# Tagged Questions

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.

39 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....
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 ...
84 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 (...
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 ...
3k 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 $x$,...
115 views

58 views

### 'Elegant' ways on solving for roots for this cubic function?

I have this interesting cubic equation, $$x^{3} - 80\alpha x^{2} + (1744\alpha^{2}-81)x + (3240\alpha-5760\alpha^{3}) = 0$$ where $\alpha$ is some constant. I went about the method of Cardano, ...
37 views

### integrate the following function

please integrate for me the following function, it comes to me in the exam, and I put my answer on it, but I don't know if it's true or not I did analyze the to and I multiply it to and the ...
28 views

### Find all the roots of this polynomial

I'm currently stuck with the following problem: Find all the roots of the equation $$1-\frac{x}{1}+ \frac{x(x-1)}{2!}-...+(-1)^n \frac{x(x-1)...(x-n+1)}{n!}=0$$ I can sort of see that the roots ...
55 views

### How to solve a polynomial $P(x) = 0$

At the moment we just learnt the factor theorem of polynomials and how if $x-a$ is a factor of $P(x)$, then $P(a) = 0$. We're then taught to find the roots of a polynomial its best to check the ...
33 views

### Isolating roots of polynomial system

I would like to isolate the regions which contain the roots of a system of two bivariate cubic polynomials. I thought I would project the solutions onto $x$ and $y$ axis by means of resultant ...
40 views

### Why is the product of the roots of $p(z) = 1 + z + az^n$ equal to $1$?

This should be very straightforward to show, but I am having some issues doing so. We have: $$1 + z + az^n = a(z-z_1)(z-z_2)\dots(z-z_n),$$ where $z_1$, $z_2$, $\dots , z_n$ are the roots of $p$. ...
76 views

### When to use Newtons's, bisection, fixed-point iteration and the secant methods?

I've been introduced more or less to these methods of finding a root of a function (a point where it intersects the $x$ axis), but I'm not sure when they should be used and what are the advantages of ...
29 views

### Showing fields are algebraically closed

Let K be a field, and let P be separable and irreducible over K. Let L be a splitting field of P over K. I want to show that the fields K(u) and K(v) are isomorphic, where u and v are roots of P, ...
35 views

### Iterative methods to find roots

I'm trying to do optional exercises for my numerical methods class. I'm stuck in this one right now: Consider the function $f(x)=-e^{-2x}+3x$. a) Prove that $f$ has an unique real root. ...
89 views

### Proving there exists such a polynomial

I'm having trouble proving the following statement: For all primes $p$, there exists a non-constant polynomial $f(x)\in \mathbb Z_p[x]$ such that f(x) does not have a root in $\mathbb Z_p$ What I ...
43 views

### Use an expression for $\frac{\sin(5\theta)}{\sin(\theta)}$ to find the roots of the equation $x^4-3x^2+1=0$ in trigonometric form

Question: Use an expression for $\frac{\sin(5\theta)}{\sin(\theta)}$ , ($\theta \neq k \pi)$ , k an integer to find the roots of the equation $x^4-3x^2+1=0$ in trigonometric form? What I have ...
25 views

### Integer values of a rational function

How does one analytically determine the integer values of a rational function $f(x)$$=$$\frac{40-8x}{8x+2}mod1$ where $x$ is an element of the rationals? I just gave the function listed as an example,...
91 views

### Cubic polynomial with three (distinct) irrational roots

I am looking for an equation $$x^3+ax^2+bx+c=0, \qquad a, b, c \in \Bbb Z,$$ of degree $3$ that has $3$ different roots. For an equation of degree $2$ it is easy---for example $x^2-2=0$---but I ...
### show that $|x_1 -x_2|=\frac{\sqrt{\Delta}}{|a|}$, where $\Delta$ is the discriminant and $x_1,x_2$ the roots of a second degree polynomial equation [closed]
Suppose that $P(x)=ax^2+bx+c, a,b,c\in\mathbb{R}, a\not=0$ has two real roots $x_1,x_2$. Show that $|x_1-x_2|=\frac{\sqrt{\Delta}}{|a|}$, where $\Delta$ is the discriminant.