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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2
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0answers
35 views

How do i find formula for the recurrence relation :$x_{n+1}= x^2_{n}-x^2_{n-1}$ with :$x_{-1}=0,x_{0}=\frac{3}{4}$?

How do i find formula for the recurrence relation according to the below initial conditions: $x_{n+1}= x^2_{n}-x^2_{n-1}$ with :=$x_{-1}=0,x_{0}=\frac{3}{4}$ ? Note:$n=0,1,2\cdots$ Thank you for ...
1
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0answers
5 views

Question on real polynomial in projective space

Hi all I was given this question and desperately in need of help. I am given a homogeneous polynomial of degree 4 of two variables x and y, with real coefficients with 4 real distinct projective roots ...
1
vote
1answer
13 views

Constructing matrices with eigenvalues equal to roots of a given polynomial

Suppose we are given a polynomial e.g. $$x^4+Ax^3+Bx^2+Cx+D,\tag1$$ and we need to construct a matrix, whose eigenvalues would be equal to roots of this polynomial. One way, rather inelegant, is to ...
0
votes
4answers
50 views

Is there any notation for general $n$-th root $r$ such that $r^n=x$?

As we know that the notation for the $n$-th principal root is $\sqrt[n]{x}$ or $x^{1/n}$. But the principal root is not always the only possible root, e.g. for even $n$ and positive $x$ the principal ...
5
votes
1answer
159 views

$15a+6b+4c+8d=0$ implies $ax^3+bx^2+cx+d$ has a positive root

Let $a,b,c,d$ be real numbers such that $15a+6b+4c+8d=0$. Show that $f(x)=ax^3+bx^2+cx+d$ has a positive root. (Komal, Problem N. 170.) I want to try to use the intermediate value theorem, showing ...
1
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3answers
1k views

How to factorize $x^3 - 7x + 6$?

How do you factorize this polynomial: $${x^3 - 7x + 6}$$ Some online solver doesn't even work saying: using GCF method doesn't work, but sites like Mathway.com gave me the answer, is there a ...
-1
votes
1answer
30 views

Any shortcut method to compare the roots of two quadratic equations? [on hold]

The given equations are(for example) $81x^2-9x-2=0$ and $56y^2-13y-3=0$. How do i compare the roots of these equation without using the Quadratic formula? Any suggestions please? Thanks.
0
votes
4answers
46 views

Prove that $f(x)=m$ has three distinct real roots for $m\in(0,8)$

We have $f:\mathbb{R}\rightarrow\mathbb{R},f(x)=x^5-5x+4$ and we need to show that $\forall m\in(0,8)$, $f(x)=m$ has three distinct real roots. How can I prove it?
2
votes
1answer
41 views

Does $\zeta(s)^2 \pm \zeta(1-s)^2$ have roots at the $\rho$s?

Maybe a strange (or stupid) question, but does $$\zeta(s)^2 \pm \zeta(1-s)^2$$ also have roots equal to the non-trivial zeros ($\rho$) ? At first sight you would expect so, however when I tried to ...
0
votes
1answer
39 views

When solving trigonometric irrational equations does the condition of existence of the radicand under an even root matter?

Hi everyone I would like to ask a thing about the following equation: $$\cos(x) + \sqrt[4]{1 - \frac{4}{3}\cos(2x) - \sin^4(x)} = 0$$ It is trigonometric and irrational, the root's index is 4 (even ...
0
votes
2answers
302 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 ...
-4
votes
4answers
74 views

Suppose that $\alpha$ root of the equation [on hold]

Suppose that $\alpha$ root of this equation: $$x^4+x^2-1=0$$ Find the value of $$\alpha ^{6}+2\alpha ^{4}$$ "I want the way, not the roots of the equation." I tried, but I couldn't find any thing.
0
votes
0answers
40 views

How many roots are there? [on hold]

I tried to find out how many roots there is for this function: $$ xe^{\frac{2}{x}}=a \quad (a\in\mathbb{R}) $$ Can anyone help by solve it/give hints?
0
votes
1answer
15 views

Expansion and factorization to determine roots of equations

Let $(a,c)$ be the roots of the equation $x ^ 2 + ax - b = 0$. Let $(b,d)$ be the roots of the equation $x ^ 2 + cx + d = 0$. Find all the possible real values for $a, b, c, d$. NOTE: I have made ...
0
votes
3answers
34 views

Find the roots of equation involving $\arctan x$

I try to find the roots of the equation: $$y=x-2\arctan\left(x\right)$$ I know that one of them is $(0,0)$ but there are two others that should solve $$\dfrac{x}{2}=\arctan\left(x\right).$$ Is ...
-1
votes
1answer
47 views

Proof exist infinitely many $n$ such that $f_{n}(x)$ has two integers roots

The two integer sequence $\{a_{n}\},\{b_{n}\}$ such $$a_{n+1}=a_{n}+1,2b_{n+1}=a_{n}+2b_{n}$$ Define function $f_{n}(x)=x^2+a_{n}x+b_{n}$, if there exisit $k$ such $f_{k}(x)=0$ has two ...
0
votes
1answer
46 views

Why use methods as Newton, ridder or secant method for root finding? [closed]

Why use methods as Newton, ridder or secant method for root finding? I am bit confused for what reason someone would use these method to determine the root of a function, as it can easily be ...
0
votes
1answer
43 views

suppose n is a natural number , prove equation $x^n+nx-1=0$ exist an unique real positive root $x_n$

suppose n is a natural number prove : equation $x^n+nx-1=0$ exist an unique real positive root $x_n$ ; and when $a>1$,$\sum_{n=1}^{\inf}x^a_n$ converges.
1
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2answers
37 views

What type of equation is this? How to solve it?

$$m^4+a^4=0$$ , the answer I obtained is $$0+i1,0-i1$$ but the answer is given as a/sqrt(2)-a/sqrt(2),a/sqrt(2)+a/sqrt(2)
1
vote
1answer
35 views

How do I calculate the values of $\zeta(0.5+ie^x)$ for large $x$ ?

In wolfram alpha the values of $$\zeta(0.5+ie^x)$$ closed to zero then How do I know the real values of $\zeta(0.5+ie^x)$ for large real number $x$ ? Thank you for any help
10
votes
3answers
214 views

Solve this tough fifth degree equation.

$$x^5+x^4-12x^3-21x^2+x+5=0$$ I think it can be solved by trigonometric ways but how?
1
vote
2answers
51 views

Why $ax^2+bx+c = a(x-r)(x-s)$, where $r$, $s$ are the roots?

When I was reading about math, I came across the following - Suppose the roots of the quadratic $ax^2+bx+c$ are $r$ and $s$. Then $ax^2+bx+c = a(x-r)(x-s)$ for all values of $x$. Is there ...
1
vote
0answers
25 views

Zeros of derivative of composition of polynomials

Let $f(x),g(x)$ be polynomials such that their derivatives $f'(x),g'(x)$ have $n$ and $m$ real roots. What is the possible minimal/maximal numbers of real roots for the polynomial $(f(g(x))'$? My ...
2
votes
3answers
36 views

Determine roots of a polynomial with variable exponent

I need to know the nature of the roots of the equation $$ x(x+a)^b -1 = 0 $$ when changing a and b, where $ a,b $ are natural numbers, I've looked around on the web but I was unable to find how to do ...
2
votes
2answers
39 views

Roots of a sixth degree polynomial

I have this question: The polynomial $f(x) = x^6 - ax^4 - ax^2 +1 $ has $(x-p)$ as a factor, where $a,p$ are real numbers. Show that $a = p^2 + p^{-2} - 1$ Here's my attempt: Let $u = x^2 ...
2
votes
1answer
39 views

Confusion about exponents like ${x^m}^{(1/n)}$.

I've been reading this post. It says that $\sqrt[m]{x^n} = x^{n\frac 1m}=x^{\frac mn}=x$ if $m=n$. Let's take $x=-2$, and $m=n=2$. Now we have, $\sqrt[2]{(-2)^2}=\sqrt[2]{4}=2$ But according to that ...
3
votes
1answer
62 views

The Passare-Tsikh solution to the principal quintic

The Bring-Jerrard quintic, $$x^5+x+t=0$$ can be solved as, $$x = -\sum_{k=0}^\infty(-1)^k\frac{(5k)!}{k!(4k+1)!}\;t^{4k+1}\tag1$$ when, $$|t|<\frac{4}{5^{5/4}}\approx 0.53\dots$$ This paper ...
0
votes
0answers
46 views

Find the number of zeroes of a function

let $f(z)=(z^2+9)(z^2+1)(z^2-1)+z^5(z^2+4)$. How many zeroes does $f$ has in $\{z|\operatorname{Re}{z}<0\}$. I want to use the argument principle, but the integral is too long. I think I need to ...
-1
votes
1answer
37 views

not easily factored quadratic expression how to find its roots [closed]

Could you please help me and explain this issue: If a quadratic equation is not easily factored then its roots can be found using quadratic formula: If $ax^2+bx+c=0$ ($a\ne0$), then the roots are ...
0
votes
3answers
101 views

Absolute value of cubic polynomial roots lower than 1

Assume we have a cubic polynomial $ x^3 +bx^2+xc+d=0 $, with $b,c,d$ real numbers. Let $x_1, x_2, x_3 $ be the roots, either real or complex. What is the relation of the coefficients $b,c$ and $d$ ...
1
vote
6answers
248 views

Find $(a,b)$ such that in $x^2+ax+b$, whenever $v$ is a root, then $v^2 - 2$ is also a root

Find $(a,b)$ such that in $x^2+ax+b$, whenever $v$ is a root, then $v^2 - 2$ is also a root $a,b$ are real numbers. Roots may or may not be real. In this question, the aim is to find values of and b ...
4
votes
1answer
186 views

Solve $2000x^6+100x^5+10x^3+x-2=0$

One of the roots of the equation $2000x^6+100x^5+10x^3+x-2=0$ is of the form $\frac{m+\sqrt{n}}r$, where $m$ is a non-zero integer and $n$ and $r$ are relatively prime integers.Then the value of ...
3
votes
2answers
377 views

Roots of a 6-degree polynomial [duplicate]

Find the roots of the equation $$2000x^6+100x^5+10x^3+x-2=0.$$ I am struggling finding a root using rational root theorem. Even if I get a root, I have to find all the roots. Please don't use ...
3
votes
3answers
1k views

$\alpha,\beta,\gamma$ are roots of cubic equation $x^3+4x-1=0$

If $\alpha,\beta,\gamma$ are the roots of the equation $x^3+4x-1=0$ and $\displaystyle \frac{1}{\alpha+1},\frac{1}{\beta+1},\frac{1}{\gamma+1}$ are the roots of the equation $\displaystyle ...
3
votes
8answers
114 views

Find the cubic equation of $x=\sqrt[3]{2-\sqrt{3}}+\sqrt[3]{2+\sqrt{3}}$

Find the cubic equation which has a root $$x=\sqrt[3]{2-\sqrt{3}}+\sqrt[3]{2+\sqrt{3}}$$ My attempt is ...
5
votes
1answer
293 views

To prove this complex polynomial has all zeros on unit circle

I'm trying to prove a self-inversive polynomial $P(z) = \sum\limits_{n=0}^{N-1}a_nz^n$ has all its roots on the unit circle. The coefficients are such that $ a_n = e^{j(n-\frac{N-1}{2})\pi u_0} - ...
0
votes
2answers
44 views

Finding the complex roots of an equation.

I feel ridiculous asking this, its something I should be able to do, however I shall ask anyway. I am doing a calculation that requires me to find the roots of the equation ...
0
votes
2answers
23 views

Improper Square Root Simplification

I'm trying to simplify a ratio to modify a vector by. Basically I want to find a constant such that the xy-components of two vectors are equal: http://math.stackexchange.com/a/1330263/194115 So I do ...
5
votes
3answers
678 views

Solution of a quartic equation.

Suppose that the equation $x^4-2x^3+4x^2+6x-21=0$ is known to have two roots that are equal in magnitude but opposite in sign. Solve the equation. This is what I have been thinking. Suppose ...
1
vote
3answers
60 views

Relation between the roots and the coefficients of a polynomial

I have studied that: For the polynomial $ax^3+bx^2+cx+d=0$, with roots $\alpha, \beta, \gamma$: We have: $$\begin{align} & \alpha + \beta + \gamma = -\frac ba \\ & \alpha\beta + \beta\gamma ...
11
votes
5answers
1k views

Finding cubic with golden ratio as root

I want to find a cubic such that it meets the following criteria: Has the golden ratio as its only real root Has integral coefficients Has a leading coefficient of $1$ and a final coefficient of ...
2
votes
1answer
30 views

Prove that the equation $az^3-z+b=e^{-z}(z+2)$ has two solutions in the right half-plane $\{z\in\mathbb{C}\,:\,\Re z>0\}$ when $a>0$ and $b>2$.

Prove that the equation $$ az^3-z+b=e^{-z}(z+2) $$ has two solutions in the right half-plane $\{z\in\mathbb{C}\,:\,\Re z>0\}$ when $a>0$ and $b>2$. This is an old qualifying exam ...
1
vote
2answers
32 views

How to approximately guess the roots of a function

My question is : How to approximately guess the root of a function... By root i mean is the starting point guess when used in case of Newton's method or any other root formulating methods. (Without ...
0
votes
2answers
23 views

Find all $a \in \Bbb {C}$ such that $F$ has at least one multiple root.

Let $F=X^{18}-8X^9+4A$. Find all $A \in \Bbb {C}$ such that $F$ has at least one multiple root. For each $A$ found determine how many different roots $F$ has and their multiplicity. My attempt: $F$ ...
1
vote
2answers
273 views

Find coefficient in quartic given product of roots

The product of two of the roots of $$x^4 -11x^3 + kx^2 + 269x - 2001=0$$ is $-69$. Find k. This is a question I have recently received, and I am required to take a test on related questions ...
3
votes
2answers
31k views

Quadratic equation - Alpha and Beta Roots

If α and β are the roots of the equation x² + 8x - 5 = 0, find the quadratic equation whose roots are α/β and β/α. My working out so far: I know that α+β = -8 and αβ = -5 (from the roots) and then i ...
0
votes
2answers
54 views

Paradox - minus one equals one using square roots [duplicate]

I was looking on Howard Eves's book "An Introduction to the History of Mathematics" and I stumbled upon a demonstration on how $-1 = 1$. The demonstration follows: $$ \sqrt{-1} = \sqrt{-1} $$ $$ ...
1
vote
1answer
58 views

Location of the roots of $f'$ (Laguerre's theorem)

Let $f \in \mathbb{R}[X]$ be a polynomial of degree $n$ having $n$ distinct roots $a_1,...,a_n$. Let $b_1<...<b_{n-1}$ be the roots of its derivative $f'$ (note that $b_i \in ]a_{i}, a_{i+1}[$ ...
1
vote
1answer
17 views

Infinite roots of a scalar function

I've been struggling with a problem for a while, I have to proove if the following proposition is true or false: Let $f:\mathbb{R^n}\to\mathbb{R}$ be a smooth funcion (i.e $f \in C¹$). Suppose that ...
1
vote
2answers
27 views

Finding Imaginary Values of a Degree 6 Polynomial

Here's the problem: $0 = x^6 - 65x^3 + 64$. I understand to insert "z" for $x^3$, which gets roots 4 and 1. That leaves you with a $4^{th}$ degree polynomial (which I have found). I know how to ...