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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1answer
85 views

Find the root of the polynomial?

Consider the root of the polynomial $p(x) = x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots+a_1x -1$. Suppose that $p(x)$ has no roots in the open unit disc in a complex plane and $p(-1)=0$. Show that ...
4
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1answer
59 views

About the zeros of $f_n(z)=\sum_{k=1}^n k^{-z}$.

Let $z$ be a complex number. Consider $f_n(z)=\sum_{k=1}^n k^{-z}$. Now I wonder : Are there infinitely many positive integer $n$ such that there exists a $z$ with $f_n(z)=0$ and $Re(z)>1$ ? I ...
1
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1answer
44 views

Repeated Eigenvalues in Systems of ODEs

Question is to find the general solution of the given system of equations below. $$ x' =\left(\begin{array}{rr}\frac{-3}{2} & \frac{-1}{4} \\ 1 & \frac{-1}{2}\end{array}\right)x $$ My ...
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1answer
109 views

Solve : $x^4 + 6x^3 -3x^2 + 2 = 0$

$x^4 + 6x^3 -3x^2 + 2 = 0$ To find the zeros, I tried this by Ferrari's method but got stuck where a value of 'lambda' has to be obtained.
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1answer
154 views

Prove $\sum_{i=0}^n \binom{n}{i}^2x^{n-i} = 0$ has $n$ negative roots

Let's $n \in \mathbb{Z^+}$, how to $\text{prove}|\text{disprove}$ that: the equation $\boxed{\sum_{i=0}^n \binom{n}{i}^2x^{n-i} = 0}$ has exactly $n$ distinct negative roots. My friend get ...
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2answers
67 views

Given $f \in \mathbb{Q}[x]$ irreducible. How many and which roots of $f$ are contained in $\mathbb{Q}[x]/(f)$?

It is a fact that struggle me for a while. When working with irreducible polynomial over $\mathbb{Q}$ it is natural to build the extension ${\mathbb{Q}[x]}/{(f)} $ in which "lives " one root of the ...
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2answers
132 views

Existence of holomorphic function with a sequence of zeros in the unit disc

The question is : Prove that there exists a holomorphic function $f$ on the open unit disc $\{z \in \mathbb{C} : |z| <1\}$ with the properties that $f(0) = 0$ and $f(1-1/n)=1$ for every integer $n$ ...
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2answers
202 views

A 3rd degree polynomial $P(x)$ has three unequal real roots. What is the least possible # of unequal real roots for $P(x^2)$

I got that if P(x) is a 3rd degree polynomial then P($x^2$) must be a 6th degree polynomial. I don't know how to proceed from here.
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4answers
132 views

Find the solution of the equation

Find all real solutions of this equation : $$x=\sqrt{2+\sqrt{2-\sqrt{2+x}}}$$
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1answer
65 views

Solution of equations of the form: $a^x+b^x+c=0$

Is it possible to solve equations of the form: $a^x+b^x+c=0,\;abc\neq0$ with analytical methods; if so, how is this done?
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1answer
213 views

zeros of a function holomorphic in the closed unit disc

Let $f$ be a holomorphic function in a neighborhood of the closed unit disc $\{z \in \mathbb{C} : |z| \leq 1\}$, and suppose that $\Re{(\bar{z}f(z))} > 0 $ when $|z| = 1$. Prove that $f$ has ...
2
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3answers
204 views

How bad, really, is the bisection method?

We know that the bisection method for root finding is slow (linear convergence), but has the advantage of always working for a continuous function, if we start with a interval which brackets the root. ...
1
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1answer
90 views

How to calculate the integral of a function with a root?

I have to solve this integral: $\int\limits_{-1}^1(3x^3-5x^2+12x-9)~dx$ I used Grapher (a nifty program that comes with Mac OS X) to display the curve of $f(x)=3x^3-5x^2+12x-9$ and it obviously has ...
4
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1answer
430 views

Roots of $f(x)=\sin(x)-ax$

How many roots are there of the function $f(x)=\sin(x)-ax$, where $a$ is a positive number? Clearly for all $a$, $x=0$ is a root; if $a>1$ that is the only root. The roots will also be symmetric ...
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5answers
219 views

Analytical solution to $a^x+b^x=x$

Maybe stupid question, but I am wondering. Is there an analytical solution to equation $$a^x+b^x=x$$ for general $a$, $b$. How should I tackle this problem, if I want to find at least one $x$. ...
4
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0answers
136 views

solution set in $\mathbb{C}$ of $ z^{\frac1{z}}=\left(\frac1{z}\right)^z$

If $z \in \mathbb{C}$ what can be said about the solution set of: $$ z^{\frac1{z}}=\left(\frac1{z}\right)^z $$ aside from the fact that it contains the fourth roots of unity? I will add as a footnote ...
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1answer
131 views

Algebraically find roots of a function composed of linear equations and trigonometric functions

I have the following equation of $t$: $\text{C0}+(\text{C1}+\text{C2} t) \cos (\text{C4} t)+\sin (\text{C4} t) (\text{C7}+\text{C8} t)+\text{C5} \cos (\text{C6} t)+\text{C9} \sin (\text{C6} t)=0$ ...
3
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2answers
92 views

What's the non-trivial root of $\lim \limits_{n\to \infty}\left(\sum_{k=0}^n x^{2^k}\right)^n$?

$$ \lim_{n\to \infty}\left(\sum_{k=0}^n x^{2^k}\right)^n=0 $$ always seems to have two real solutions. One trivial $x_0=0$ and another around $x_1=-0.65862...$ (see W|A @ $n=13$). Where does this ...
2
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2answers
35 views

Polynomial With Imaginary Roots

Working on question 1 here http://www.sosmath.com/cyberexam/precalc/EA2002/EA2002.html Find a polynomial with integer coefficients that has the following zeros: ...
4
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2answers
232 views

Use $\alpha, \beta, \gamma $ roots of a polynomial to construct another polynomial [duplicate]

Let $\alpha, \beta, \gamma $ be roots $\in \mathbb{C}$ of $x^3-3x+1$. Determinate a monic polynomial, degree $3$, witch roots are $1- \alpha^{-1},1-\beta^{-1},1-\gamma^{-1}$ The catch is that i can't ...
4
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1answer
110 views

Number of zeros equal number of linearly independent analytic functions

I'm trying to read this paper and I'm stuck on a particular point. The authors are constructing an analytic function $f(z)$ which have to satisfy the following boundary conditions: ...
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1answer
75 views

how to prove roots quadratics

the quadratic equation $3(k+2)x^2+(k+5)x+k=0$ has real roots show $(k-1)(11k+25) \geq 0 $ If $\Delta$ greater than $0$ it has real roots so, $$\Delta = (k+5)^2 - 4 \cdot (3(k+2))\cdot k$$ ...
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1answer
177 views

If f is n-times differentiable, and $f^n$ is never 0, then f has at most n zeros in R

Let $n \ge 0$, let $f:\mathbb{R} \rightarrow \mathbb{R}$ be n-times diff erentiable on $\mathbb{R}$, and assume that $f^{(n)}(x) \neq 0$ for all $x \in \mathbb{R}$. Show that $f$ has at most $n$ zeros ...
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2answers
149 views

Polynomial divisibility

Given $p(x) \in \mathbb Q[x] $ an irreducible polynomial, and $\alpha \in\mathbb C $ root of $p(x)$. Prove that if $q(x) \in \mathbb Q[x]$ it's a polynomial, such $q(\alpha) = 0$ then $p(x) \mid ...
4
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2answers
86 views

Solution of $Ax^5+Bx^3=C$

I have to find the positive solution of the type $Ax^5+Bx^3=C (A,B,C>0)$. It is well known that a polynomial of degree greater than $4$ does not admit an expression for the roots but I hope :D In ...
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2answers
792 views

Use a given zero to write P(x) as a product of linear and irreducible quadratic factors

The polynomial in question is: $x^4 - 8x^3 - 19x^2 + 288x - 612$ and the zero is $4 - i$. What I don't understand is how to go from the given zero to factorizing, especially as it's imaginary. ...
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0answers
68 views

How to prove that there are $O(T\ln T)$ zeros in the critical strip of the Riemann zeta function?

Define $F(T)$ as the number of solutions to $\zeta(a+ ti) =0$ for $0\le t\le T$ and $0<a<1$. How to show that $F(T)= O(T\ln T)$? For clarity, $\zeta$ is the Riemann zeta function, $i$ is the ...
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0answers
65 views

How does the polynomial transformation $P(x) \mapsto P(x) + c$ alter the roots of that polynomial? Specifics inside.

Consider a real quadratic polynomial $Q_k(x) = (x-\nu)(x-\omega_k) - g_k^2$. I can interpret $Q_k(x)$ as a translation of the polynomial $$ (x-\nu)(x-\omega_k) = ...
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1answer
318 views

Prove there are 3 real roots to this equation using Rolle's Theorem

I need to prove there are $3$ real solutions to $x^5 - 4x + 2 = 0$. I know $f(-2)$ is negative, $f(0)$ is positive, $f(1)$ is negative, $f(2)$ is positive so that by IVT there are at least $3$ roots. ...
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2answers
93 views

Show these approximations of $\cos$, $\sin$ and $\tan$ are exact.

A while back I was looking for an approximation to $\cos(x)$ and I constructed a polynomial with zeros in the same places as the first few zeros of $cos(x)$: $$c_n(x) = \frac{\prod_{i=1}^n ...
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1answer
73 views

Let $p$ be a prime in $\mathbb{Z}$, find all roots of $x^{p-1}-1$ in $\mathbb{Z}_p$.

Let $p$ be a prime in $\mathbb{Z}$. Find all roots of $x^{p-1}-1$ in $\mathbb{Z}_p$. Attempt at Solution I have to solve $x^{p-1}-1=0(\text{mod }p)$ for $x\in\mathbb{Z}_p$. This becomes ...
1
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2answers
68 views

root of exponantial equation

How to find the solution/root of following equation? $$\sum_{n=1}^N\big(1-e^{-q(n)t}\big) = C$$ where $C$ is constant and $q(n)$ is given, we need to solve the equation for $t$.
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0answers
112 views

Number of roots of a polynomial

I would like to know if anything can be said about the number of roots of a polynomial whose coefficients depend on the $x$, particularly, $$x^2(f(x))^2-2xf(x)+g(x)=0$$ We further know that $f(x)$ ...
0
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1answer
197 views

Find the minumum using Newton-Raphson

I have the following function: $f(x) = 100(x_2 - x_1^2)^2 + (1-x_1)^2$ I have to find the minimum of this function using the Newton Raphson method. The point where I have to start is $x = [1.2$, ...
2
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1answer
45 views

Prove that $x_1^n+x_2^n$ is an integer and is not divisible by $5$

If $x_1$ and $x_2$ are the roots of the polynomial $x^2-6x+1$ then , for every non-negative integer, prove that $x_1^n+x_2^n$ is an integer and is not divisible by $5$ . My trying: $ x_1 = ...
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6answers
4k views

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

I encountered the following problem in a high-school math text, which I wasn't able to solve using factorization/factor theorem: Solve $x^3+3x^2+1=0$ Am I missing something here, or is indeed a more ...
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1answer
62 views

Prove $f(x)=9x^2-5y^2-34$ has no integral roots

Prove $f(x)=9x^2-5y^2-34$ has no integral roots. I have tried working this mod 2, 3, 4, 5, and 17, and some random others, to no avail. It is for a graduate course, so I am thinking maybe I tried to ...
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2answers
168 views

Roots of $e^z=1+z$ on complex plane

What are the roots in the complex plane of $e^z=1+z$? Clearly $z=0$ is one root. On the real line, we can show that $e^x>1+x$ for all $x\neq 0$. But what about the rest of the complex plane?
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3answers
74 views

$\left(\frac1\alpha-\frac1\beta\right)^2$ for $p(x)=x^2+x-2$

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $p(x)=x^2+x-2$, then $\left(\frac1\alpha-\frac1\beta\right)^2 is:$ A) $\frac94$ B) $\frac{-9}4$ C) $\frac25$ D) $\frac{-2}5$ This ...
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0answers
49 views

Show equation has at most two solutions on (0,2*Pi)

$ \sin (\text{ha})\text{ = } \text{dec}'(\text{ha}) (\tan (\text{lat})-\cos (\text{ha}) \tan (\text{dec}(\text{ha}))) $ I want to show this equation has at most two solutions for 0 < ha ...
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0answers
137 views

How to find the number of zeros in the left half plane?

Given a rational function $P(s)/Q(s)$ with $deg(Q(s))\geq deg(P(s))$. How to show that $ Q(s)$ and $P(s)-Q(s)$ have same number of roots in the left half plane using Rouche's theorem? Instead of ...
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2answers
64 views

Find the $8^{\text{th}}$ root of $1$ in the form $x+iy$.

I have squared each side $3$ times (not sure on the correct word but made it so it's $1=(x+iy)^8$ and expanded, is this the answer or is there a step to simplify everything?
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1answer
139 views

Finding the solutions of $\cos (x) +x = a$

What is the approach to finding the solutions of the following function? I was not able to analytically resolve the solutions - but rather resorted to a graphical approach. $$\cos (x) + x = 1$$ or in ...
2
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1answer
23 views

Number of positive roots of sparse polynomial

When $a<b<c$ are three positive integers, let $$ P_{a,b,c}(x)=x^c-(x^a+x^b)+1 $$ and denote by $N(a,b,c)$ the number of positive real roots of $P_{a,b,c}$ (note that $1$ is always a root). ...
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1answer
82 views

If all convex combinations of $p(x)$ and $q(x)$ have real roots, then $p,q$ have a common interlacing poly

I heard this result in a talk the other day: Suppose $p$ and $q$ are polynomials. Suppose $p$ is a polynomial of degree $n$ and $q$ a polynomial of degree $n-1$. Call $q$ and interlacer of $p$ if the ...
0
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2answers
287 views

Graeffe's root finding method

What are the practical applications of Graeffe's root finding method?I searched a lot but couldn't find.I found that it is used in aerodynamics and electric circuit analysis.But don't know much about ...
2
votes
2answers
130 views

The polynomial $P(x)=x^4 -\sqrt{7} x^3 + 4x^2 - \sqrt{22} x+15$ has four different roots. Prove that not all zeros of polynomial $P(x)$ are real

This is from my real analysis class. I know how to show a function has exactly one root but im not sure how to go about this.
7
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3answers
300 views

All roots of the quartic equation $a x^4 + b x^3 + x^2 + x + 1 = 0$ cannot be real

Problem Prove that all roots of $a x^4 + b x^3 + x^2 + x + 1 = 0$ cannot be real. Here $a,b \in \mathbb R$, and $a \neq 0$. Source This is one of the previous year problem of Regional ...
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3answers
253 views
3
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3answers
214 views

How find this equation solution $2\sqrt[3]{2y-1}=y^3+1$

find this equation roots: $$2\sqrt[3]{2y-1}=y^3+1$$ My try: since $$8(2y-1)=(y^3+1)^3=y^9+1+3y^3(y^3+1)$$ then $$y^9+3y^6+3y^3-16y+9=0$$ Then I can't.Thank you someone can take hand find the ...