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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0
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1answer
38 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
votes
1answer
26 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: $$ x^...
0
votes
0answers
31 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
votes
1answer
49 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 ...
0
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1answer
44 views

roots of a polynomial with zero coefficient summation [closed]

Consider a polynomial, for which the summation of the coefficients is zero.What do we know about its roots?
0
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2answers
24 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: $$ X_{n+1}=X_{n}-\frac{e^{x}-x^{2}+1}{e^...
1
vote
0answers
37 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
33 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 $\pm\...
0
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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.) ...
1
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4answers
37 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 we ...
4
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1answer
49 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 ...
0
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0answers
25 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 ...
6
votes
1answer
32 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 ...
1
vote
2answers
67 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
5answers
121 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.
3
votes
3answers
105 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 ...
0
votes
0answers
55 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 ...
1
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1answer
54 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 ...
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: $$0 = at^2+bt+c+\sin(mt^2+nt+o)$$
2
votes
1answer
32 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) x^...
1
vote
1answer
39 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 ...
4
votes
3answers
108 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 $...
-3
votes
1answer
28 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 :)
2
votes
2answers
56 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 ...
1
vote
1answer
32 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 ...
7
votes
2answers
127 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 ...
0
votes
1answer
22 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 $$\Delta\left(\dfrac{az+b}{cz+d}\right)=(cz+d)^{12}\...
3
votes
3answers
106 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} a^...
2
votes
0answers
42 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
26 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 ...
0
votes
2answers
51 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
41 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 $f$ has no ...
1
vote
1answer
49 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 $$f(...
8
votes
3answers
110 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 $$ x^{a+2}-x^...
-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
85 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 $\...
3
votes
1answer
52 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 ...
0
votes
1answer
30 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 $w$, ...
1
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0answers
46 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
33 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
4
votes
1answer
77 views

A contest math problem

Let $P(x)$ be a polynomial with integer coefficients of degree $d>0$. If $\alpha $ and $\beta $ are two integers such that $P(\alpha)=1$ and $P(\beta)=-1$, then prove that $|\beta -\alpha | ...
1
vote
0answers
35 views

Roots of the Taylor approximation of the exponential

While answering another question, I looked at the roots of the $n^{th}$ degree Taylor approximation of the exponential. $$e^x\approx E_n(x):=\sum_{k=0}^n\frac{x^k}{k!}.$$ Apparently, these root are ...
2
votes
3answers
70 views

The number of distinct real roots of a polynomial of degree 4

Suppose I have a equation of a degree of 4 and I don't know a proper method of solving this type of equation (like completing the square is a proper method to solve the quadratic equation) so how or ...
0
votes
2answers
27 views

Roots of a Non-Monic Cubic Polynomial

Find all roots of $f(x)=231x^3+68x^2-9x-2$ I cannot use the cubic formula or Viete's theorem here because the polynomial is not monic. The only other way I can think of doing this is by the rational ...
3
votes
1answer
77 views

Find real roots of the equation

Find all real solutions to $$\dfrac{\sqrt{x+1}}{2+\sqrt{2-x}} - \dfrac{\sqrt{x^2-x+2}}{2+\sqrt{-x^2+x+1}} = x^3-x^2-x+1$$ This question is very similar to one of my previous problem, ...
2
votes
1answer
50 views

Solving a mixed radical and quadratic equation

Solve for $x \in \mathbb{R}$ $$4x^2(x+2) +3(2x^2-4x-3)\sqrt{4x+3} +6x = 0$$ I tried taking square by isolating the radical, but the resultant equation couldn't be solved. Any help ...
10
votes
1answer
148 views

Finding all real roots of the equation $(x+1) \sqrt{x+2} + (x+6)\sqrt{x+7} = x^2+7x+12$

Find all real roots of the equation $$(x+1) \sqrt{x+2} + (x+6)\sqrt{x+7} = x^2+7x+12$$ I tried squaring the equation, but the degree of the equation became too high and unmanageable. I ...
5
votes
1answer
101 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 tried ...
5
votes
1answer
107 views

Solving a radical equation for real $x$

Solve for $x \in \mathbb{R}$ $$\dfrac{\sqrt{x^2-x+2}}{1+\sqrt{-x^2+x+2}} - \dfrac{\sqrt{x^2+x}}{1+\sqrt{-x^2-x+4}} = x^2-1$$ I tried squaring the equation but it became a sixteen degree ...
2
votes
0answers
23 views

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 ...