropy is $2\ln p$, where ...
1
vote
4answers
79 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?
1
vote
1answer
60 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 ...
4
votes
0answers
291 views

Prove equation has only one root in a specific interval

Prove that the following equation has only one solution in the interval $[-\text{min}(a_i), +\infty]$: $f(x) = \left(\sum_{i=1}^n \frac{1}{a_i + x}\right)\times \left(\sum_{i=1}^n \frac{a_i b_i}{(a_i ...
6
votes
2answers
113 views

Degree Polynomials and Zeroes

"Find a degree $3$ polynomial that has zeros $-3, 4$ and $8$ and in which the coefficient of $x^2$ is $-18$." I've been trying to solve this problem, but I keep getting it wrong. I've worked with ...
5
votes
0answers
58 views

Does this simple problem using Vieta's formulas have deeper connections to elliptic curves?

A friend posed the following question to me: Suppose $p(x)=x^3+ax+b$ has one real root, $x_1$, and two non-real roots, $x_2$ and $x_3$. Compute $x_1$ in terms of $x_2$. By Vieta's formulas, ...
1
vote
3answers
194 views

Polynomials in one variable with infinitely many roots.

Can a non-zero polynomial in one variable have infinitely many roots ? Can a non-zero polynomial in one variable have uncountably many roots ? Motivation : over $\mathbb Z/12\mathbb Z$, ...