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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5
votes
4answers
94 views

What is the minimum value of $abc$

If the roots of the equation $$ax^2-bx+c=0$$ lie in the interval $(0,1)$, find the minimum possible value of $abc$. Edit: I forgot to mention in the question that $a$, $b$, and $c$ are natural ...
4
votes
2answers
64 views

Evaluate $a+b+c+d$

If $a$, $b$, $c$, and $d$ are distinct integers such that $$(x-a)(x-b)(x-c)(x-d)=4$$ has an integral root $r$, what is the value of $a+b+c+d$ in terms of $r$? I tried to analyze graphically by ...
0
votes
2answers
50 views

Elliptic Curves and “roots”

Given elliptic curve $\omega$ in $\mathbb{R}^2$ such that $y^2 = x^3 + ax + b$, how can you find how many solutions (and what they are) of $x^3+ax+b$ have a $y$ value of $0$; or as they call it, a ...
12
votes
3answers
166 views

Why is this polynomial a monomial?

Let $p$ be a polynomial of degree $n$ such that $|p(z)| = 1$ for all $|z| = 1$. Why is it that $p(z) = az^n$ for some $|a| = 1$? I've noticed that we could easily prove this by induction if we ...
3
votes
2answers
56 views

Find the value of $\left | b-c \right |$

Given that $a, b, c \in \mathbb{Z}$, $a>10$ and $$(x-a)(x-12)+2=(x+b)(x+c)$$ Find the value of $\left | b-c \right |$ NOTE: The answer to this problem (as given on the last page of my book) is ...
0
votes
1answer
23 views

How to prove a nonlinear tracendent equation has two positive roots?

How to show (but do not use numerical software such as Mathematica, Matlab...etc.) that this equation \begin{equation} \frac{u (83811 u-88223)+18076}{396-3276 u}-\frac{10 \log ...
2
votes
1answer
31 views

Finding Complex Zeros

I have to find how many zeros $3e^z - z$ has in $abs(z) < 1$. I know a function has a zero of order m if $f(z) = (z-z_0)^mg(z)$, where $g(z)$ does not equal 0. I was thinking of maybe applying ...
8
votes
1answer
58 views

Existence of root of a polynomial over $\mathbb F_p$.

I came accross the following question and I can't find an easy proof of this fact : Let $p\geq 17$ be a prime number such that $p\equiv 1 \pmod 4$. Show that for any $z\in \mathbb ...
2
votes
3answers
72 views

Find amount of roots

We are given equation $$\frac {e^x}{x^2} = a$$. Task is to find how many solutions equation would have depending on values of a. Let's illustrate a(x): It's easy to conclude that there aren't no ...
-4
votes
2answers
123 views

Select the approximate values of x that are solutions to $f(x) = 0$, where $f(x) = -7x^2 + 6x + 9$? [closed]

These are the answer choices: $$\begin{align*}\\ A&~~\{–0.78, 0.67\}\\ B&~~\{-7, 6\}\\ C&~~\{–0.86, –1.29\}\\ D&~~\{–0.78, 1.64\} \end{align*} $$
2
votes
2answers
72 views

Zeroes of polynomial

$$c_1,c_2 \text{ are polynomial's }g(x)=x^2+ax+b \text{ roots } \Leftrightarrow \begin{cases} g(c_1)=c_1^2+ac_1+b=0 \\ g(c_2)=c_2^2+ac_2+b=0 \end{cases}$$ Prove that for every polynomial with integer ...
0
votes
1answer
36 views

Quadratics and roots

The question I am trying to solve is this: $4 x^2 - 3 x - 3 = 0$ has roots $p, q$. Find all quadratic equations with roots $p^3$ and $q^3$. I was able to answer this question by simply finding the ...
2
votes
0answers
19 views

Extensions of the Hermite Bielher and Hermite-Kakeya Theorem

A stable polynomial is one with zeros in the upper half plane or lower half plane. Interlacing polynomials are polynomials with only real zeros, where between every two zeros of one polynomial lies a ...
1
vote
1answer
32 views

Constant function with maximum modulus [duplicate]

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is ...
6
votes
2answers
149 views

All roots of polynomial inside the open unit disc

I know from here that for a polynomial $p(z)=a_0+a_1z+...+a_nz^n$ with $0<a_0\leq a_1\leq...\leq a_n$ all roots are in the closed unit disk. What condition do we need to get that all roots are in ...
1
vote
1answer
63 views

Minimum Modulus Principle for a constant fuction in a simple closed curve

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is ...
0
votes
0answers
22 views

Counting the roots of nonlinear systems of equations

I have a "nice" function (vector field) $$\mathbf{f}: \mathbb{R}^n \rightarrow \mathbb{R}^n$$ and I need to find how many roots (zeros) it has in a certain domain (hopefully prove that it has at most ...
3
votes
2answers
117 views

Real zeros of the zeta function

How does one show that the negative even integers make up all the real zeros of the zeta function? That is, how does one show that there are no real zeros on the interval [0,1]? I am aware that you ...
1
vote
3answers
27 views

Roots of Polynomial Equation?

$y=1/x$ so I plugged in $x=1/y$ into the equation above and got $y^{4}+y^{3}+y^{2}/c+y/4-1/2$, but apparently it's wrong, when I looked up the answer below. What am I missing?
0
votes
2answers
41 views

Distinct polynomials with exactly one shared root

If $\alpha_1,\ldots,\alpha_n$ are roots of a polynomial with rational coefficients is there a polynomial with rational coefficients for which $\alpha_1$ is a root but $\alpha_2,\ldots,\alpha_n$ are ...
1
vote
1answer
39 views

Determine all real solutions of the system of n equations

For $n\geq3$, determine all real solutions of the system of $n$ equations: $x_{1}+x_{2}+...+x_{n-1}=\frac{1}{x_{n}}$ ... ...
0
votes
2answers
26 views

Possibility of integral quadratic with these roots

If x and w are the roots of a quadratic equation with integral coefficients then is this possible: ${x = w = \frac{2}{3}}$. The correct answer says it is, but how is that so if it means: ...
4
votes
4answers
190 views

Can $x^3+x^2+1=0$ be solved using high school methods?

I encountered the following problem in a high-school math text, which I wasn't able to solve it: $x^3 + x^2 + 1 = 0$ Am I missing something here, or is indeed a more advanced method necessary to solve ...
1
vote
1answer
32 views

description of the function whose graph corresponds to Figure

Consider f be a real continuous function , $f(0) = 0$ , and whose graph has the form shown in the figure: a) How can a give description of the function whose graph corresponds to Figure. b) Sketch ...
0
votes
1answer
45 views

Using Rouche's theorem to find number of roots.

I am still unsure how exactly one applies Rouche's Theorem to find the roots of polynomials. For example, to find how many roots $z^9+z^5-8z^3+2z+1$ has in between the circles $|z|=1$ and $|z|=2$. I ...
0
votes
1answer
47 views

Two factored polynomials and a parameter

I need to solve $p(x)=aq(x)$ with multiple real $a$, where $p(x)$ and $q(x)$ are the two polynomials in $x$ (with real coefficients). The roots of $p(x)$ and $q(x)$ were found previously, i.e. these ...
1
vote
2answers
86 views

Evaluation of complex real numbers

The much anticipated math.SE community blog will $\tiny\mathrm{hopefully}$ contain a contribution from Alex Becker with the topic The Complex Real Roots of $x^3-3x+1$, which I'm really looking forward ...
0
votes
1answer
33 views

problem regarding theory of equations

given quadratic equation : ${x^2+bx+c=0}$ let the roots of the equation be ${u}$ and $v$. let ${S_0 = u^0+v^0}$ let ${S_1 = u^1+v^1}$ let ${S_2 = u^2+v^2}$ show that : ${S_2+bS_1+S_0 = 0}$
2
votes
2answers
53 views

finding the value of u of equation 5u^2 = 10u

I was solving a question, and while solving that problem I noticed something $5u^2 = 10u$ (solving this) this can be solved as: $5 \cdot u \cdot u = 10 \cdot u$ $u = \dfrac{10u}{5u}$ $u = 2$ ...
1
vote
2answers
41 views

Finding matching roots

If ${4 + \sqrt{2}}$ is one root of a quadratic equation given by ${x^2 - Px + Q =0}$ where P and Q are rational numbers then find the missing root. The answer is ${4 - \sqrt{2}}$. And I'm a bit ...
0
votes
1answer
21 views

Quadratic roots question

If $3.5 - {\sqrt 2}$ and $3.5 + {\sqrt 2}$ are the roots of a quadratic equation ${ax^2 + bx + c = 0}$; then which of the following is not correct? A. a is nonzero - I ruled out this because if was 0 ...
1
vote
0answers
48 views

How to explain this result due to Pôlya

How to explain this result due to Pôlya: An entire function is determined uniquely by the inverse images, counting multiplicities of three distinct non omited values. I cannot understand how this ...
3
votes
1answer
33 views

Finding zeros of maps between manifolds with different dimensions

One of the questions for which the notion of degree is useful is: does this map have a zero. For example, one can prove the Fundamental Theorem of Algebra using the following fact involving the ...
1
vote
1answer
45 views

Laguerre's method and zero division

I'm trying to understand Laguerre's method for root finding and I have hit one road block. Suppose I have a polynomial $p(x) = x^4 + 1$ and an initial guess $x_0 = 0$. This results in division by ...
2
votes
1answer
37 views

Solving $\operatorname{ctg} x=x/b$

I have no problems finding first solution (both: $b \to 0$ and $b \to \infty$). My solutions on photos. I got stuck trying to find solution when $x \to \infty$. As I think, solution for $x$ will have ...
1
vote
2answers
80 views

Discriminant of $f(x)=x^3+ax+b$

Suppose we have the polynomial $f(x)=x^3+ax+b$, with roots $\alpha, \beta, \gamma$ in $\mathbb{C}$, and let $\Delta = (\alpha - \beta)(\beta - \gamma)(\gamma - \alpha)$. Is there any quick way of ...
2
votes
4answers
71 views

$f(x^2 + 1) = f(x)g(x)$ $\forall$ x $\in\mathbb{R}$ $\Rightarrow$ no. of roots of $f(x)=$?

If two real polynomials $f(x)$ and $g(x)$ of degrees $m$ $(\ge2)$ and $n$ $(\ge1)$ respectively, satisfy $f(x^2 + 1) = f(x)g(x)$; for every $x \in \mathbb{R}$ , then what can be said about the ...
3
votes
4answers
112 views

Find roots of $3^x+x^3=17$

Find $x$ in the following equation: $$3^x+x^3=17$$
0
votes
2answers
83 views

Roots of $x^4-3x^2+x-\sin x$

Is there a logical/qualitative argument to find out the number of roots of $x^4-3x^2+x-\sin x$ ? I tried plotting graphs for $x^4-3x^2+x$ and $\sin x$ to check the points of intersection but it only ...
1
vote
1answer
36 views

Basic question from LA: Why do we find Roots of a polynomial?

This may sound like a basic question, but I am sorry to say that I did not find it's answer which completely satisfy my query. Here is the question: "What is the need to find roots of a polynomial ?" ...
0
votes
0answers
35 views

Prove that $x^3+px+q$ has multiple root $\Longleftrightarrow$ the discriminant is $0$. Is my solution correct?

$f'=x^2+p$ $f'=0,p=-3x^2$ Using $p=-3x^2,f=x^3-3x^3+q$ $f=0,q=2x^3$ Then the discriminant is: $$D=-108\cdot\big((2x^3/2)^2+(-3x^2/3)^3\big)=-108\cdot(x^6-x^6)=0$$ Even if it's correct, can you ...
3
votes
1answer
91 views

Restrictions on the coefficients that allow a polynomial in a field of characteristic 0 to be solvable by radicals and the associated formula.

We know that a general polynomial $p(x) \in \mathcal{F} \left[ x \right]$, $\deg{ p } = n$, (char(${\mathcal{F}}) = 0$) is not solvable by radicals if $n \geq 5$. However, I was wondering what ...
13
votes
2answers
702 views

Can we introduce new operations that make quintics solvable?

I have heard from various sources that the typical arithmetic operations (addition, subtraction, multiplication, division, rational exponentiation) are not sufficient to express in general the roots ...
9
votes
2answers
272 views

Are rational numbers + numbers constructed with a root = algebraic numbers?

I'm a math newbie, so an intuitive explanation is the most helpful for me, but don't pull your punches with the formulas, if you feel like it. We can construct the rational numbers using the division ...
0
votes
0answers
31 views

Finding roots of a fractional exponential equation.

If we consider a polynomial equation its easy to find the number of roots associated with the expression by applying Descartes Rule. This method, however, doesn't work with non integer exponents. ...
0
votes
2answers
58 views

n'th and (n-1)'th derivative of $\frac{1}{x^n - a}$

I have a function of the form $f(x) = \frac{1}{x^n - a}$, and I need to programmatically find the n'th and (n-1)'th derivative of the function. Since the function has this specific form and that the ...
1
vote
1answer
29 views

Effect on roots of function on taking the derivative of the function

Suppose there is a function $$f(x)=(x-1)^{15}(x-2)^{20}(x-3)^{25}(x-4)^{30}$$ As we take the derivatives of the function, what will happen to the number of real roots and the number of distinct real ...
1
vote
0answers
25 views

Roots and Weights

I use a Mathematica package to compute roots and weights (and other things) but the package gives me only the expression of the roots in $\omega$-basis (basis of fundamental weights) and in the ...
3
votes
2answers
56 views

Solutions for quartic

Suppose I have an equation in the form $(x-a)^4 + (x-b)^4 = c$. What is a clever way to find all four solutions? I have tried expanding and then used long division. However, I believe a better way is ...
0
votes
4answers
83 views

Why all such polynomials have $-1$ as a root?

Why all polynomials of this form have $-1$ as a root? $ x^5+x^4+x^3+x^2+x+1 $ and similar polynomials like $ x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1$