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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30 views

Use Galois theory to find all complex roots of $T^4-2T^2-\sqrt{6}T+\frac{3}{4}$

I am currently studying Galois theory and a question that often comes up is "find all complex numbers which are roots of the polynomial $T^4+aT^2+bT+c$" where the coefficients are of the form ...
0
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0answers
26 views

Cubic polynomial with 1 real root and 2 complex conjugated roots (real coefficients)

I am stuck on this problem about cubic polynomials. I rely on the Wikipedia page on the topic. Using wikipedia notations (chapter "General formula for roots") : For the case where $\Delta > 0$, ...
0
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1answer
23 views

Bombelli's solution to a cubic

On p151 of my edition of Ian Stewart's "The Problems of Mathematics", he describes early work with imaginaries and Cardano's noting that Tartaglia's formula for solving a cubic, when applied to: $$ ...
0
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0answers
26 views

Calculation the root function

Given that $$g'(x)=4+xe^{-x} $$ I want to find an $x$ where $g'(x)=0$ holds. The solution is supposed to be $x=-1.2$, but im not able to find it . thanks
-1
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1answer
47 views

Number of roots $f(x)(\log(f(x)))^\prime$

EDIT: I am interested in derivative of a function $f(x)$. However, since the function is logarithmically convex/concave it is easier to analyze $\log f(x)$. Therefore, I rewrote the derivative and ...
3
votes
2answers
109 views

For a fixed and small $\epsilon$, finding the number of real roots of $x^{2}+e^{-\epsilon x}-2+\sin(\epsilon x)$

I saw the following question in an introduction to applied mathematics exam (this is only the first part of the question): Assume $0<\epsilon\ll1$ . Denote $$ f(x,\epsilon):=x^{2}+e^{-\epsilon ...
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1answer
44 views

roots of a polynomial with zero coefficient summation [on hold]

Consider a polynomial, for which the summation of the coefficients is zero.What do we know about its roots?
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2answers
23 views

Newton Raphson Example help

For h:= $\mathbb{R} \rightarrow \mathbb{R}, x \rightarrow e^{x}-x^2+1$ I know the formula as $$X_{n+1}=X_{n}-\frac{f(X_{n})}{f'(X_{n})}$$ so this would give me: $$ ...
1
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0answers
27 views

How does the Riemann Hypothesis show the prime spectrum with zeros?

I learned that dependent on the Riemann Hypothesis $$d(x)=-\frac{1}{\pi}\sum_{p^n}\frac{\ln(p)}{p^{\frac{n}{2}}}\cos(x\ln(p^n))$$ has peaks converging at the real points $t$ where $\zeta(\frac{1}{2} + ...
0
votes
0answers
7 views

Proving that largest root (obtained via P.C.A.) is a symmetric function

Suppose, we are given $\textbf{X} = (X_1, X_2, \ldots,X_m)$ and $\textbf{Y} = (Y_1, Y_2, \ldots, Y_n)$. Also, we are given, S = pooled variance. If we implement Principal Component Analysis (P.C.A.) ...
0
votes
1answer
66 views

Polynomials with roots having the same module and linear dependent arguments

Is it possible for a polynomial with integer coefficients to have some of its roots: $$m_1e^{i\theta_1 \pi}, m_2e^{i\theta_2 \pi}, \ldots, m_ke^{i\theta_k \pi}$$ such that there exist nonzero integers ...
7
votes
2answers
123 views

Roots of Sum of Two Polynomials (with Known Roots)

I am writing a piece of software and I'm trying to avoid root finding polynomials for efficiency purposes. I have two polynomials with complex coefficients, where the roots of both polynomials are ...
1
vote
4answers
33 views

Find the condition such that the roots of the polynomial are in AP

$f(x)=x^3+3px^2+3qx+r$ has roots in AP.Find the relation between $p,q$ and $r$. [Answer:$-2p^2-3pq+r=0$] My attempt:- Taking $d$ as the common difference of the roots in AP ...
1
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2answers
64 views

Solve $z^6+7z^3-8=0$

I want to find the solutions $z^6+7z^3-8=0$ but I don't know where to start because of the high degree of the equation. This is an exercise that involves complex numbers, so I have to transform the ...
4
votes
1answer
44 views

Graphically solving for complex roots — how to visualize?

So recently we've been doing the complex roots of quadratics, cubics and polynomials in general in school. But my question is, is there a way to see where these roots are, just like you can see where ...
6
votes
1answer
29 views

Iteration of polynomial has only positive roots

Let $P(x)$ be a real polynomial with a positive leading coefficient, and $k\geq 2$ an integer. Suppose that $Q(x)=P(P(\dots(P(x))\dots))$, where there are $k$ iterations of $P$'s, has at least one ...
0
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0answers
23 views

Explicit form of a generating function.

Let $q \geq p$ be natural numbers both larger than or equal to two. Let $u(z):=z^p+z^{p+1}+...+z^q$ and $p(z)=\frac{z u'(z)}{1-u(z)}$. Since $p(z)$ is rational, one can write (by the theory of ...
4
votes
5answers
116 views

Roots of $x^{101}-100x^{100}+100=0$

I do not know how to prove that $x^{101}-100x^{100}+100=0$ has exactly two positive roots. Some can give me hint for solving this please. Thanks for your time.
5
votes
3answers
435 views

Algorithm to find the exact roots of solvable high-order polynomials?

It is not generally possible to determine the roots of a polynomial whose grade is bigger than 4 in terms of roots and basic operations. But I heard that it is possible to give a criteria whether a ...
0
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0answers
54 views

Find a polynomial such that this proposed root finding algorithm fails.

Is this polynomial root finding algorithm below known, and under what conditions for the choice of polynomial coefficients does it find at least one root? Description of the algorithm: Consider the ...
3
votes
3answers
69 views

Need help solving $x^4-3x^3-11x^2+3x+10=0$

Solve $x^4-3x^3-11x^2+3x+10=0$ I have tried to solve this equation using 'general formula from roots' from https://en.wikipedia.org/wiki/Quartic_function. $$ax^4+bx^3+cx^2+dx+e=0$$ $$x_{1,2}=-\frac ...
1
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1answer
51 views

What mathematical notation can use for this formula

I just played around with archimedes $\pi$ formula and ended up with $\pi = \lim\limits_{n \to \infty} 6 \cdot 2^n \cdot \sqrt{2 - \sqrt{2 + \sqrt{2 + ...n times... \sqrt{2 + \sqrt{3}}}}}$ I want to ...
27
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2answers
699 views

Something strange about $\sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$ and its friends

This post can be generalized to, $$\begin{align} \sqrt{ 2+ \sqrt{ 2 + \sqrt{ 2-x}}}=x&,\quad\quad x = -2\cos\left(\frac{8\pi}{9}\right)=1.8793\dots\quad\quad\quad \\ \\ \sqrt{ 4+ \sqrt{ 4 + ...
0
votes
0answers
34 views

Is there an analytic solution to find zeroes of a polynomial plus sin()?

Is there an analytic solution to find the zeroes of an equation of the form: $$0 = at^2+bt+c+\sin(mt^2+nt+o)$$
1
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1answer
36 views

Find zeros of a function or at least say things about their location?

Let $a>0$ be a fixed parameter. I would like to find the (I think there are only two) $x\in \mathbb{R}$ such that $$(x-a)e^{-\frac{1}{2}(x-a)^2} = (x+a)e^{-\frac{1}{2}(x+a)^2}.$$ I know this might ...
2
votes
1answer
31 views

When are the limits of roots of a polynomial identical to the roots of the limit of the polynomial?

I have a univariate polynomial of degree $n$ (where $n$ is larger than $4$). The real-valued coefficients of the polynomial depend on a parameter $\psi$, i.e. $$p_\psi(x)=a_n(\psi) x^n+a_{n-1}(\psi) ...
4
votes
3answers
105 views

Is this an equivalent statement to the Fundamental Theorem of Algebra?

Is the following equivalent to the usual statement of the fundamental theorem of algebra: Let $$f(z)=c_nz^n+\cdots+c_1z+c_0$$ be a polynomial with complex coefficients. For all but finitely many ...
8
votes
3answers
109 views

Largest root as exponent goes to $+\infty$

Let $a\geq 1$ and consider $$ x^{a+2}-x^{a+1}-1. $$ I am interested to see what is the largest root of this polynomial as $a\to +\infty$. In order to find a root, we surely have to have $$ ...
2
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2answers
55 views

Rouche's Theorem application for $z^6-5z^4+3z^2-1$ in $|z|\leq 1$

Find the number of roots of $f(z)=z^6-5z^4+3z^2-1$ in $|z|\leq 1$ Taking $g(z)=1$ would be the obvious choice, but it's not the right one. The next choice would be $z^6-1$ because we know the roots ...
-3
votes
1answer
27 views

$\forall x,y\in \mathbb{R}\colon\forall n\in \mathbb{N}\colon [Odd(n)\lor Even(n) \land y\geq 0\implies [x^\frac{1}{n} =y\iff x=y^n ]]$ [closed]

Prove the following theorem : $\forall x,y\in \mathbb{R}\colon\forall n\in \mathbb{N}\colon [Odd(n)\lor Even(n) \land y\geq 0\implies [x^\frac{1}{n} =y\iff x=y^n ]]$ Thank you :)
0
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1answer
19 views

roots of modular forms in the complex field

For $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL_2(\mathbb{Z})$ the modular discriminant $$\Delta(z)=(2\pi)^{12}\eta(z)^{24}\qquad(1)$$ holds ...
3
votes
1answer
50 views

Roots of the equation $x^2+1=0$ in $\Bbb Z/p^{n}\Bbb Z$

Let $p$ be an odd prime number and $n$ be a positive integer. I want to consider roots of the equation $x^{2}+1=0$ in the ring $\Bbb Z/p^{n}\Bbb Z$. Suppose $n=1$. Find a condition on $p$ such ...
5
votes
1answer
96 views

Solve $ 1 + \dfrac{\sqrt{x+3}}{1+\sqrt{1-x}} = x + \dfrac{\sqrt{2x+2}}{1+\sqrt{2-2x}} $

Solve for $x \in \mathbb{R}$ $$ 1 + \dfrac{\sqrt{x+3}}{1+\sqrt{1-x}} = x + \dfrac{\sqrt{2x+2}}{1+\sqrt{2-2x}} $$ I tried some substitutions and squaring but that didn't help. I also ...
1
vote
1answer
30 views

Unique generating element of all integer polynomials that have $1+\sqrt 2$ as a root.

I have to find a polynomial $p(X)$ with root $1+\sqrt2$, so that no matter with what $q(X)\in\mathbb Z[X]$ it is multiplied, it again becomes a polynomial with that root. And probably that every ...
3
votes
3answers
100 views

If $a$ and $b$ are roots of $x^4+x^3-1=0$, $ab$ is a root of $x^6+x^4+x^3-x^2-1=0$.

I have to prove that: If $a$ and $b$ are two roots of $x^4+x^3-1=0$, then $ab$ is a root of $x^6+x^4+x^3-x^2-1=0$. I tried this : $a$ and $b$ are root of $x^4+x^3-1=0$ means : $\begin{cases} ...
2
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0answers
40 views

Monotonic roots

Consider we have a stricktly increasing positive sequence $\lambda_n$ and the following sixth order algebraic equation for every $n\in \mathbb{N}$, $$\zeta s^6-s^4+\lambda_n^2=0,$$ where $\zeta$ is a ...
1
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0answers
25 views

Is there an application to NOT assuming that a square root is positive?

Further to the question here:Why is the even root of a number always positive? If it is mere "convention" (agreement) that we use positive real numbers as the even-powered-roots of positive real ...
1
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1answer
46 views

Degree of the field extension

I need to determine the degree of the field extension $\mathbb{Q}(\sqrt{(2+\sqrt{2})(3+\sqrt{3}}))/\mathbb{Q}$. I've determined that the minimal polynomial of $\sqrt{(2+\sqrt{2})(3+\sqrt{3}})$ is ...
0
votes
2answers
48 views

Show that the set of polynomials with 1 as a root form a linear subspace

Let $\mathbb{C}(x)$ be the vector space $\mathbb{C}$ of polynomials $p\left(x\right)$ in one variable $x$ with coefficients in $\mathbb{C}$. Is the set $p(x) \in \mathbb{C}\left(x\right)$ such that ...
0
votes
2answers
34 views

Zeroes of the polynomial $f(x)$ over the field $F$ of order 256.

Let $F$ be a field with 256 elements and $f \in F[x]$be a polynomial with all roots in $F$. Then (1) $f \neq x^{15} -1$. (2) $f \neq x^{63} - 1$ (3) $f \neq x^2 + x + 1$ (4) if $f$ ...
5
votes
2answers
6k views

Finding the discriminant and roots of a polynomial

How is the discriminant of a polynomial determined? I know that for a quadratic function, the roots (where $f(x)=0$) are found by $$x=\frac{-b\pm\sqrt{\Delta}}{2a}$$ and here $\Delta$ is the ...
1
vote
4answers
85 views

Find $\alpha^3 + \beta^3$ which are roots of a quadratic equation.

I have a question. Given a quadratic polynomial, $ax^2 +bx+c$, and having roots $\alpha$ and $\beta$. Find $\alpha^3+\beta^3$. Also find $\frac1\alpha^3+\frac1\beta^3$ I don't know how to proceed. ...
-2
votes
3answers
51 views

$ \sqrt[n]{b} =a \Leftrightarrow a^{n} =b$ [closed]

Why the two-way relationship is established: $$ b^{ \frac{1}{n} }=\sqrt[n]{b} =a \Leftrightarrow a^{n} =b$$
2
votes
1answer
83 views

Why is $\varepsilon x^5 \sim -x$?

I'm trying to understand what's going on in this lecture on perturbation (the link brings you to 1h 08m 12s). The original problem is to find the real root of $$x^5+x=1.$$ We have inserted ...
0
votes
1answer
85 views

Approximate roots of nonlinear equation (non-integer polynomial)

In case of pulsating bubble arising from underwater explosion, bubble radius satisfies the following equation. $x^3\dot{x}^{2} + x^3 + \frac{k}{x^{3(\gamma-1)}} = 1$ The minimum and maximum bubble ...
0
votes
1answer
28 views

Analyticity of roots of a polynomial in terms of coefficients

Suppose that $f(z,w)$ is a non-constant polynomial in $z,w$ with coefficients in $\mathbb{C}$. Fix $z$, we define $p(w)=f(z,w)$. From Liouville's theorem, we know that $p(w)=0$ is solvable for $w$, ...
6
votes
3answers
204 views

Why four roots to this equation: $(7x+1)^{1 \over 3}+(8+x-x^2)^{1 \over 3}+(x^2-8x-1)^{1 \over 3}=2$

$$(7x+1)^{1 \over 3}+(8+x-x^2)^{1 \over 3}+(x^2-8x-1)^{1 \over 3}=2$$ I figured the roots are $0$, $1$, $-1$, and $9$. But why?
0
votes
1answer
513 views

Inverse Quadratic Interpolation and the secant method

I am currently completing a maths project that aims to approximate the roots of functions using MATLAB. The two root finding methods that I have used are inverse quadratic interpolation and the ...
1
vote
0answers
45 views

To find root of $x^n+1=0$ [duplicate]

If $ \alpha_1,\alpha_n, ....\alpha_n $ be the roots of the equation $x^n+1=0$, then $(1-\alpha_1)(1-\alpha_2)...(1-\alpha_n)$ is equal to a) 1. b) 0 c)n d)2 when I put n=3,and directly evalutae ...
1
vote
1answer
31 views

finding root of an equation with real coefficient.

If the equation $x^4 + ax^3 + bx^2 + cx+ 1=0 $ (where a,b,c are real numbers) has no real roots and if at least one root is of modulus one, then a)b=c b)a=c c)a=b d)none of the above