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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0
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2answers
50 views

roots written as exponents

It appears that i'm not quite sure anymore how to write roots as exponents, and how to work with them. I know $\sqrt[3]{a}$ is written $a^{\frac{1}{3}}$, but I don't know how to handle them in things ...
4
votes
4answers
1k views

How do I cube/square a logarithm?

Btw, please don't give me the answer. I just wanna know how to raise a logarithm to its cube cause I'm stuck in this part, but don't solve it for me. $$\log \sqrt[3]x = \sqrt[3]{\log x}$$ I tried ...
2
votes
3answers
396 views

Relations between the roots of a cubic polynomial

How do I solve the last two of these problems? The roots of the equation $x^3+4x-1=0$ are $\alpha$, $\beta$, and $\gamma$. Use the substitution $y=\dfrac{1}{1+x}$ to show that the equation ...
0
votes
1answer
39 views

How would you find the roots to this question?

I have a homework problem that I arrived. With Mathematica, the limit is 0. So by using $\epsilon= 10^{-6}$ (it is -6, not -0, sorry for the cutoff). $\sin(n^2)/\sqrt{n} <\epsilon =10^{-6}$ ...
0
votes
0answers
264 views

Integral of absolute value of polynomial?

Let $a(x)$ and $b(x)$ be integer irreducible polynomials where $b$ is U-shaped in the interval mentioned below and has 2 distinct real zeros. The zeros of $b$ cannot be expressed by radicals. Also $b$ ...
6
votes
2answers
1k views

roots of a cubic polynomial

Consider a cubic polynomial of the form $$f(x)=a_3x^3+a_2x^2+a_1x+a_0$$ where the coefficients are non-zero reals. Conditions for which this equation has three real simple roots are well-known. What ...
0
votes
1answer
90 views

Discriminant and roots of $ x^{n^2} \pm (x-1)^{n^2}$?

When considering the polynomials $x^{n^2} \pm (x-1)^{n^2}$ ( $n$ integer > 1 ) i noticed some things that appeared weird to me. Discriminant($x^{n^2} + (x-1)^{n^2}) = (n^2)^{n^2}$. ...
2
votes
2answers
94 views

bound the distance of two roots of multivariate polynomial systems

Consider a system of multivariate polynomial equations $\vec{x}= f(\vec{x})$ with integer coefficients, $f$ is at most of degree 2. Suppose $\vec{x}_1$ and $\vec{x}_2$ are two real roots of $f$, is ...
4
votes
3answers
456 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
267 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
86 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
151 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
757 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
136 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
155 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
426 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
111 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
186 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
204 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 ...
2
votes
2answers
772 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
894 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
62 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
463 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 ...
1
vote
1answer
317 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
131 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
93 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
47 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
171 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
583 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
84 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
169 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
190 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
3k 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?
6
votes
1answer
268 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
107 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 ...
5
votes
1answer
114 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
163 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
125 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
220 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
104 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
627 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
205 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
903 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
174 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 ...