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
votes
2answers
31 views

Quartic with $4 $ equidistant roots

Today I got the problem $(x^2 -1)(x^2 -4)=k$, and I have no idea how to prove this algebraically. $K$ is a real, non-zero number that makes the equation have $4$ distinct real equidistant roots. Solve ...
0
votes
0answers
24 views

Solving $\mathbb{E}_X[\log (a + X)] = b$ to $a$

I'm trying to solve the equation $$ \mathbb{E}_X[\log (a + X)] = b$$ to $a$ where $b>0$ and $X$ is a positive random variable distributed according to $P(X)$. The solution can be written as a ...
0
votes
2answers
54 views

Sum of Reciprocal of a sum of square roots

I am trying to find a closed form expression of the following sum: $\sum_{k=1}^{N}\frac{1}{\sqrt{k}+\sqrt{k+3}},\; N>1.$ I tried to determine whether methods used for evaluating more conventional ...
0
votes
1answer
42 views

Partial fraction integration with unclear roots

Let's look at a simple example like $\frac{1}{x^3+2x+1}$ here. We know that the denominator has a real root between $0$ and $-1$ (could go closer, but that's not the point). By the concept of slope of ...
1
vote
2answers
52 views

Set of solutions to $\sqrt{x^2}=-x$

The question is: The set of all real numbers x such that $$\sqrt{x^2}=-x$$ consists of: A: zero only B: nonpositive real numbers only C: positive real numbers only D: all real ...
0
votes
0answers
19 views

Counting zeros of trigonometric functions of functions

There is not any context for this problem, it is a general question: In General: If you are given a trigonometric function of a function $\sin(f(x))$ with an arbitrary function f(x), is there any ...
2
votes
1answer
43 views

When is it more appropriate to use multiple branches for roots?

Often times, more often then I believe should be, I will see a question under the tag algebra-precalculus that asks about odd solutions, which many answers will note to be extraneous solutions, when ...
0
votes
0answers
42 views

Finding a mirror point on a parabola

What is the height of the ball at a point of 3 metres beyond where it was thrown, measured horizontally? How far is the ball from where it was thrown when its height has this value again? ...
1
vote
1answer
46 views

Finding all roots of multivariate polynomial using Newton's method

I read that it is possible to find a solution to a nonlinear system of equations using Newton method and Jacobian matrix. But if I understood correctly, this finds just one solution, and which one ...
0
votes
3answers
51 views

Cubic equation (polynomial)

A cubic polynomial with real coefficients, $a x^3 + b x^2 + c x + d$, has either three real roots, or one real root and a pair of complex conjugate ones. If the latter happens, what is the explicit ...
0
votes
1answer
60 views

How to guess initial intervals for bisection method in order to reduce the no. of iterations?

SO, A function $f(x)$ is given to me and but the initial intervals are not given. I need to find the root of the equation using Bisection method. Sometimes when I randomly guess the initial interval ...
1
vote
0answers
36 views

Analytic closed-form solution

I have the following equation: $$\left(\frac{x}{\cosh(x)}\right)^2-x\tanh(x)+\ln\cosh(x)=0$$ and I would like to know if there is some analytic closed form solution. WA gives me two non-zero ...
0
votes
2answers
65 views

Calculating $a_1^4+a_2^4+a_3^4$ of the roots of a polynomial

We have a polynomial $f=X^3+19X^2+12X+3\in\mathbb{C}[X]$ with roots $a_1,a_2,a_3$. What is $a_1^4+a_2^4+a_3^4$? And how do I know that these roots are all different? Edit: How can I show that ...
-1
votes
3answers
87 views

Functions Mapping Integers to Zero?

I am looking for functions such that: $z∈$ Z ⇔ $f(z)=0$ That is to say, functions that map from Z to the zero set. One example is $f(z)=\sin(πz)$. EDIT: To narrow the possible group of ...
2
votes
0answers
31 views

Finding the roots of a polynomial with limited information

Let $\ f(x) \in \mathbb{R}[x]$ be a 7-th degree polynomial, such that. $$ f(0)=0 \land f(i)=-3i $$ $$ f'(0)=0 \land f'(i)=-21 $$ Find all the complex roots of $ f(x)-3x^7$. Find all possible ...
3
votes
1answer
42 views

Let $a,b,c,d$ be distinct integers such that the equation $(x-a)(x-b)(x-c)(x-d)-9=0$ has an integer root $r$,then find the value of $a+b+c+d-4r.$

Let $a,b,c,d$ be distinct integers such that the equation $(x-a)(x-b)(x-c)(x-d)-9=0$ has an integer root $r$,then find the value of $a+b+c+d-4r.$ As $r$ is the integer root of the equation ...
2
votes
1answer
58 views

How do to find all the roots of $x^4-2yx^3+3y^3x-2y^4=0$?

The main question is how do we factorize $x^4-2yx^3+3y^3x-2y^4=0$ where $y$ is a parameter. I thought we could use Vietta's formula and solve the following system: $$\begin{cases} \begin{split} ...
1
vote
2answers
67 views

Find all complex roots of $T^4-{1/2}T^2-\sqrt{15}T+{69/16}$

I want to find all complex roots of $T^4-{1/2}T^2-\sqrt{15}T+{69/16}$. The only way I can think to do it is to find 1 complex root, $\alpha$, by inspection, so we can rearrange the polynomial to ...
4
votes
2answers
57 views

Solution to Sextic Polynomial with Two Real Roots

I have the polynomial $$f(x;a)=3ax^6+6x^5-9ax^4-4x^3+9ax^2+6x-3a$$ where the variable $a$ is a random variable from the uniform distribution in the range $[0,1)$. When I analyze this function using ...
1
vote
2answers
41 views

Polynomials and common roots

When dividing $f(x)$ by $g(x)$: $f(x)=g(x)Q(x)+R(x)$. How to find the quotient $Q(x)$ and the remainder $R(x)$? For example: $f(x)=\ 2x^4+13x^3+18x^2+x-4 \ $ , $g(x)=\ x^2+5x+2 \ $ At first $g(x)= ...
2
votes
0answers
115 views

Finding a solution to $\sum _{n=1}^{n=k} \frac{1}{n^x}+\sum _{n=1}^{n=k} \frac{1}{n^y}=0$

Scroll down to the update to see what I am meaning. The Mathematica program below finds a solution to the equation: $$\sum _{n=1}^5 \frac{1}{n^x}+\sum _{n=1}^5 \frac{1}{n^y}=0$$ My question is if you ...
0
votes
2answers
40 views

Show that sum of roots is rational.

If $f(X)$ in $\mathbb{Q}[X]$ is an irreducible polynomial polynomial of degree $n \geq 2,$ with roots $\alpha_1, \alpha_2,\ldots,\alpha_n$ in $\mathbb{C},$ show that $\displaystyle ...
0
votes
0answers
16 views

Count the number of positive real solutions of a polynomial of arbitrary degree?

Let $$P(x) = a_0 + a_1 x + \dots + a_n x^n$$ be a generic polynomial of order $n$. I need to know the number of positive real solutions to the equation $P(x)=0$. Specifically, I need to determine if ...
5
votes
0answers
64 views

Geometry of the zeros of a power series.

This is probably a basic question that is easily googlable, but it seems that I dont have the right keywords. So my question is, having some power series $$ f(z)=\sum_{k=0}^{\infty}C_{k}z^{k}, ...
-1
votes
1answer
42 views

Is there a general formula the solutions of a polynomial equation of the form $Ax^n + Bx^{n-1} + C = 0$?

Is there a general formula for the solutions of a polynomial equation of the form $$Ax^n + Bx^{n-1} + C = 0,$$ where $A$, $B$, $C$, and $n$ are constants?
0
votes
2answers
40 views

How would you find the roots of the following equation?

How would you find the roots of the following equation? $$x^{13} + 1 = 0.$$ I am absolutely new at this and have no clue.
-2
votes
1answer
54 views

Can someone explain to me how to find zeros of a function? $10x^2+20x+19x+97^1$

I got this function right here and my teacher wants me to find all real number zeros $$10x^2+20x+19x+97^1.$$ I looked up this video on how to find it and they were using the $P/Q$ and I found ...
2
votes
0answers
40 views

Does $y=\lim_{n\to\infty}x\uparrow\uparrow n-f(x,n)$ have a root at $2$?

While messing around on Desmos calculator, I found an interesting factorial/tetration function, where we are using Nutch arrow notation. $$y=\lim_{n\to\infty}x\uparrow\uparrow ...
2
votes
2answers
31 views

prove that $x_{n+1}=\frac{x_n(x_n^2+15)}{3x_n^2+5}$ is cubic order of convergence near $x_0=\sqrt{5}$

To solve the equation $$x^2-5=0$$ There exitsts a iteration method $$x_{n+1}=\frac{x_n(x_n^2+15)}{3x_n^2+5}$$ I know that it is cubic convergence but I don't know how to prove it. I have tried the ...
3
votes
4answers
108 views

Minimal polynomial over $\mathbb Q(\sqrt{-2})$

Find the minimal polynomial for $\sqrt[3]{25} - \sqrt[3]{5} $ over $\mathbb Q$ and $\mathbb Q(\sqrt{-2})$. I have done the first part of this, over $ Q$, and have a polynomial. But I do not know ...
0
votes
1answer
29 views

Describe the rational solutions to this quartic.

Consider the quartic $-(a^5 + b^5)x^4 + (4a^5 - b^5)x^3 - (6a^5 + b^5)x^2 + (4a^5 + b^5)x - (a^5 + b^5) = 0$ where $a$ ...
2
votes
0answers
38 views

Find all such $a$ that $x+2\lvert x-3 \rvert = 7\lvert x-a \rvert + 3 \lvert x-a-4|$ has at least one root.

In the equation, $a$ is a parameter and $x$ is a variable: $$x+2\lvert x-3 \rvert = 7\lvert x-a \rvert + 3 \lvert x-a-4|.$$ I want to find all values of $a$ that make the equation have at least one ...
1
vote
1answer
42 views

Is this true of all linear recurrences?

Is it true that any linear recurrence $f_n$ can be written as: $$f_n = \sum_{i=1}^{k} \alpha_i r_i^n$$ where $f_n$ is a linear recurrence of degree $k$ and $r_i$ represents a root of the ...
1
vote
1answer
34 views

Monic polynomal $f \in \mathbb{Z}[X]$ such that $f(2)=13$ have at most three distinct roots in $\mathbb{Q}$?

how can I show that given a monic polynomial $f \in \mathbb{Z}[X]$ such that $f(2)=13$ have at most three distinct roots in $\mathbb{Q}$? I know that these roots has to be in $\mathbb{Z}$, but I ...
0
votes
1answer
51 views

Find $a$ such that $p(x)\geq 0$

The problem is: Let $p(x)=x^4-2x^3+ax^2-2x+1$, let a and x be real numbers, find a such that $p(x)\ge0$. My intent to solve it: We see that $(x^2-x+1)^2-3x^2+ax^2\ge0$ then ...
3
votes
2answers
88 views

Find the number of real roots of the derivative of $f(x)=(x-1)(x-2)(x-3)(x-4)(x-5)$ [duplicate]

Find out the number of real roots of equation $f'(x) = 0$, where $$f(x)=(x-1)(x-2)(x-3)(x-4)(x-5)$$ How can I differentiate this function without expanding it to the polynomial form. Am I ...
4
votes
4answers
785 views

How to find root of derivative of any polynomial/equation?

Let $$f(x) = (x-1)(x-2)(x-3)(x-4)(x-5),\quad -\infty< x<\infty.$$ The number of distinct roots of equation $$\frac{d}{dx}f(x) = 0$$ is exactly ? Source. The only method that I know ...
4
votes
2answers
57 views

Polynomial whose roots are not integers but almost so

Let $\varepsilon \in (0,\frac{1}{2})$. Say that a real number $x$ is an $\varepsilon$-pseudointeger if it is not an integer but at distance at most $\varepsilon$ from some integer (thus $|x-i|\leq ...
1
vote
0answers
59 views

How can I apply Rouche's Theorem here?

How many solutions lie in the left half-plane? $$f(z) = z^3+2z^2-z-2+e^z=0$$ My work so far: Factoring the polynomial, moving the exponential term over to the RHS, and taking the modulus of both ...
2
votes
1answer
123 views

how to find the roots of the following floor-equation: [closed]

How to find the roots of $$\lfloor x\rfloor+\lfloor 2x\rfloor+\lfloor 3x\rfloor=6$$
1
vote
3answers
28 views

Showing that $1$ is the only root in $f=x^5 -1 \in \mathbb{F}_p[x]$ if $p-1$ is not divisible by $5$

I am trying to show that if $p-1$ is not divisible by $5$ then $1$ is the only root in $f=x^5 -1 \in \mathbb{F}_p[x]$, where $p$ is a prime. I can see that $f = (x-1)(x^4 + x^3 + x^2 + x + 1)$ so it ...
3
votes
0answers
100 views

Why does Ridders' method work as well as it does?

I've just read section 9.2.1 in Numerical Recipes Ed. 3 (Press et al. 2007), which describes Ridders' method of root finding. I understand that allowing for some curvature of the function by ...
0
votes
0answers
28 views

Upper bounds for the number of roots of polynomials, over finite fields, lying in given extensions

Let $F$ be the finite field with $q$ elements, where $q$ is a power of a prime, and let $E$ be its degree $n \geq 2$ extension. Let $f(x) \in F[x]$ such that $f(E) = F$. Clearly the number of distinct ...
1
vote
3answers
41 views

If $x=2+i$, $gcd(a,b,c)=1$, and $ax^4+bx^3+cx^2+bx+a=0$, then what is $|c|$?

Suppose $$a(2+i)^4 + b(2+i)^3 + c(2+i)^2 + b(2+i) + a = 0,$$ where $a,b,c$ are integers whose greatest common divisor is $1$. Determine $|c|$. So I first simplified the exponents and ...
0
votes
2answers
40 views

Which equation has roots -2c, 2c, and 2?

This is a multiple choice question $$-4c^2 -2c=0$$ $$-4c^2+2c=0$$ $$x^3 - 2x^2-4x+8=0$$ $$x^3 - 2x^2-4c^2x +8c^2=0$$ I know roots mean solutions, so do I plug in the given roots and see if they ...
-2
votes
1answer
40 views

How do I find the roots from this? [closed]

Problem: We are give that $2+7i$ is a root of $x^4 - 8x^3 + 73x^2 - 228x + 212=0$ I got stuck on this one: how do I find all the roots? $\left(i=\sqrt{-1}\right)$
4
votes
4answers
471 views

Prove the roots of a complex polynomial are imaginary

For an equation $f(z) = z^5 - 6z^4 + 15z^3 - 34z^2 +36z -48$ show that roots $f(z) = 0$ of this equation include 2 purely imaginary roots, and find them. I thought to substitute in $z=x+iy$ to show ...
2
votes
1answer
50 views

How many roots lie inside the disk |z|<1

How many roots of $f(z) = z^3 +cz^2 + z + 1$ lie inside $|z|<1$ if Part 1) $c>3$ Part 2) $3\ge c >2$ Part 3) $2\ge c >1$ Part 4) $c=1$ I am able to solve Parts (1) and (4) by Rouche's ...
1
vote
2answers
83 views

Is the complex square root of $z^2 = \pm z$?

Is $\sqrt{z^2} = \pm z$, for $z$ complex? I think it is, since either $-z$ or $+z$ satisfies the definition $\sqrt{z^2}= e^{\large \frac{1}{2}\log(z)^2}$ but I just wanted to make sure. It's a bit ...
6
votes
3answers
236 views

$\sqrt[31]{12} +\sqrt[12]{31}$ is irrational

Prove that $\sqrt[31]{12} +\sqrt[12]{31}$ is irrational. I would assume that $\sqrt[31]{12} +\sqrt[12]{31}$ is rational and try to find a contradiction. However, I don't know where to start. Can ...