Questions tagged [roots]
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 the (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.
6,922 questions
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$\zeta_r(s) = \sum_{n=1}^{\infty} \frac{\operatorname{rad}(n)}{n^s} = \prod_p (1 + \frac{p^{1-s}}{1 - p^{-s}}) = 0$?
In number theory, the radical $\operatorname{rad}(n)$ of a positive integer $n$ is defined as the product of the distinct prime numbers dividing $n$.
$$\DeclareMathOperator{\rad}{rad}
\rad(n) = \prod_{...
- 19.7k
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Inequalities with powers [duplicate]
Given this inequality:
$$
x^2 > 1
$$
What is the solution for x?
If I take the square root I would end up with this:
$$
x > \pm 1
$$
So
$$
x > 1,\
x > -1
$$
Is that correct?
And can that ...
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Succinct conditition for multivariate polynomial to have two geometric zeros along a/any line through origin
Given a multivariate polynomial with real numbered coefficients $f(x_1, \dots, x_n)$ with $f(\vec 0 ) < 0 $ and a non zero vector $\vec c \in \mathbb{R}^n$. Is there a succinct condition for the ...
- 351
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Can it be stated that no root with index $n > 2$ will be a number greater than half the subscript amount? [closed]
I am simplifying an algorithm for a program in pseudocode that guesses the root of any index, looking for a number raised to the power of the index as the result.
And I thought that if it's a root ...
- 119
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How to map square roots as a linear progression on a circle?
I have been exploring geometric constructions with roots. It seems that the fact that the circle is the locus of precise square root dispositions is often overlooked, as I haven't seen it used much in ...
- 483
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3
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For $\frac{3}{4}< a < 1$. Prove that $x^{3}(x+1)-(x+a)(2x+a)$ has $4$ distinct real roots and find a explicit form of it.
Let $\frac{3}{4}<a<1$. Prove that $$x^{3}(x+1)=(x+a)(2x+a)$$ has four distinct real solutions and find these solutions in explicit form.
My attempt:
The existence of the four roots can be seen ...
- 971
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Is there always at least a double root to a polynomial if it is the case that immediately to the left and right the polynomial has the same sign?
Every function $f(x)$ I have ever seen which has a root $r$ with the property
$$\text{sign}(f(r + \epsilon)) = \text{sign}(f(r + \epsilon))$$
has at least a double root at $r$, i.e. a function that ...
- 67
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1
answer
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Is there any possible way to solve this polynomial equation?
Let
$$ G(y,n) := y {\left( -1 + 3 y + y^2 + y^3 \right)^{n-1}}{\left( -1 - 4 y^3 + y^4 \right)^n} - {\left( -1 - y - 3 y^2 + y^3 \right)^{n-1}}{\left( -1 + 4 y + y^4 \right)^n} $$
and
$$ F(y, n) := y\...
- 21
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1
answer
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Asymptotic equivalence of the root of $f_n(x) := x^n + x - 1$?
The exercise consisting of finding the asymptotic equivalence of the root of the function $g_n(x) := x^n +nx - 1$ where $x\in[0,1]$, is a classic exercice, and its equivalence is $1/n$.
Now, take $f_n(...
- 431
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3
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views
$\Delta(u)+\kappa\tau(u)^2$ has at most two real roots on an interval where $\Delta\ne0$, with $\deg \Delta=4$ and $\deg \tau=2$
This is the final stage to finish an earlier question. I isolated it from the original geometric setting.
Let $\Delta(u)$ be a real quartic polynomial and let $\tau(u)$ be a real quadratic polynomial ...
- 551
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votes
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answers
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views
How to prove $(z-k)^n + (-z)^n + k = 0 \implies Re(z) = k/2$?
Let integer $n>0$ and integer $k$ with $k^2 > 1$.
Then we have:
$$(z-k)^n + (-z)^n + k = 0 \implies Re(z) = k/2$$
For instance
$$
\begin{align}
(z+2)^{15} + (-z)^{15} - 2 = 0 &\implies Re(z) ...
- 19.7k
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views
About the nature of the roots of $\frac{1}{x-a} + \frac{1}{x-b} + \frac{1}{x-c} = 0$, for real $a$, $b$, $c$
Let $a, b, c$ be three real numbers. Then the equation $\frac{1}{x-a} + \frac{1}{x-b} + \frac{1}{x-c} = 0$
(A) always has real roots
(B) can have real or complex roots depending on the values of $a, b$...
- 409
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answers
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views
Find all integers $a$ such that $(x-a)(x-12) + 2$ can be factored as $(x+b)(x+c)$ where $b$ and $c$ are integers. [duplicate]
Problem: Find all integers $a$ such that $(x-a)(x-12) + 2$ can be factored as $(x+b)(x+c)$ where $b$ and $c$ are integers.
If $(x-a)(x-12) + 2$ can be factored as $(x+b)(x+c)$, then $b$ and $c$ must ...
- 39
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How to find the largest root of this sextic polynomial?
I am trying to find the largest root of the sextic polynomial
$$ p(x) := \left( x^2 - a \right) \left( \left( x^2 -b -1 \right) \left( x^2 - a - 1 \right) - \left( x^2 - a \right) \right) - \left( x^2 ...
- 13.1k
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How do you handle $\frac{x^2-25}{x-5}=10$ when $x=5$?
I'm engaged in a dispute concerning the equation
$$\frac{x^2-25}{x-5}=10$$
Solution by factoring the LHS numerator to get $x+5=10$, or cross multiplying to get the quadratic equation $x^2-25=10(x-5)$ ...
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