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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52 views

Am I thinking about the Argument Principle in the correct way?

I know that, by the Argument Principle, the number of zeroes of a function f inside a simple closed contour $\gamma$ minus the number of poles inside $\gamma$ = the number of times that the image ...
0
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0answers
41 views

How to compute the argument of z= i(y^5+3y)+1,

I'm trying to find the number of zeroes of $z^5+3z+1$ in the first quadrant. I've already shown that there are no roots along the half lines -- the boundary -- of the first quadrant. Now I want to ...
4
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3answers
111 views

The polynomial $a z^n+z+1$ has at least one root in $|z| \leq 2$

I am trying to solve this problem, but I don't have any idea. The problem is: Prove that for arbitrary $a \in \mathbb{C}$ and $n \geqslant 2$, polynomial $P(z) = a z^n+z+1$ has at least one root ...
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0answers
47 views

Prove that a polynomial of this form has a real root of this form.

Prove or disprove that for positive coefficients: $$a_k>0$$ this polynomial: $$a_1x^0+a_2x^1+a_3x^2+a_4x^3+...+a_nx^{n-1}-x^n=0$$ has a real root $x$ that is the fraction from the following ...
2
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2answers
97 views

How many roots have modulus less than $1$?

If roots of the equation $$x^7 - 4x^3 + x + 1=0$$ are plotted on the Argand plane, how many of them have distance from the origin less than $1$? I found, by plotting the rough curve of $y=...
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1answer
42 views

Building a cubic function with integer coefficients and trigonometric roots

I want to find the answer to the following problem: Construct a cubic polynomial with integer coefficients, whose roots - $\cos{\frac{2 \pi}{7}}$, $\cos{\frac{4 \pi}{7}}$ and $\cos{\frac{6 \pi}{7}}$. ...
3
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1answer
91 views

Number of roots smaller than degree of polynomial

Let $R$ be a commutative ring and $f\in R[X]$ a polynomial with $f\neq 0$ and suppose $a_1,...,a_n\in R$ are roots of $f$ with $a_i-a_j\in R^*$ for all $i,j$ with $1\leq i<j\leq n$. How do I ...
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1answer
80 views

Complex Analysis Questions - $|z + 2| + |z - 2| = \sqrt{10}$

I'm just starting in a Complex Analysis course, and I am stuck on a couple questions. The questions are as follows: If $z = a + bi$ is a point on the curve $|z + 2| + |z - 2| = \sqrt{10},$ find ...
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1answer
25 views

Find the sum of roots of the polynomial

If $\alpha$ and $\beta$ are the zeros of a quadratic polynomial, what is the value of $\alpha - \beta$ ? I have to find the answer in terms of the coefficients of the polynomial and I have to use ...
2
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1answer
51 views

Integral eigenvalues of a matrix

Suppose $M$ is a $n \times n$ matrix such that, $$M=\left[\begin{array}{ccccccc} 0 & k_2&k_3&\cdots &k_{n-1}&k_n\\ k_{1} & 0 & k_3&\cdots &k_{n-...
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4answers
71 views

Finding root of $x^3-36x^2+405x-1458=0$

How can I solve the following equation : $$x^3-36x^2+405x-1458=0$$ I have tried as : $$x^3-36x^2+405x-1458=0$$ $$\Rightarrow x^2(x-36)+405(x-3.6)=0$$ How is to proceed ?
1
vote
1answer
102 views

Finding the root of the equation using Newton's Method

Find the Root of the equation: $x^3 = x^2 + x + 1$ in the interval [1,3] by Newton's method using double precision. Use $x_0 = 1.5$ and iterate 5 steps. Make a table that shows the number of correct ...
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1answer
24 views

Reducing a complex expression to a real expression

I need to find a real root of the cubic equation $$\frac{-x^3 + 3 x + 2}{4}=p$$ where $0<p<1$ I am only interested in the real root in the interval $[-1;1]$ When I try to solve this ...
0
votes
2answers
19 views

Interval in which roots lie given the interval for constant

The question says-if roots of $x^2 -2mx +m^2 -1$ lie in $(-2,4)$, Prove that $m$ lies in $(-1,3)$. This was pretty easy, but then I tried to work backwards. What if the question read-"If $m$ lies ...
-1
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1answer
54 views

Evaluating expressions involving roots of cubic equations [closed]

I'm having trouble understanding the concept of "roots"; could someone please solve this problem and explain the logic behind their method? If $\alpha$, $\beta$ and $\gamma$ are the roots of $x^3+2x^...
0
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1answer
104 views

Find the range of values of p for the equation to have real roots?

Question 7a for my homework said to show that $x=-1$ is a solution of $x^3+px^2+px+1=0$. I just did that by using synthetic division. Question 7b however says "hence find the range of values of $p$ ...
5
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1answer
57 views

Not a root of unity - how to prove?

How does one prove that a given complex number is not a root of unity for some positive integer power n? Say, I want to prove that there does not exist a positive integer $n$ such that $(2i)^n = 1$, I ...
3
votes
5answers
213 views

Real Roots and Differentiation

Prove that the equation $x^5 − 1102x^4 − 2015x = 0$ has at least three real roots. so do i sub in values of negative and positive values of x to show that there are at least three real roots? the ...
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1answer
62 views

Finding unknown variable in a polynomial

$n^2x^{2n+3}-25nx^{n+1}+150x^7$ This polynomial has $x^2-1$ as a factor for: 1) no values of n; 2) n=10 only 3) n=15 only 4) n=10 and n=15 only ^which one's correct? Could someone please tell me ...
4
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2answers
74 views

Condition for quartic polynomial coefficients given at least one real root

Find the minimum possible value of $a^2+b^2$ where $a$, $b$ are two real numbers such that the polynomial $$x^4+ax^3+bx^2+ax+1,$$ has at least one real root. My attempt: Let p be a real root. ...
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0answers
19 views

Number of root clusters

Is there any way to tell how many clusters there are with respect to all the roots of a polynomial? Specifically, I'm after the multiplicity of each root but since I would like to work in floating-...
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1answer
14 views

Implication of two statements

my question is with regards to two different problems (both containing a statement A and statement B) that are quite similar. The objective is to decide how the implication arrow is supposed to be ...
0
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1answer
78 views

Finding roots of polynomial using companion matrix

The standard method for finding roots of a polynomial is to form the companion matrix, balance it, then compute the eigenvalues by double shift QR algorithm. This method is used by Matlab ROOTS ...
3
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1answer
50 views

What is the solution sets for $(5x-2)^{x-5}=(5x-2)^{2x+1}$?

Here is my attempt: For every exponent equation like this, we can use 4 equations to find its solution sets: The solution sets of $$(a(x))^{f(x)}=(a(x))^{g(x)}$$ is satisfy: f(x) = g(x) a(x) = 1 a(...
3
votes
5answers
619 views

Solving Quartic Equations

Given the following quartic equation: $$x^4-2x^3-7x^2+8x+12=0$$ Could anyone give some techniques required to solve any quartic equation (apart from this one) if they exist?
5
votes
2answers
182 views

Proving a Polynomial Identity

Prove that $$\sum_{i=1}^{n} \dfrac{{r_{i}}^k P(x)}{P'(r_{i})(x-r_{i})} = x^k$$ where $P(x)$ is an $n$ degree polynomial having distinct roots $\{ r_{i} \}_{i=1}^{n}$ and $k$ is an ...
5
votes
0answers
126 views

Finding closed-form approximations of the solutions of $f(x,y)=0$

Consider $$f(x,y)=\sum_{i=1}^n\dfrac{\sin(\omega_ix)}{\sin(\omega_iy)}r_i$$ where $n,\omega_i,r_i>0$ are known parameters. Restrict to domains where $ \sin(\omega_iy)\neq 0$, and by symmetry $x,y&...
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0answers
14 views

how many linear inequalities on coefficients needed to locate complex root of polynomial in bounded domain

It is a remarkable fact that in $\mathbb R$, from just two (carefully chosen) linear inequalities on the coefficients of a (square-free, say) polynomial we may deduce that the polynomial has a root ...
0
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1answer
60 views

Why 3 root 2 is equal to 3 divided by root 2 in this quadratic equation

As part of my research into quadratics, I am trying to show algebraically the following equation : $${x^2-3\sqrt2 x + 4} = 0$$ alternatively shown as $x^2 - 3 \cdot 2^{\frac{1}{2}}x + 4 = 0$ Now, ...
2
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1answer
92 views

What are the conditions on $a$ such that the polynomial $x^4-2ax^2+x+a^2-a$ has four distinct real roots?

I was assisting with a local competition for high school students which was being run today, and one of the questions on the question paper was: What are the conditions on the real number $a$ such ...
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1answer
33 views

An Algebra Problem: [closed]

An Algebra Problem: If $a, b, c, \text{ and } d$ are the solutions of the equation $x^4-bx-3=0$, then an equation whose solutions are $$\frac{a+b+c}{d^2}, \frac{a+b+d}{c^2}, \frac{a+c+d}{b^2}, \...
0
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0answers
52 views

Mean Value Theorem Problem - finding $\xi$ given $f$ and $[a,b]$

For the function $f(x) = x^{6}+x^{4}-1$ and the interval $[0,1]$, I need to find the number $\xi$ that occurs in the Mean Value Theorem: $\displaystyle \frac{f(b)-f(a)}{(b-a)} = f^{\prime}(\xi)$. ...
4
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3answers
123 views

What is the largest possible value from $x_1^{2014}+x_2^{2014}$ from this following problem?

Given quadratic equation $x^2+px+q+1=0$ with two distinct roots $x_1$ and $x_2$. If $p$ and $p^2+q^2$ is prime numbers, what is the largest possible value from $x_1^{2014}+x_2^{2014}$? My ...
1
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0answers
53 views

Asymptotic estimates for roots of a transcendental equation

In computing asymptotic estimates for the eigenvalues of a spectral problem, I have been dealing with the a characteristic equation for the eigenvalue parameter $\lambda$ that I have couched into the ...
0
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0answers
23 views

Possible to find roots of a function involving heaviside?

Consider the function $f(t) = 3+t-(t-3)^2\theta(t-3)$, where $\theta(t)$ is the heaviside step function. Can i solve the equation $f(t) = 0$ somehow?
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2answers
117 views

Asymptotic expansion of the solution of $f_n(x)=x^5+nx-1=0$

Let's call $u_n$ the solution (it is unique) of $f_n(x)=x^5+nx-1=0$. I'd like to find an asymptotic expansion of $u_n$ of order two. I found $$u_n=\frac{1}{n}-\frac{1}{n^6}+o(\frac{1}{n^6})$$ The ...
2
votes
1answer
29 views

How many (positive real) zeroes can $f(x)=x^a-b(c^x)$ have?

Let $a,b,c$ be positive real numbers. Define $f:(0,\infty) \to \mathbb R$ by $f(x)=x^a-b(c^x)$. What is the maximal numbers of zeroes that $f$ can have ? My guess is that the answer is $2$ (note that ...
6
votes
1answer
137 views

What are the roots of the polynomial $x^{3}+3x-2\pi$ $?$

By using Descartes's sign rule , I can tell this polynomial $$x^{3}+3x-2\pi$$ has one real root. But I want to know what that root is and what the factorization of it is....
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1answer
138 views

Why is this approximation of polynomial root so accurate?

I have an engineering problem where I have to find the smallest positive real root of a polynomial in $x$: $$Ax^5+Bx^3 - C = 0$$ Instead of solving numerically, I want simple approximative formulas ("...
0
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0answers
28 views

Symbolic solution to a non-linear equation

Is it possible to solve following univarate ($y$) equation symbolically: $$ a_{2} \left(- \frac{b_{1} y + d_{1}}{2 a_{1}} - \frac{1}{2 a_{1}} \sqrt{- 4 a_{1} c_{1} y^{2} - 4 a_{1} e_{1} y - 4 a_{1} ...
6
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2answers
230 views

Find the limit analytically when the sine functions have square roots?

Find the limit analytically of the following: $\lim \limits_{x \to 0} \frac {\sin(\sqrt{2x})}{\sin(\sqrt{5x})} $ The closes thing we learned in class about this was that $\sin(x)$ over $x$ will ...
1
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0answers
27 views

Showing polynomials as products of roots

How do I show rigorously that any polynomial $a_nx^n+a_{n-1}x^{n-1}+...a_1x+a_0$ can be written as $a_n(x-b_1)(x-b_2)...(x-b_n)$ for real $a_i$ and real or complex $b_i$
2
votes
1answer
132 views

Polynomial tending to infinity

Take any polynomial $(x-a_1)(x-a_2)\ldots(x-a_n)$ with roots $a_1, a_2,\ldots,a_n$ where we order them so that $a_{i+1}>a_i$ is increasing so $a_n$ is the biggest root. It doesn't matter whether ...
1
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2answers
113 views

find the number of solutions to $p(z) = z^6 + 9z^4+z^3+2z+4$

Let $p(z) = z^6 + 9z^4+z^3+2z+4$ find then number of roots in each quadrant of the complex plane find in which quadrant exists a root which is inside the unit circle using the Argument priniciple ...
2
votes
1answer
44 views

Find a rational function of the roots of a given polynomial by transforming the polynomial

In the AOPS vol 2 problem solving book, it states that you can find the sum of the reciprocals of a polynomial by flipping the coefficients(first -> last, last -> first etc). The book summarized the ...
3
votes
0answers
132 views

What methods are known to visualize patterns in the set of real roots of quadratic equations?

I came across a previous awesome question about the visualization of the distribution of polynomial roots and tried to do a simpler version applied to the set of real roots of quadratic equations $ax^...
1
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2answers
68 views

Find $\prod\limits=(\alpha_1+1)(\alpha_2+1)…(\alpha_n+1)$ where $\alpha_i$ are complex roots of a complex polynomial

The complex roots of a complex polynomial $P_n(z)=z^n+a_{n-1}z^{n-1}+\cdots+a_1z+a_0$ are $\alpha_i$, $i=1,2,...,n$. Calculate the product $(\alpha_1+1)(\alpha_2+1)\cdots(\alpha_n+1)$ By the ...
3
votes
2answers
57 views

Polynomial with real roots

Consider the polynomial: $$f=X^4+4X^3+6X^2+aX+b$$ We know that $f$ has four real roots. Let $x_1,x_2,x_3,x_4$ be the roots of this polynomial. How can one compute $$x_1^{2015}+x_2^{2015}+x_3^{2015}+...
4
votes
2answers
201 views

Is $\sqrt[3]{-1}=-1$?

I observe that if we claim that $\sqrt[3]{-1}=-1$, we reach a contradiction. Let's, indeed, suppose that $\sqrt[3]{-1}=-1$. Then, since the properties of powers are preserved, we have: $$\sqrt[3]{-1}=...
0
votes
3answers
56 views

How to find the value of $(a+b+c)(a+b+d)(a+c+d)(b+c+d)$ from the following equation?

I have a question about polynomial. Given a polynomial: $$x^4-7x^3+3x^2-21x+1=0$$ Given too that the roots of this polynomial are $a, b, c,$ and $d$. Find the value of $(a+b+c)(a+b+d)(a+c+d)(b+c+...