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.
6,571
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Can we say that $x=0$ is a double root of $f(x) = (e^x-1)(\ln( x+1))$?
Let $$f(x) = (e^x-1)(\ln( x+1))$$
So far , I've only seen examples of textbooks referencing repeated roots if we can write the function with linear factors raised to some natural exponent ( like if $a$...
- 2,329
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0
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58
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Roots of the polynomial $z^4+4z^3+6z^2-4z+4$ in the circle $|z+1|<1$.
I came across the following problem:
Find the number of zeros of $P(z)=z^4+4z^3+6z^2-4z+4$ in the circle $|z+1|<1$. I know that I can't apply Rouche's Theorem directly since we don't have a ...
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1
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36
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Minimal distance between zero's $d = D(f(z)) = \inf_{i \neq j} |(z_i - z_j)| s.t. f(z_n) = 0 $?
Let $f(z)$ be a transcendental entire function.
Hence $f$ is not a polynomial.
Assume $f(z)$ has infinitely many zero's $z_n$
$$f(z_n) = 0$$
Lets say that $f(z)$ is given by a taylor series.
Im ...
- 15.2k
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0
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35
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Is the fact that complex roots of polynomials with real coefficients occur in pairs of any use if we have a polynomial and know one real root only?
Consider the following theorem appearing in Linear Algebra Done Right
4.15 Polynomials with real coefficients have zeros in pairs.
Suppose $p\in P(\mathbb{C})$ is a polynomial with real coefficients.
...
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0
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37
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Reverse Engineering Squared Squared [closed]
So i'm working on an encryption module and im having a wee bit of trouble.
Lets say I have multiple numbers between 2 and 8.
Now I square the numbers with each other.
a * a * b * b * c * c * d * d = ...
- 1
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1
answer
96
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Permutating the coeffecients of equations with roots of unity.
I"ve noticed the roots of
$$(x^k-1 =0) = (1-x^k=0)$$
This is simple enough since changing +/- signs gives you the additive inverse which is equal at zero.
However this property can be also be ...
1
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1
answer
93
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Find the sum of cube root of the roots of a cubic equation.
Question. Given roots $a,b,c$ of the cubic equation $x^3+3x^2+11x+1=0$, find $$s=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}.$$
Note that this polynomial is irreducible over $\Bbb Q$.
Vieta's formula was my ...
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Find range of constant part of 4th degree polynomial
Suppose there is a 4th degree polynomial ax^4 + bx^3 +cx^2 + dx + e then what will be the range of e so that this equation will have four real roots.
I thought about using differentiation to try it ...
1
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0
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Finding roots of a partially scaled function using the roots of the original function
I was playing around with trig functions and came across the Dottie Number $D \approx 0.73$ as the root of $f(x) = \cos{x}-x$. Then some time later I came across a similar function: $g(x) = \sqrt2\:\...
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0
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30
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Result involving monic polynomial, its roots and the gamma function
Do you know any reference where this result is proved? We have already proved it but since we know that this is a known result, we want to omit the proof and cite a proper reference. Here you have a ...
user1257320
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0
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44
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Can a nonzero complex power series have an uncountable set of complex roots?
Following Can a real power series have an uncountable number of real roots? and this essentially equivalent question about a sort of linear independence of powers of a real function, the natural thing ...
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1
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Why does a complex function that is real-valued when given real values as arguments have conjugate zeros? [duplicate]
I am currently reading Hamming's Numerical Methods for Scientists and Engineers. On pg. 79, it is stated that if a complex function is such that for every real argument produces a real value, then its ...
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67
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Determine how many $k$ have a square root $\mod n$
How can I easily determine how many $k$ have a square root $\mod n$ without having to write them out?
For example, I could ask how many $1\le k\le 143$ have a square root $\mod 143$. I can look at $\...
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0
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67
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Finding the 8th and 10th sum of a polynomial. roots of polynomial equations [closed]
The cubic equation has roots $$ x^3-x+4=0$$ has roots α, β, Γ find the cubic equation that has the roots $α^2, β^2, Γ^2$ hence or otherwise determine the values of S6, S8 and S10 where Sk is the sum ...
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2
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the zeros of a polynomial depend in a continuous way on its coefficients. [closed]
We use $P_n$ to denote the vector space of complex polynomials of degree $\leq n$, and we write
$$ \mathcal{P}^{*}_n = \mathcal{P}_n \setminus \mathcal{P}_{n-1} $$
for the set of polynomials of exact ...
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0
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Proving $f(x)=\sum_{k=1}^n a_k \cos k x$ has at least $n$ roots, for $0 \leq x \leq \pi$ and real $a_i$, when $|a_n|>\sum_{k=1}^{n-1}|a_k|$ [closed]
How do you prove this question? Thanks for helping!
The number of roots of $f(x)=\sum_{k=1}^n a_k \cos k x$, for $0 \leq x \leq \pi$, is as least $n$, when $\left|a_n\right|>\sum_{k=1}^{n-1}\left|...
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Finding coefficients using roots of polynomials [closed]
For the polynomial $x^4+ax^3+bx^2+c=0$, with zeros $\alpha,\beta,\gamma,\text{ and }\sigma$, it is given that
(i) sum of all roots is $2$,
(ii) the product of all roots is $1$, and
(iii) the ...
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1
vote
2
answers
80
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Sum of roots of an irreducible polynomial in Q[x]. [closed]
Set $f(x)\in \mathbb{Q}[x]$ is irreducible, and $\deg f(x)=2n+1$. Prove that sum of any two different roots of $f(x)$ can't be a rational number.
- 21
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4
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Finding $a$ such that $\tanh(x)-a\sin^2(x)=0$ has a double root
Given $f(x) = \tanh(x)-a\sin^2(x)$, what is the value of $a$ for which $f(x) = 0$ has a double root, and what is the value of that double root?
My Work :
Using MAPLE , I plotted a few graphs and ...
0
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1
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38
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A Question on prove square root equality formulation [closed]
I am looking for a proof of the following equality
$$\sqrt{2+\sqrt{3}} + \sqrt{4-\sqrt{7}}= \sqrt{5+\sqrt{21}} $$
- 19
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0
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43
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About the degree of a polynomial that has only real roots and all coefficients are 1 except one
I have a polynomial P with real coefficients of degree n, where all coefficients are equal to 1 except one. I need to prove that if the roots of P are all real, then necessarily n < 5. I just want ...
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1
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50
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On the lacunary function $f(z) = \frac{r + f(-1)}{\sqrt 5}$ [closed]
Consider the lacunary series
$$ f(z) = \sum_{n=1}^{\infty} a_n z^n $$
with radius $1$ and natural boundary at the unit circle.
The $a_n$ are real and strict positive, and also
$$\sum_{n=1}^{\infty} ...
- 15.2k
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0
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$ f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 0$?
Let $p_n$ be the $n$ th prime number.
Let $f(s)$ be a Dirichlet series defined on the complex plane as :
$$f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 1 + \frac{2^{-s}}{2}+ \frac{3^{-s}}{3} + \...
- 15.2k
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$ 0 = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{\ln(n)} \implies Re(s) \leq \frac{1}{2}$?
Define $f(s)$ as
$$ f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{\ln(n)}$$
where we take the upper complex plane as everywhere analytic.
Notice this is an antiderivative of the Riemann Zeta function, ...
- 15.2k
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3
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78
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Limits with Square root [duplicate]
I have this ridiculous doubt, why don't we take $+$ or $-$ while taking the square root of a number at the last step of finding a limit?
For example,
$$\lim_{x\to 0} \frac{\sqrt{1-x^2}- \sqrt{1+x^2}}{...
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Prove that a Kirchhoff polynomial have only real roots
I read Kirchhoff's original article from 1847 and tried to generalize his result to the case of the transcient regime of $R$, $C$ networks.
The original article is limited to a network of $n$ ...
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1
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A function with negative discontinuous derivative and many zeros
I want to construct an example of a function $f:\mathbb{R} \to \mathbb{R}$ with the following properties. By the way, I am not sure if it can exist.
$f(0) = 0$
$f'(0) < 0$
$f$ is continuous at $x=...
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About real roots of A polynomial with real coefficients that has three consecutive equal
How to prove : A polynomial with real coefficients that has three consecutive equal (non-zero) coefficients of successive powers of the variable cannot have all its roots real.
I know how to prove it ...
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Sum of the roots of the equality. [closed]
Sum of the roots in the range $\left(-\frac{\pi}{2},\frac{\pi}{2} \right)$ of the equation $\sin x\tan x=x^2$ is
$\frac{\pi}{2}$
$0$
$1$
None of these
This is a contest problem so I do not wish to ...
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2
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63
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Finding a complex number when given its roots
Given 3 complex numbers a, b, and c, does there always exist another number N where taking the cube root of N gives a, b, and c? If not, when is that possible? How do I find N? How does this extend to ...
1
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1
answer
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Do simple roots on most slices imply a polynomial is square free?
Consider a complex polynomial $p(x,y)$ in two variables.
I am interested in what it implies for $p$ if for all $y\in\mathbb C$ the one-variable polynomial $p(\cdot,y)$ has or does not have multiple ...
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Understanding Gradient descent with momentum equations
I'm working on implementing gradient descent with momentum for root finding but I am slightly confused about a part of the equation, it's said that you can replace your regular gradient step by doing: ...
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0
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Root of a negative real number - mistake in GRE Question?
I already apologize for the banality of the question but I've come across the following GRE question and wanted to ask whether this is really a well-defined exercised.
One has to compare quantities ...
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Find the equation whose roots are twice the negative reciprocal of the roots
$f(x) = 3x^6+x^4-2x^3-x+1$
Let the roots of $f(x)$ be $\alpha_1,\alpha_2,...,\alpha_6 $
Find the equation whose roots are twice the negative reciprocal of $f(x)$
My question is do I need to find an ...
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When is the square root differentiable?
Let $f$ be a non negative differentiable function (defined on $\mathbb R$) and $g(x)=\sqrt{f(x)}$. Can you characterise the points $x_0$ where $g$ is differentiable at $x_0$?
It is clear that when $f(...
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Why for a real number $x$ and an even integer $n, x^{1/n}$ is not real if $x<0$?
Why for a real number $x$ and an even integer $n$, $\;x^{1/n}$ is not real if $x<0$?
What I'm really asking is how do you define a complex number without referring to imaginary numbers.
How do you ...
user1243185
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2
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Find an expressions for $\alpha^3+ \beta^3$ in terms of $k$ [closed]
Find an expression for $\alpha^3+ \beta^3$ in terms of $k$ for the quadratic equation $3x^2+kx-4=0$
This is a questions from my Roots of Polynomials class.
Using the facts of $\alpha+\beta= -b/a$ and $...
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0
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Reference for Zassenhaus bound of roots of polynomial
I remember author of some textbook alluding to Zassenhaus and Knuth's bound of the zeros of a complex polynomial.Unfortunately after even after days of search I could not find some reference or ...
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How many roots the equation $ax^2+bx+c$ has in $(1,2)$ if $4a+3b+2c$ and $a$ have the same sign? [duplicate]
Let $a, b, c \in \mathbb R$ and $a \neq 0$ such that $a$ and $4a + 3b
+ 2c$ have the same sign. Then the equation $ax^2 + bx + c = 0$ must have
$1)$ both roots in $(1,2)\quad$ $2)$ no root in $(1,2)\...
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0
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Finding root of nonlinear equation in interval
I'm trying to understand finding root questions and I do not quite understand how you should approach them. I've searched online and I've come across posts suggesting to use substitution where you ...
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1
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Zeros of $f(z) = \sum_{n= 0}^{\infty} \frac{z^n}{n!\exp(\ln^2(n+1))} $ and $\lim_{x \to -\infty} f(x) = A $?
Consider the entire function
$$ f(z) = \sum_{n= 0}^{\infty} \frac{z^n}{n!\exp(\ln^2(n+1))}, $$
where $\ln^2(n+1)$ stands for $(\ln(n+1))^2$.
Where are the zero's of this function? That is, what are ...
- 15.2k
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0
answers
105
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Bisection Method : Estimating Iterations needed to approximate a root within three decimal places
I have been working on a problem involving the bisection method and the estimation of the number of iterations required to approximate the root of a given function to three decimal places. I have made ...
- 11
2
votes
3
answers
118
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Rolle's Theorem in proving exactly one real zero
The question asks to use the Rolle's Theorem to prove $f(x)=(x-8)^3$ has only one real zero.
I have already used IVT to prove that there is a zero, but I'm stumped on how to use the Rolle's Theorem to ...
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1
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98
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Prove that $a_n\ge 1$ for all $n\ge 1$ with equality iff $n=1,2,4,5$
Let $\alpha, \beta,\gamma \in \mathbb{C}$ be the three roots of $x^3 + x+1$. For any $n\in\mathbb{N}$, let $a_n = \dfrac{(\alpha^n-1)(\beta^n-1)(\gamma^n-1)}{(\alpha-1)(\beta-1)(\gamma-1)}$. Prove ...
- 1,195
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0
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Is the number of roots for a polynomial dependent on the degree or the field
If you have a polynomial for example:
$$z^{2} + 1 =0$$
We know the roots are $i$ and $-i$ in the $\mathbb{C}$ but if we got $\mathbb{H}$ we will get 3 roots $i,j$ and $k$.
But this violates the ...
1
vote
1
answer
89
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Is there a field $k$ in which $x^3+3$ has double roots?
I am doing a problem in which I am looking at the polynomial $x^3+3$ in a field $k$. I have separated my answer in three cases:
$x^3+3$ has no roots in $k$
$x^3+3$ has exactly one root in $k$
$x^3+3$ ...
- 1,640
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1
answer
31
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Relationship of the roots of two polynomials both which have the same constant and x term but vary in the x^3 term by a function of the x coefficient [closed]
Given that we know the roots of the first equation is there anyway to use them or the relationship between these two quintic polynomials to discern the roots of the second equation.
is there any ...
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1
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266
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Explore the relationship between $\sum\limits_{n = 1}^{2x} \frac{1}{{n^s}^x}$ and $\sum\limits_{n = 1}^{2x-1} \frac{(-1)^{n-1}}{{n^s}^x}$ [closed]
I am trying to find an algorithm with time complexity $O(1)$ for a boring problem code-named P-2000 problem. The answer to this boring question is a boring large number of $601$ digits. The DP ...
user1236730
1
vote
0
answers
84
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How many roots in a system of quartic equations
I am considering three general tori in 3D space, each defined by a quartic equation. My primary question revolves around the number of real solutions that arise from the system of these three ...
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2
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75
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What is $i^{\frac13}$?
When asked the value of $i^{\frac13}$, should I present one solution, or the three solutions of $z^3=i$, $z\in\mathbb{C}$? If only one solution, what is the criteria? It seems principal roots can be ...
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