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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4
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1answer
73 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
votes
2answers
97 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
vote
0answers
64 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
votes
0answers
52 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
votes
1answer
87 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
votes
2answers
79 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
vote
1answer
53 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
vote
2answers
55 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 $t$.
1
vote
0answers
89 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
66 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
votes
1answer
77 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
42 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 = ...
17
votes
6answers
2k 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
vote
1answer
57 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
votes
2answers
108 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
votes
3answers
48 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
votes
0answers
41 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
votes
0answers
39 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
votes
2answers
62 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
vote
1answer
82 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
18 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
votes
0answers
55 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
votes
1answer
102 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
113 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.
6
votes
3answers
129 views

Nature of roots of quartic equation [closed]

Prove that all roots of a$x^4$ + b$x^3$ + $x^2$ + $x$ + 1 = 0 cannot be real. a,b $\in$ $\Re$. a $\neq$ 0.
0
votes
2answers
137 views
3
votes
3answers
161 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 ...
3
votes
3answers
2k 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 $f(x) = x^7+x^5+x^3+1$ ...
9
votes
3answers
284 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 ...
0
votes
2answers
262 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: ...
3
votes
1answer
110 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?
1
vote
0answers
56 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
vote
1answer
41 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
vote
2answers
62 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
vote
3answers
517 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
67 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
32 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 ...
0
votes
0answers
34 views

Formula for roots on an arbitrary polynomial

This question is related to a previous question I have posted: Solution for a Mixture of Two Exponential Equations I have reduced my problem to the following equation: $\left( ...
3
votes
1answer
124 views

Finding roots of a function in an interval

Does the equation $x^3-12x+2=0$ have three solutions in the interval $[-4,4]$? We know that this is a continuous function because it's a polynomial, and so we can use the Intermediate Value ...
0
votes
3answers
64 views

Find the max value of a function at a given interval

Trying to determine local max for a function at interval $[-4, 6]$. $$f(x)= x^3 -3x^2-24x + 7$$ Is the proper next step to take the derivative of $f(x)$ and find the roots, set roots = to zero?
1
vote
1answer
57 views

Fields of polynomials . Proving that a belongs to k as a root

if $f(x)\in k[x]$, where $k$ is a field, then $a\in k$ is a root of $f(x)$ iff $x-a$ divides $f(x)$ in $k[x]$. My result ... If $a$ is a root of $f(x)=q(x)(x-q)$ and if we let $f(x)=q(x)(x-a)$,then ...
1
vote
1answer
124 views

Is my simple (in my opinion) way of solving cubic equations correct?

I've been analyzing ways of solving cubic equations and I've come up with this one. I've tried to make it as simple as possible. So I'll show you a way of solving cubic equations when none of the ...
0
votes
0answers
29 views

Conversion of roots of a polynomial

I'm wondering, given a polynomial $P(x)$ with roots $r_i (1\le i\le n)$, how to determine the polynomial $Q(x)$ such that its roots are $r'_i=f(r_i)$. For example, if $P(x)=x^2-x-6=(x-3)(x+2)$ and ...
3
votes
1answer
1k views

Find all real zeros of $f(x)=2x^3+10x^2+5x-12$

Hey guys I'm having a little trouble with one problem: Find all real zeros of $$f(x)=2x^3+10x^2+5x-12.$$ I got $x=-4,(2x^2+2x-3)$. I'm just having trouble using the quadratic formula to get ...
2
votes
0answers
41 views

Is there a known algorithm for approximating all the real and imaginary zeros of any well behaved equation of a single variable?

Does there currently exist a general algorithm (or set of algorithms used together) that will approximate all the zeros of any well behaved non-differential equation of a single variable which has a ...
1
vote
2answers
105 views

The largest root of $-3x^3+24x^2+6x-9=0$

Since the polynomial has three irrational roots, I don't know how to solve the equation with familiar ways to solve the similar question. Could anyone answer the question?
0
votes
0answers
379 views

The Conjugate Roots Theorem for Irrational Roots

The Conjugate Roots Theorem for Irrational Roots states that for a polynomial $f(x)$ with integer coefficients, if a root of the equation $f(x) = 0$ is expressed as $a+\alpha$, where $a\in\mathbb{Q}$ ...
1
vote
0answers
161 views

Third degree polynomial with integer coefficient and three irrational roots

There are some polynomial with the above characteristic, and real roots of such polynomials cannot be found using rational number theorem and irrational conjugate theorem. The example of such function ...
4
votes
1answer
139 views

Roots of some modified Bernoulli polynomials

Update The polynomials are generated as follows: Where $B_n(x) = \sum_{k=0}^n {n \choose k} b_{n-k} x^k$ is used to generate standard Bernoulli polynomials, top plot is generated as follows: ...
0
votes
5answers
320 views

Why doesn't $1/x=0$ have any solution?

Just out for curiosity ! Why $1/x=0$ doesn't have any solution? Or is it that the solution takes you to $1=0$ situation which would nullify mathematical principle that we stood for years Educate ...