# 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.

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### 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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### 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 :)...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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....
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### 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 ...
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### 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 (...
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### 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 ...
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 $x$,...