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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.

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28 views

Number of roots of $p_n-\lambda p_{n-1}$ where $p_n$ are orthogonal polynomials

Let $(p_n)_{n\in\Bbb N}$ be the orthonormal sequence of polynomials associated to a tempered weight $w$ on an interval $I$ (so that for example, $\deg(p_n)=n$). Show that $p_n-\lambda p_{n-1}$ has $n$ ...
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2answers
25 views

New roots from old

The roots of $2x^2 − 8x − 1$ are $\alpha$ and $\beta$. Suppose another quadratic, $x^2 + qx + r$, has roots $1/(\alpha^3\beta)$ and $1/(\beta^3\alpha)$. What are $q$ and $r$? What I did to solve this ...
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0answers
35 views

What are some good recommendations of nonlinear equations/functions?

I have a project for my matlab course. I need to find a nonlinear equation to use to find the roots of it using various root-finders. I then have to write up a paper talking about the these different ...
4
votes
1answer
59 views

How to show that two trigonometric polynomials of degree $n$ combined have at most $2n$ zeros?

I am already aware of this question: Prove the following trigonometric polynomial has $2n$ zeros But it's not the same. Let be $P(x) = \sum_{k=0}^{n} a_k \cos (kx)$ and $\tilde{P}(x) = \sum_{k=0}^n ...
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vote
3answers
113 views

Why is $49^{-\frac{1}{2}}=\frac{1}{7}$?

Why is $49^{-\frac{1}{2}}=\frac{1}{7}$? I know that $\sqrt[n]{m^p}=m^{\frac{p}{n}}$ so I figured I can state the above is $-\sqrt{49}=-7$, but that is incorrect. I can't put the negative inside the ...
3
votes
1answer
47 views

Show that $\psi(n)$ has finitely many roots

Define $\psi(n)=\pi(n)-\phi(n)$ where we have the prime counting function and totient function respectively. I'm interested in where $\psi(n)=0$. Specifically is it possible to prove that there are ...
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0answers
15 views

Fundamental theorem of algebra and quaternions

I'm not sure if the fundamental theorem of algebra extends to every possible and imaginable numbers (real, complex, quaternions, etc.) but here's my question anyway. Let $f(x) = x^2-2ax+(a^2+b^2+c^2+...
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2answers
37 views

Show that $c^2 + a^2d=abc$ for a monic quartic polynomial

I apologize in advance for asking a homework question, but I have genuinely no idea on how to approach part b). The question is as follows: Consider the polynomial equation $\rm{P}(x)=x^4 + ax^3 + ...
0
votes
1answer
26 views

Prove that the roots of $0=z^3+3z+5$ are outside the circle $S(0,1)$

Prove that the roots of the complex equation $0=z^3+3z+5$ are outside the circle $S(0,1)$. I tried to guess some roots but failed. Thanks.
2
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1answer
34 views

Determining the number of real roots of a certain function

I apologize in advance for asking a homework question, but I have genuinely no idea on how to approach part b). The question is as follows: (a) Show that the polynomial expression $x^4 -x^2 + x +\...
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votes
0answers
31 views

Symmetry in the roots of a quadratic [closed]

Given $m$ is a root of $x^2+ax+b=0$ find all the possible values of $(a,b)$ such that $m^2-2$ is also a root.
5
votes
1answer
62 views

Continuous function on $[0,1]$ such that its zeros form a nowhere dense set of positive measure?

I know few facts, if $f : [0,1] \to \mathbb{R}$ is continuous, $Z(f) \triangleq f^{-1}(\{0\})$ is closed, there are continuous functions whose zeros are nowhere dense, there are nowhere dense sets of ...
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4answers
60 views

Finding the eigenvalues of $A=\left(\begin{smallmatrix} a & 1 & 1 \\ 1 & a & 1 \\ 1 & 1 & a \\ \end{smallmatrix}\right)$

I would like to calculate the eigenvalues of the following matrix $A$, but the factorization of the characteristic polynomial does not seem to be easy to compute. $A=\pmatrix{ a & 1 & 1 \\ 1 &...
3
votes
3answers
61 views

Solution to $\sqrt{\sqrt{x + 5} + 5} = x$

There are natural numbers $a$, $b$, and $c$ such that the solution to the equation \begin{equation*} \sqrt{\sqrt{x + 5} + 5} = x \end{equation*} is $\displaystyle{\frac{a + \sqrt{b}}{c}}$. Evaluate $a ...
2
votes
1answer
34 views

Computation of the complex roots of the Laplace transform of a function?

I have a function $f \in L^1(\mathbb{R}_+, \mathbb{R})$ with Laplace transform $$ \forall \Re(z) \geq 0,~~ \hat{f}(z) := \int_{\mathbb{R}_+} { f(t) e^{-zt } dt}.$$ I know explicitly the expression of ...
1
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1answer
19 views

Singular points of a matrix when the entries are restriced to a Lie Group

Let $\mathsf{SO}(3)$ be the set of $3 \times 3$ rotation matrices. Let $R\in\mathsf{SO}(3)$ and $r_{ij}$ represent the entry of $R$ sitting at the $i^{th}$ row and $j^{th}$ column, i.e., $$ R \in\ \...
0
votes
1answer
30 views

Quintic polynomial with three real roots

I want to get a quintic polynomial $f(X) \in \mathbb{Q}[X]$ whose Galois group $\mathrm{Gal}(L/\mathbb{Q}) \cong S_5$ where $L$ is the splitting field of $f(X)$. One of strategies to get it is ...
4
votes
2answers
92 views

Number of real roots of $p_n(x)=1+2x+3x^2+…+(n+1)x^n$ if $n$ is an odd integer

If $n$ be an odd integer. Then find the number of real roots of the polynomial equation $p_n(x)=1+2x+3x^2+....+(n+1)x^n$ $$ p_n(x)=1+2x+3x^2+....+(n+1)x^n\\ x.p_n(x)=x+2x^2+....nx^n+(n+1)x^{n+1}\\ p(...
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0answers
34 views

Why can you use substitution to find a polynomial with roots g(a), g(b)

In a maths textbook is this statement which pertains to roots of polynomials: If an equation in x has a root x=p, and if we make a substitution u = f(x), then the resulting equation in u has a root ...
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votes
0answers
31 views

Roots along a homotopy

Suppose we have two functions $f,g:\mathbb{R}^n\rightarrow \mathbb{R}$. Assume the existence of unique roots to both function, i.e. $x^f,x^g\in \mathbb{R}^n$ such that $f(x^f)=0=g(x^g)$. Define the ...
0
votes
1answer
22 views

How can I find the roots of this for k?

$$ln(1-e^{-kx})(1-e^{-kx})+kxe^{-kx}=0$$ I need to find $k$ in this equation , it should be a function of $x$. Any hints on how should I do it ?
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2answers
66 views

Let $P (x )=x^4+ax^3+bx+c=0$ and have real coefficient and have all real roots . Prove that $ab \leq 0$

Let $P (x )=x^4+ax^3+bx+c=0$ and have real coefficient and have all real roots . Prove that $ab \leq 0$ First Let the roots of this polynomial (call it P(x)) be $q,r,s,t$ By Vieta's, $a=-(q+r+s+t)$ ...
1
vote
0answers
31 views

Find a set of polynomials whose common zero set is $\{(1, 2), (0, 5)\}$.

Find a set of polynomials $\{P_1, \dots, P_n\}$, all of whose coefficients are real numbers, whose common zero set is the given set. I know what a zero set is, but I think my confusion comes from ...
5
votes
5answers
305 views

Quickest way to find $a^5+b^5+c^5$ given that $a+b+c=1$, $a^2+b^2+c^2=2$ and $a^3+b^3+c^3=3$

$$\text{If}\ \cases{a+b+c=1 \\ a^2+b^2+c^2=2 \\a^3+b^3+c^3=3} \text{then}\ a^5+b^5+c^5= \ ?$$ A YouTuber solved this problem recently and, though he spent some time explaining it, took over 40 ...
0
votes
1answer
46 views

A particular cubic can be written as $y_1^2 = (x_1 - e_1)(x_1 - e_2)(x_1 - e_3)$. Show that $e_1, e_2,$ and $e_3$ are distinct.

I am working through Algebraic Geometry: A Problem Solving Approach and am stuck on exercise 2.4.22. The previous problem was to consider $y^2 = 4x^3 + b_2x^2 + 2b_4x + b_6$ and transform this with $...
0
votes
3answers
51 views

Find all values of $a$ for which equation $ e^{x^2} = ax$ always has exactly two roots. [closed]

How shall I solve the following question? Please help! "Find all values of '$a$' for which equation $ e^{x^2} = ax$ always has exactly two roots."
0
votes
1answer
31 views

Reconstruct Quartics from roots

Not sure how to reconstruct an equation from roots. The roots are $-1/2, 1/2, 2, 3$ and the equation is $4x^4-20x^3+23x^2+5x-6.$
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2answers
51 views

Does the degree of a polynomial give the number of roots?

I am aware of the fundamental theorem of algebra, i.e., the degree of a polynomial is the number of roots of the polynomial. For example, $x^2 - 9 = 0$ would have two solutions: $x=3$ and $x=-3$. ...
2
votes
1answer
65 views

How to find the real positive root of $x^{k/2} - x - 1$.

Let $k$ be a large even integer. The following polynomial has exactly one real positive root. $$x^{k/2} - x - 1$$ How can one determine what it is asymptotic to, as a function of $k$?
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4answers
46 views

$x^2-2mx+m^2-1=0$ Find the range of m when one root lies in (-2, 4)

Let: $x^2-2mx+m^2-1=0$. For all $m$ there exist real roots for the above equation. If one root lies in between $(-2,4)$ Find the value range for $m$ My Work Since coefficient of $x^2$ is 1 the graph ...
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1answer
47 views

Solving $\tan (2x) = 6\cos^2(x) - 4\sin(x)\cos(x) - 2\sin^2(x)$

I need to solve the following trigonometric equation: $$\tan (2x) = 6\cos ^2(x) - 4\sin (x)\cos (x) - 2\sin ^2(x)$$ My attempt: $$\frac{\sin(2x)}{\cos(2x)} = 3(\cos(2x)+1) - 2\sin(2x) -2(1-\cos^2(...
2
votes
2answers
156 views

Determine the real numbers $a$, $b$, $c$ such that $1$, $\frac1{1+\omega}$ and $\frac1{1+\omega^*}$ are zeroes of the polynomial $p(z)=z^3+az^2+bz+c$

I am stuck on this question: Let $1$, $\omega$ and $\omega^*$ be the cube root of unity. a. Show that $\dfrac1{1+\omega}=-\omega$ and $\dfrac1{1+\omega^*}=-\omega^*$. b. Determine the ...
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votes
3answers
89 views

What are the integer coeffcients of a cubic polynomial having two particular properties?

Let $f(x) = x^3 + a x^2 + b x + c$ and $g(x) = x^3 + b x^2 + c x + a\,$ where $a, b, c$ are integers and $c\neq 0\,$. Suppose that the following conditions hold: $f(1)=0$ The roots of $g(x)$ are ...
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votes
0answers
30 views

Math terminology roots

I need to know the correct maths terminology in the following case: I find the roots (zeroes) of the first derivative of the Laguerre polynomial to give the positions (of an object) in a field. I ...
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0answers
10 views

Tower relation for field degrees and separable polynomial in splitting field

I have the following example exercise: Let $K$ be a field and $L$ the splitting field of a separable polynomial $f\in K[X]$ of degree $n$. Denote the zeros of $f$ in $L$ by $\alpha_1,\alpha_2,...,\...
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2answers
73 views

If $a$ is an $n$-th root of $z$ , $b$ is an $n$-th root of $w$, and $c$ is an $n$-th root of $zw$, then $ab=c$.

How do I prove that for all natural numbers $n$ and complex numbers $a, b, c, z, w$ if $a$ is an $n$-th root of $z$ , $b$ is an $n$-th root of $w$, and $c$ is an $n$-th root of $zw$, then $ab=c$. ...
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votes
0answers
7 views

Finding Maximum Value of CST Parameterization over an interval

I have a CST parameterization for a shape over an interval (0,1), so I have y as a function of x like so $$y = C(x)*s(x)$$ where $$C(x) = x^{n1}*(1-x)^{n2}$$ and $$S(x) = \sum_{i = 0}^{n} A_i(x)^i(1-x)...
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votes
1answer
112 views

Prove that $f(x)$ and $g(x)$ do not have any roots in common.

Suppose that $a(x)f(x) +b(x)g(x) = 135$ where $a(x), b(x), f(x)$ and $g(x)$ are polynomials over $F$. Prove that $f(x)$ and $g(x)$ do not have any roots in common. Any help is appreciated; thanks!
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votes
2answers
78 views

Find all $z\in{C}$ such that $z^6+(-8+8i)z^3-64i=0$. [closed]

Find all $z\in{C}$ such that $z^6+(-8+8i)z^3-64i=0$. For now I have that let $w=z^3$ such that the equation becomes a quadratic equation, $w^2+(-8+8i)w-64i=0$. Not sure what to do next; any help is ...
6
votes
3answers
117 views

How to prove that expessions like $\sqrt{93+63\sqrt{85}} - \sqrt{143} \notin \Bbb{Z}$?

The Problem: There are multiple "rooty" equations that can be simplified to a whole number, for example: $$\sqrt{19 + 6\sqrt{2}} - \sqrt{18} = 1$$ Because: $$\sqrt{19 + 6\sqrt{2}} - \sqrt{18} = \...
4
votes
1answer
51 views

Largest real root of a degree six characteristic polynomial

I am studying the growth rate of a weighted variant of self-avoiding walks and came up with a linear recurrence in terms of the fixed weights $a$ and $b$ whose characteristic polynomial contains the ...
1
vote
1answer
20 views

Algorithm for determining all the zeros of a complex rational function without initial guesses

Given a rational function R(x) = P(x)/Q(x), where P and Q are polynomials which can have complex coefficients, is there an algorithm which allows us to determine the zeros of R without an initial ...
4
votes
1answer
53 views

If a,b,c are positive rational numbers such that a>b>c then tell which of the following statement are correct following quadratic equation

I am solving following question based on quadratic equation If $a,b,c$ are positive rational numbers such that $a>b>c$ and the quadratic equation $(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$ has a root ...
-1
votes
1answer
30 views

Proof of root decomposition

How can I prove that if $\alpha$ is a root of the polynomial $S$ then $S(x) = Q(x)(x-\alpha)$, where the highest exponent $(Q)=n-1$ being $n$ the highest exponent of $S$. This can by generalized as: $...
1
vote
1answer
42 views

Find all the monic polynomials P(x) ∈ C [X] satisfying the below condition

What are all the monic polynomials $P(X) \in \mathbb C [X]$, with simple roots such that $$P(X^n) = \pm P(X) P(\zeta X) P (\zeta^2 X) …P (\zeta^{n-1} X),$$ where ζ is a primitive n-th root of unity?
1
vote
3answers
39 views

How can you determine if a root of a polynomial is a repeated one?

Let's say we have the polynomial $x^3 − 12x − 16$, and we know there are 2 roots, $x = -2$, and $x = 4$, and one of those roots appears twice. How can we determine which of those two roots is the ...
1
vote
3answers
81 views

Solve the equation $\sqrt{x + 2} - \sqrt{3 - x} = x^2 - 6x + 9$.

Solve the equation: $$\sqrt{x + 2} - \sqrt{3 - x} = x^2 - 6x + 9$$ Here's what I've done. Let $\sqrt{x + 2} = a$ and $\sqrt{3 - x} = b$ $\implies \left\{ \begin{align} a^2 + b^2 &= 5\\ a^2 - b^...
1
vote
0answers
46 views

Criterion for real roots of 4th order polynomial

Suppose we are given a polynomial equation of the form $$ax^4 + bx^3 + cx^2+dx +e = 0 $$ with $a,b,c,d,e\in\mathbb{R}$. Does there exist conditions on the coefficients $a,b,c,d,e$ such that the ...
1
vote
1answer
21 views

Roots of polynomial in $\mathbb{F}_{101}$ using symmetric polynomials.

I want to calculate the linear factorisation of the polynomial $f=x^5+5x^4+10x^3+10x^2+5x+70$ in the ring $\mathbb{F}_{101}[x]$. Using Eisenstein $p=5$ and Gauss' lemma, the polynomial is irreducible ...
1
vote
0answers
39 views

Root objects and the simplest possible analytic continuation of the Riemann zeta function.

The equation I am trying to solve is: $$\lim\limits_{k \rightarrow 3} \left( \sum\limits_{n=1}^{n=k} \frac{1}{n^s}+ \frac{1}{k^{s - 1} \cdot (s - 1)}\right)=0 \tag{1}$$ The simplest possible ...