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 (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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4
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3answers
507 views

solution to equation $a \cdot \cos(\theta) - b \cdot \sin(\theta) = c$

Does the equation $$ a \cdot \cos(\theta) - b \cdot \sin(\theta) = c$$ have a closed-form solution for $\theta$? What about the case where $a^2 + b^2 = 1$?
1
vote
0answers
311 views

Transforming root-equations into polynomials

Let's define special polynomials as polynomials in $\mathbb{Q}[X]$, where we allow to make roots, too. Examples: $\sqrt{X^4+1}$, $\sqrt[3]{X}+\sqrt{X+1}$, $\sqrt{X+\sqrt{X+1}}$ How can I transform a ...
1
vote
2answers
88 views

Find $y=\sqrt{x}$ where $x$ and $y$ positive integers in polynomial time?

Let $x$ be a positive integer and let $y$ be a real number such that $$y=\sqrt{x}$$ Objectives: If $y$ is an integer, find it in polynomial time. If $y$ is not an integer, prove that there is no ...
1
vote
1answer
153 views

two functions $ f(x) $ and $ g(x) $

let be two functions $ f(x) $ and $ g(x) $ with an infinite set of roots $ a_{n} $ and $b_{n} $ so $ f(a_{n}) =0= g(b_{n}) $ also they satisfy the same functional equation $ f(1-s)=f(x) $ and $ ...
15
votes
1answer
1k views

Showing that the roots of a polynomial with descending positive coefficients lie in the unit disc.

Let $P(z)=a_nz^n+\cdots+a_0$ be a polynomial whose coefficients satisfy $$0<a_0<a_1<\cdots<a_n.$$ I want to show that the roots of $P$ live in unit disc. The obvious idea is to use ...
2
votes
2answers
137 views

Solving polynomials in $\mathbb{Q}[X]$ exactly

I wanted to write an equation solver for rational polynomials in one variable $X$. However, such solutions do not need to be in $\mathbb{Q}$. What I wanted was to display solutions "lossless", i.e. ...
1
vote
3answers
157 views

Equations - Solving for x

I have this problem: $$9x^3 - 18x^2 - 4x + 8 = 0$$ However, I'm not sure how to find the values of $x$. I moved the 8 over and factor out an $x$, but the trinomial it created can't be factored. ...
5
votes
1answer
480 views

Finding all roots of polynomial system (numerically)

I want to numerically find all the roots of a system of polynomials (n equations in n variables). Since I can compute the Jacobian for the system (analytically or otherwise), I can use the Newton ...
3
votes
4answers
112 views

Easy way to find roots of the form $qi$ of a polynomial

Let $p$ be a polynomial over $\mathbb{Z}$, we know that there is an easy way to check if $p$ have rational roots (using the rational root theorem). Is there an easy way to check if $p$ have any roots ...
1
vote
2answers
187 views

Irreducible polynomials with integer coefficients over Q

Suppose p(x) is an irreducible polynomial over Q of degree n, with integer coefficients. If p(x) has two roots r1 and r2 satisfying r1r2 = 5, prove that n is even. Attempt at solution: Because the ...
5
votes
1answer
1k views

Find the number of real roots of the polynomial

Find the number of real roots of the polynomial $$f(x)=x^5+x^3-2x+1$$ If I use Descarte's Rule then I get $$f(x)=x^5+x^3-2x+1$$ there can't be more than two positive real roots. Again ...
4
votes
2answers
209 views

Why isn't this square root $+$ or $-$?

I was tasked with proving the identity $\tan(\frac x 2) = \dfrac {\sin(x)}{1+\cos(x)}$ I used the quotient identity for tangent and the half angle identities for sine and cosine to get $ \pm \dfrac ...
4
votes
2answers
995 views

root-finding methods to invert numerically a function

let be the equation $ y-f(x)=0 $ the idea is to get $ s=g(y) $ that is x as a function of 'y' can this be made by a root finding algorithm ?? i mean you treat $ y $ as a numerical free parameter and ...
4
votes
1answer
1k views

What are the best methods for solving cubic and quartic equations by computer programs?

We know that there are closed form formulas for real roots of degree 4 and 3 polynomials, but people sometimes advise to use numerical (e.g. Newton) methods anyway. They claim that closed formulas ...
1
vote
1answer
67 views

Understanding a theorem of Marden's on the moduli of zeros of polynomials

My question is concerning Theorem 3.2 in this paper of Marden's. The gist of the theorem is stated below. Theorem 3.2. Every polynomial of the form $$ f(z) = \sum_{j=0}^{n} (b_j - ...
2
votes
4answers
479 views

Approximate solution for the root of a non-linear function

I have been working with a system which involves computing the roots of functions that look like \begin{equation} e^t (g\cos(\omega t) + b) = c \end{equation} where $t$ is the independent variable ...
3
votes
1answer
431 views

Multiple choice question - number of real roots of $x^6 − 5x^4 + 16x^2 − 72x + 9$

The equation $x^6 − 5x^4 + 16x^2 − 72x + 9 = 0$ has (A) exactly two distinct real roots (B) exactly three distinct real roots (C) exactly four distinct real roots (D) six distinct real roots
-1
votes
1answer
137 views

Find a degree 5 polynomial $f \in \mathbb Z_5[x]$ with exactly 4 distinct roots

Find a degree 5 polynomial $f \in \mathbb Z_5[x]$ so that it has exactly 4 distinct roots and factorize it as product of irreducible factors. I'm really struggling in finding such polynomial, so ...
0
votes
1answer
113 views

quadratic polynomial investigation

in my mathematics textbook,i have found one interesting problem and i have one question.textbook asks following problem deduce all possible value of $a$,for which equation $4*x^2-2*x+a=0$ has ...
3
votes
0answers
48 views

Existence of a Root of Elementary Monomials

Let $m_\lambda(X_1(t),X_2(t),...X_N(t))$ be a monomial symmetric function with partition $\lambda$. For example: $$ m_{(3,1,1)}(X_1(t),X_2(t),...X_N(t)) =X_1^3X_2X_3 + X_1X_2^3X_3 + X_1X_2X_3^3 $$ ...
2
votes
4answers
191 views

Proving all roots of a sequence of polynomials are real

Let the sequence of polyominoes $R_n(z)$ be defined as follows for $n\geqslant1$: $$R_n(z)\;= \;\sum_{r=0}^{\lfloor\frac{n-1}{2}\rfloor} \tbinom{n}{2r+1}(4z)^r.$$ I would like to prove that all the ...
0
votes
4answers
134 views

How do I form this equation?

If $A$ and $B$ are the root of the equation $3x^2-4x-9=0$, what is the equation whose roots are $(A+3)/(A-3)$ and $(B+3)/(B-3)$
0
votes
1answer
59 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 ...
9
votes
4answers
590 views

Solve for $x$: $2^x = x^3$

What category of equation is this? What methods are available to solve it? $2^x -x^3 = 0$ where $x\in\Bbb R$
4
votes
1answer
2k views

How do you prove an equation has one root?

I have this equation: $$9x + \cos x = 0$$ but I need to write out and prove why it has one real root. Could someone maybe give me a few pointers or what do I do exactly?
4
votes
2answers
87 views

Number of times two rescaled, 'fully' monotonic functions can cross

Consider two functions $f: [0,1) \rightarrow \mathbb{R}$ and $g: [0,1) \rightarrow \mathbb{R}$. Suppose $f(x) > g(x)$ for all $x \in [0,1)$. Suppose further that $f$ and $g$ are infinitely ...
4
votes
1answer
173 views

Is it assumable that $2^{1/12}$ is irrational because $2^{1/2}$ is?

I need to prove that $2^{1/12}$ is irrational but I need to connect this to $2^{1/2}$ being irrational. I know how to prove that $2^{1/2}$ is irrational, but can I assume that $2^{1/12}$ is irrational ...
2
votes
2answers
206 views

How to determine if a polynomial is of a particular order: 3rd degree (cubic), 4th degree (quartic) etc.

I am working on a math puzzle that results in the answer setting up a pair of equations for corresponding sides of similar triangles, then solving the first for y and substituting in the second that ...
5
votes
4answers
3k views

Is it true that a 3rd order polynomial must have at least one real root?

We were solving a problem with a friend and he said - look this polynomial is 3rd order (looks like ax^3+bx^2+cx+d), so it must have a real root. I didn't want to ...
25
votes
4answers
4k views

Is it possible for a quadratic equation to have one rational root and one irrational root?

Is it possible for a quadratic equation to have one rational root and one irrational root? Yes, a pretty straightforward question. Is it possible?
7
votes
1answer
297 views

Continuous root map of the coefficients of a polynomial

I have a set of polynomials $P_t(z)= z^n+ a_{n-1}(t)z^{n-1}+\cdots+ a_0(t)$ which depends on a real parameter $t \in [a,b]$ and where $a_{n-1}(t),\ldots, a_0(t)$ are real continuous functions. May I ...
3
votes
1answer
111 views

Root bracketing in complex space

I have some function $F(\omega): \mathbb R\to\mathbb C$. The function $F(\omega)$ has both roots and singularities. Fortunately, I can calculate positions of singularities analytically. So my ...
6
votes
1answer
120 views

Prove the equation has a root.

Assume that $f$ is a bounded and differentiable function in $(0,1)$. If $f({1\over 2})=0$, prove that the equation, $$2f(x)+xf'(x)=0,$$ has at least one root in $(0,{{1}\over{2}})$. I tried to do it ...
2
votes
3answers
177 views

Root equation - How to deal with two unknowns within some root terms and isolate one from another?

$$\sqrt{x+a} - \sqrt{x-a} = 2\sqrt{a}$$ Squaring both sides of the equation doesn't get rid of the root. How do I isolate $x$ from $a$?
2
votes
0answers
137 views

References for “closed form” numeric solutions of $\tan x=-a x$

I am looking for references that discuss solutions of the equation $\tan x=-a x$ (for $x,a\in \mathbb{R}$). I know about the graphical approaches, and any number of numerical solution approaches, ...
5
votes
2answers
227 views

Number of Complex Roots of a Complex Polynomial

This is related to the question I asked regarding finding the complex roots of $z^3+\bar{z}=0$. It turned out that there were 5 complex roots, but because the equation was of degree 3 I was only ...
6
votes
4answers
2k views

Help with Cardano's Formula

I'm trying to understand how to solve cubic equations using Cardano's formula. To test the method, I expand $(x-3)(x+1)(x+2)=x^3-7x-6$. My hope is that the formula will produce the roots $-1,-2,3$. ...
2
votes
3answers
106 views

Root equation - What am I missing?

There's a problem of which I know the solution but not the solving process: $(\sqrt{x} + 7)(\sqrt{x} - 1) = \frac{105}{4}$ I'm convinced that up to: $x + 6\sqrt{x} - 7 = \frac{105}{4}$ ...
10
votes
6answers
668 views

Complex roots of $z^3 + \bar{z} = 0$

I'm trying to find the complex roots of $z^3 + \bar{z} = 0$ using De Moivre. Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know ...
2
votes
1answer
231 views

Analytical method for root finding

Is there an analytical method to find the roots of the following equation? $$y = -\frac{1}{2}{x}^{2}-\cos(x)+1.1$$ I'm sorry for the trivial question, I'm new at math! :)
4
votes
3answers
1k views

How to tell if a quartic equation has a multiple root.

Is there any way to tell whether a quartic equation has double or triple root(s)? $$x^4 + a x^3 + b x^2 + c x + d = 0$$
5
votes
1answer
176 views

Root of a special polynomial

Given a polynomial $P(x)=\sum_{n=0}^{d}a_nx^n\in\mathbb{R}[x]$ with all roots on the unit circle. Question: Is it true that all the roots of $Q(x)=\sum_{n=0}^{d}a_n{{x+d-n}\choose{d}}$ lie on a ...
5
votes
1answer
76 views

How to extract roots in a complete local ring using binomial series

Let $A$ be a local ring with maximal ideal $m$ that is $m$-adically complete, and assume $1/2 \in A^\times$. I've read in several places that for any $x \in m$, a square root of $1 + x$ in $A$ is ...
1
vote
1answer
1k views

Factorising polynomials and Equating Coefficients

I am studying Factorising polynomials and equating coefficients at the moment, and up until now all has been well. I have no problem working these out up as high as the third degree as there are ...
25
votes
2answers
670 views

Countability of the zero set of a real polynomial

This is the question from my calculus homework: Is it possible for a polynomial $f\colon\, \mathbb{R}^{n}\to \mathbb{R}$ to have a countable zero-set $f^{-1}(\{0\})$? (By countable I mean countably ...
1
vote
1answer
40 views

Formula to scale a series that is being bent with a root / power.

I have a reference number, Rx, and a series of numbers, Sx[], to compare to it. Let's call the output Ox[]. I am using a simple square root to accelerate the apparent difference between the reference ...
6
votes
2answers
788 views

How was Euler able to create an infinite product for sinc by using its roots?

In the Wikipedia page for the Basel problem, it says that Euler, in his proof, found that $$\begin{align*} \frac{\sin(x)}{x} &= \left(1 - \frac{x}{\pi}\right)\left(1 + \frac{x}{\pi}\right)\left(1 ...
1
vote
6answers
831 views

Show that $z^6 + 5z^4 - z^3 + 3z$ has at least two real roots given that all roots are distinct.

Show that $z^6 + 5z^4 - z^3 + 3z$ has at least two real roots given that all roots are distinct. Also, show that $|3z - z^3 + 5z^4| < |z^6|$ when $|z| > 3$. I can see that 0 is a real ...
1
vote
2answers
106 views

Showing $2x=\left( 2n+1\right) \pi \left( 1-\cos x\right) $ has $2n+3$ roots when $n\in \mathbb{Z}_+$

I am struggling to show that the equation $$2x=\left( 2n+1\right) \pi \left( 1-\cos x\right) $$ where n is a positive integer, has $2n+3$ roots and no more and also if it possible to indicate their ...
4
votes
3answers
558 views

complex zeros of the polynomials $\sum_{k=0}^{n} z^k/k!$ inside balls

this is a question from a Temple prelim exam, and i'm trapped in it! We have $p_n(z)=\sum_{k=0}^n\frac{z^k}{k!}$ and we have to prove that $\forall r>0 \quad \exists N\in\mathbb{N}$ s.t. $p_n(z)$ ...