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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1answer
89 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
341 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
216 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
132 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 ...
0
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1answer
121 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
90 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
188 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
109 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: ...
2
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1answer
67 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, $$\Delta = (k+5)^2 - 4 \cdot (3(k+2))\cdot k$$ ...
0
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1answer
159 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 ...
0
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2answers
138 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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1answer
76 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 ...
0
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2answers
546 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. ...
1
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0answers
67 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 ...
5
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0answers
60 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) = ...
0
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1answer
243 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. ...
2
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2answers
90 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 ...
1
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1answer
66 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
64 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 $t$.
2
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0answers
103 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
votes
1answer
102 views

roots of cubic - descartes and viete [closed]

Consider the equation $y^3 - 8y^2 - y + 8 = 0$. According to Descartes, how many roots does the equation have and how many are false roots? According to Viete, what is the product and what is the ...
0
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1answer
158 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
votes
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
3k 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 ...
1
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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 ...
3
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2answers
149 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?
2
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3answers
65 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 ...
0
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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 ...
0
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0answers
108 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 ...
0
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2answers
63 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?
1
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1answer
131 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
votes
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). ...
6
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0answers
69 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
242 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
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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.
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3answers
245 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
237 views
3
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3answers
210 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 ...
4
votes
3answers
7k views

Prove using Rolle's Theorem that an equation has exactly one real solution.

So the question is; Prove that the equation $x^7+x^5+x^3+1=0$ has exactly one real solution. You should use Rolle’s Theorem at some point in the proof. And I have, Since $f(x) = x^7+x^5+x^3+1$ ...
10
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3answers
321 views

How prove this Polynomial $g(x)=\sum_{i=1}^{n}a^m_{i}x^i$have only real roots?

Question 1: let Polynomial $f(x)=\displaystyle\sum_{i=0}^{3}a_{i}x^i,$ have three real numbers roots,where $a_{i}>0,i=1,2,3$. show that: $$g(x)=\sum_{i=0}^{3}a^m_{i}x^i$$ have only real ...
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2answers
511 views

Finding polynomal function with given zeros and one zero is a square root

I've been having trouble with this problem: Find a polynomial function of minimum degree with $-1$ and $1-\sqrt{3}$ as zeros. Function must have integer coefficients. When I tried it, I got this: ...
4
votes
1answer
119 views

Given the polynomial $(x-1)(x-8)(x-31)-1$, how do you conclude that its roots are irrational?

Example $(x-1)(x-8)(x-31)-1$. Just by looking at this polynomial how do you conclude that the roots are irrational?
0
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1answer
66 views

Prove that the roots of $2x^3 - x + 5 = 0$ are irrational

We want to prove that for the equation $2x^3 -x + 5 = 0$, any root must be irrational. How can this be done? Seems like plugging in $x = a/b$ doesn't really help at all.
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0answers
69 views

Zeta zeros by recurrence of zeta function, but this is useless isn't it?

One more useless question of mine can't do this site any harm. So here we go. The following Mathematica program converges to most of the riemann zeta zeros, by using an approximation as a starting ...
1
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1answer
74 views

Solving roots of a sum of sinusoids

Suppose I have a sinusoid with fundamental frequency $f_0$ and $N$ harmonics (all with distinct amplitudes $a_k$. Each harmonic also has it's own corresponding phase $\phi_k$ and offset $c_k$. $y(t) ...
1
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2answers
66 views

Solving for the zero of a multivariate

How does one go about solving the roots for the following equation $$x+y+z=xyz$$ There simply to many variables. Anyone have an idea ?
1
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3answers
2k views

Use the intermediate value theorem to show a function has a root [closed]

Let $f$ be a function defined on $(-\infty, 0)$ by $$f(x) = x^3 + \frac{4}{x^2} + 7 \ .$$ Use the Intermediate Value Theorem to show that the given function has at least one zero in the ...
2
votes
1answer
73 views

How to prove that this polynomial has no more than $s$ repeated roots

Let $\beta_{1},\beta_{2},\cdots,\beta_{s+1}\in R$,and $\alpha_{0},\alpha_{1},\cdots,\alpha_{s}$ be postive integers, with $\alpha_{0}>\alpha_{1}>\cdots>\alpha_{s}$. Show that: the ...
0
votes
1answer
37 views

$\frac{x^4 - x^3 + ax^2 + bx + c}{x^3 + 2x^2 - 3x + 1}$, remainder $3x^2 - 2x + 1$. Find $(a + b)c$.

Given the polynomials $P(x) = x^4 - x^3 + ax^2 + bx + c\\ Q(x) = x^3 + 2x^2 - 3x + 1\\ R(x) = 3x^2 - 2x + 1$ such that $P(x) = D(x)Q(x) + R(x)$, find $(a + b)c$. I would normally apply little ...