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
72 views

How to solve cubic equations with given coefficients?

I have a large data set that requires a cubic equation to be solved for each point. There are too many points to use Goal Seek (numerical Excel method) on them all. For example: $$y=7\cdot ...
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3answers
56 views

approximate the root of perturbed polynomial

Approximate the root of $$f(x)=(x-1)(x-1)(x-3)(x-4)-10^{-6}x^6$$ near $r=4$. Do I have to use iterative method of finding the root, such as Bisection, Secant, etc? Is there other way?
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0answers
39 views

Analytic formulas for recursive sequences of polynomials

I have a sequence of polynomials $\{ f_{m} \}$ with $f_{m} \in \mathbb{Z}[x]$ that satisfies an order 2 linear recurrence relation. In particular, I have a polynomial $g \in \mathbb{Z}[x]$ such that ...
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0answers
46 views

Asymptotic behavior of zeros of a function

Let $f(x,m)=(2m-1)\Gamma(m)\,x^{-m}$ where $x>0$ and $\Gamma(z)$ denotes the Gamma function. Let $g(x,m)=f(x,m)+f(x,-m)$. I'm interested in the solution $m=m(x)>0$ of the equation $g(x,m)=0$ ...
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2answers
64 views

How does this factoring work?

$$ (z^2 - 2i ) = (z -1 -i)(z + 1 +i) $$ I see if you multiply out the right-hand side, you obtain the left-hand side, but how does one know to factor like that or this? $$ (z^2 − 3iz − 3 + i) = (z − ...
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0answers
23 views

Question on polynomials

What I know is that if the above said equation has three equal roots that meant that the above equation can be written as (x-q)^3*(x-w)*(x-e)... Now do we have to compare this equation with the ...
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0answers
24 views

What are the necessary and sufficient conditions to guaranty this property for sequences?

In this page we know that if a function $f:ℝ→ℝ$ or a sequence has infinitely many zeros, then it is not necessary to changes its sign infinitely many times. My question is: What are the necessary ...
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2answers
50 views

Prove that a polynomial of odd degree in $ \Bbb R[x]\\$ with no multiples roots must have an odd number of real roots.

The coefficients of the polynomial are in the ring of real numbers. Prove that a polynomial of odd degree in $ \Bbb R[x]\\$ with no multiple roots must have an off number of real roots. I hate to ask ...
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2answers
56 views

Existence of real roots for vector quadratic equations

I have a vector quadratic equation of the form $\boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{x} + \boldsymbol{x}^{T} \boldsymbol{b} + c = 0$ where $\boldsymbol{A}$ is symmetric and for my particular ...
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2answers
44 views

Finding all the roots of a function - Numerical methods

I'm trying to find all the roots of a function f(x) without algebraically solving the function. What I have done so far: I attempted to use a numerical method, more specifically the secant method to ...
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4answers
144 views

Are there real solutions to $x^y = y^x = 3$ where $y \neq x$?

I need to solve the following equation for (x,y) $$x^y = y^x = 3$$ Everytime I run a numerical method for this problem, I get $$ (x,y) = (1.82546...,1.82546..) $$ I expect there to be a solution ...
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4answers
36 views

Can Extraneous Roots be Introduced by Elimination?

Suppose you have two equtaions: $$2xy + y^2 = 0$$ $$x^2 + 2xy + 1 = 0$$ Subtracting the second from the first yields $y^2 - x^2 - 1 = 0$. Isolating y, we discover that $y = \pm\sqrt{x^2 + 1}$. ...
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1answer
32 views

Non-linear equations Numerically

I have no clue on this question. Please help me. Find an approximate to $80^{1/3}$ correct within $10^{-5}$ using the Bisection method and false position method and compare the rates of convergence. ...
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1answer
34 views

Solving Non-linear equation for Computational math

Question is, the curve $y = x^3 - 2x^2 + x -1$ intersects the parabola $y = 2x^2 + 3x + 1$. Locate the intersection then use both False Position Method and Secant Method. (Numerically) I really need ...
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0answers
47 views

Convergence results for Durand-Kerner method

I've been using the mpmath library's (in Python) implementation of the Durand-Kerner method to find the roots of some "not small" polynomials (this is the polyroots function). Typically my ...
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2answers
38 views

Solve equation with exponentials

I'm trying to obtain $x$ in the following equation: $$ 1= ae^{bx}+ce^{dx} $$ with a,b,c,d known. What I did was to take the ln of all the equation, so I have: $$ \log\frac{1/ac}{b+d} = x $$ ...
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1answer
38 views

sum and product of roots of polynomials: finding equations for roots

this question pertains to the sum and product of roots. $x^2 + 2x + 5 = 0$ has roots $\alpha$ and $\beta$, hence $\alpha + \beta = 2$ and $\alpha \beta = 5$. Find the equation which has roots ...
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0answers
36 views

How to prove $a_{1}\sqrt[b_{1}]{c_{1}}+a_{2}\sqrt[b_{2}]{c_{2}}+…+a_{n}\sqrt[b_{n}]{c_{n}}$ is irrational?

Let's define the number $$A=a_{1}\sqrt[b_{1}]{c_{1}}+a_{2}\sqrt[b_{2}]{c_{2}}+.....+a_{n}\sqrt[b_{n}]{c_{n}}$$ where $a_{1}, a_{2}, ..., a_{n}$ are positive integers and $b_{1}, b_{2}, ..., b_{n}, ...
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4answers
120 views

How to solve a 6th-degree polynomial

How would you solve a polynomial with the equation $$ax^6 + bx^5 + cx^4 + dx^3 + ex^2 + fx + g = 0, \text{ ?}$$ with all of the coefficients being positive except for the constant.
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1answer
44 views

Correct Terminology for “Smallest” Number which could be Negative

I'm writing a paper and I'm doubting myself with a piece of terminology. I am finding roots of an equation, say {x1, x2}, where it's ambiguous which is greatest from the perimeters involved. So ...
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1answer
72 views

Coefficents of cubic polynomial and its least root

Let $x^3-(m+n+1)x^2+(m+n-3+mn)x-(m-1)(n-1)=0$, be a cubic polynomial with positive roots, where $m,n \ge2$ are natural nos. For fixed $m+n$, say $15$, it turns out that least root of the polynomial ...
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60 views

if $x^k-x\in\mathbb{Z}$ and $x^l-x\in\mathbb{Z}$, then $x\in\mathbb{Z}$?

is it true that for any $k,l\in\{2,3,4,\dots\}$, $k\neq l$, if $x\in\mathbb{R}$ satisfies $x^k-x\in\mathbb{Z}$ and $x^l-x\in\mathbb{Z}$, then $x\in\mathbb{Z}$? This is a generalisation of if ...
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1answer
19 views

Bounding the Roots of a Complex-Valued Function

Roots: $Z_1$= $\frac{v(1+ \alpha)+ \sqrt{v^2(1+\alpha)^2 -4 \alpha}}{2}$ $Z_2$= $\frac{v(1+ \alpha)- \sqrt{v^2(1+\alpha)^2 -4 \alpha}}{2}$ It is clear that $|Z_2| \leq|Z_1|$ However I'm stuck on ...
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4answers
81 views

Show that $8x^4 −16x^3 +16x^2 −8x+k = 0$ has at least one non-real root for all real $k$. Find the sum of the non-real roots

Show that $8x^4 −16x^3 +16x^2 −8x+k = 0$ has at least one non-real root for all real $k$. Find the sum of the non-real roots. Since this polynomial looks so symmetric, I think factoring it might ...
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1answer
34 views

Finding the number of solutions of the equation $2x^5- 6x^3 + 2x = 4x^4 - 6x^2 +1$ in the interval $I = [-2, 2]$

I have to find the number of solutions for the following equation on the interval $I=[-2,2]$ $$2x^5- 6x^3 + 2x = 4x^4 - 6x^2 +1$$ Now I know I have to put them all on one side and then use the ...
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1answer
63 views

Show that no ring containing R can contain a root of g(x) = 3x +1

Show that if $R = \mathbb Z_6$ and $g(x) = 3x + 1 ∈ R[x]$, then $R[x]/(g(x)R[x])$ does not contain a root of $g(x)$. More generally, show that no ring containing $R$ can contain a root of $g(x)$. ...
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1answer
53 views

Does the resolvent cubic of the quartic equation always have at least 1 positive real root

I have written some code to solve for the roots of a 4th order polynomial, and in the process, I noticed that the resolvent cubic always has at least one positive real root. I can't find anywhere ...
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2answers
58 views

Roots of trigonometric function.

I have a function: $ f(x) = (ax+b) \cdot \sin(x) + (cx+d) \cdot \cos(x) + e$ for which I want to determine the roots. I know that for $ax \cdot \sin(x) + cx \cdot \cos(x)$ the roots are ...
0
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1answer
45 views

why are the Bisection and Newton Method for finding roots complementary to each other?

my lecture note states that the bisection and newton method for finding roots are most of the time complementary to each other but I can not figure out why. I have basic understanding of both of the ...
3
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1answer
103 views

Closed form for the sole positive root of the polynomial ${x^\alpha } + {x^{\alpha - 1}} + \cdots + {x^3} + {x^2} -p$, $p > 0$

For a study I'm making about the minimum and maximum radial values of bounded orbits in a central force system with general power law forces, I came across this special polynomial equation: ...
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4answers
56 views

finding real roots by way of complex

I was given $$x^4 + 1$$ and was told to find its real factors. I found the $((x^2 + i)((x^2 - i))$ complex factors but am lost as to how the problem should be approached. My teacher first found 4 ...
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2answers
23 views

Finding a root approach with a polynomial

So, i'm solving last's year's exams in Mathematical Analysis and i've found one interesting. It says: The equation $e^{-4x}=5x^2$ has one root close to (nearby) 0. By approaching $e^{-4x}$(close to ...
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0answers
23 views

Zeros of this function?

Let $$f(z)=\gamma + z^{\beta_2-\beta_1}$$ where $\gamma\in \mathbb{R}$, $\beta_1\in \mathbb{Z}$, $\beta_2 \in \mathbb{Z}$ and $\beta_2 > \beta_1$. The variable $z$ takes complex values. Is there a ...
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1answer
129 views

Roots of iterations of polynomials

Let $f \in \Bbb Q[X]$ a polynomial, and let denote by $f^n$ the composition $\underbrace{f \circ \cdots \circ f}_{n \text{ times }}$. Let $R(f^n) \subset \Bbb C$ the roots of $f^n$. I'm interested in ...
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2answers
42 views

Find multiple roots of a three-dim system

Consider the three equations $$ y-x^2=0,\quad z+xy=0,\quad -y-z+x^2-xy+y^2+z^2-x^4=0. $$ How can I find multiple roots of this? Is it allowed to reduce the system as far as possible and then to find ...
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1answer
58 views

Is it possible to solve the following equation without using the Rational Root Theorem?

Given $f(x)=x^4+2x^3+2x^2-2x-3$, where $x-1$ is a factor of $f(x)$, how is it possible to solve $f(x)$ without the Rational Root Theorem? Here's my progress: $$f(x)=x^4+2x^3+2x^2-2x-3$$ ...
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2answers
198 views

Is $x$ irrational when $2^{x}+3^{x}=6$?

Is $x$ rational or irrational when $2^{x}+3^{x}=6$. How to show that?
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2answers
182 views

How to find $x$ when $2^{x}+3^{x}=6$?

$$2^{x}+3^{x}=6$$ How to find the real number x? I mean it's approximately $1.19$ bur can we write $x$ as the form of $a, b, c$ when $a^{x}+b^{x}=c$ in general. Maybe an infinite sum?
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1answer
29 views

Rational Points on Fibonacci-like Sequence of Polynomials

Let $\{a_n\}$ be a sequence of polynomials in $\mathbb{Q}[x,y]$ with $a_0=0,a_1=1$, and $$a_n=xa_{n-1}+ya_{n-2}$$ The first few look like $$a_3:y+x^2$$ $$a_4:2xy+x^3=x(2y+x^2)$$ $$a_5:y^2+3x^2y+x^4$$ ...
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3answers
57 views

Find the two values of $k$ for which $2x^3-9x^2+12x-k$ has a double real root.

Find the two values of $k$ for which $2x^3-9x^2+12x-k$ has a double real root. I've found one method which is to equate $$2x^3-9x^2+12x-k=2(x-r)^2(x-c)$$ Expanding and equating coefficients I ...
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1answer
76 views

Solution to the equation $x^3-3=2\sqrt{x+2}$

Solve the equation $x^3-3=2\sqrt{x+2}$. I have tried to let $t=\sqrt{x+2}$ then we have $$\begin{cases} x^3-3&=2t \tag 1\\ t^2 &=x+2 \end{cases}$$ But I've stuck here... Any help ...
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2answers
41 views

All zeroes of monic cubic $x^3+ax^2+bx+c$ are negative reals and $a\lt3$. Range of $b+c$?

$a,b,c$ are real numbers. I have to find the range of values of $b+c$. So, I started off by assuming $\alpha , \beta , \gamma$ as the roots. This gives us $\alpha \beta \gamma = -c$ and ...
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1answer
48 views

Prove that the equation $sin(x) = ax + b$ has at least one real root

I came across a question earlier this day, that I did not manage to solve. I have been asked to prove that the equation $\sin(x) = ax + b$ has at least one real root, for all $a, b$, where: 1) $a$ is ...
2
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1answer
28 views

Roots of polynomials combined with Trigonometric Functions

If $$ f(x) = x^2 + ax + d \cos x $$, where $a$ is an integer and $d$ is a real number, what are all possible values of the tuple $(a,d)$ such that $f(x)$ and $f(f(x))$ have the same set of real roots? ...
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1answer
31 views

Simplifying Rational Expressions in a Finite Field Extension

In Dummit and Foote's textbook one of the exercises is: Let $\theta$ be a root of $x^3-2x-2$ over $\mathbb{Q}$. Compute $\frac{1+\theta}{1+\theta+\theta^2}$ in $\mathbb{Q}(\theta)$. My approach ...
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2answers
51 views

magnitude of vector in algebra

I am trying to solve the following equation for x, in plain algebra this was easy $ y = x - \frac{1}{ x} $ $ x^{2} - yx - 1 = 0 $ $ x = \frac{-y \pm \sqrt (y^{2} + 4)}{2} $ However, throwing ...
1
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3answers
70 views

Find $\alpha^3 + \beta^3$ which are roots of a quadratic equation.

I have a question. Given a quadratic polynomial, $ax^2 +bx+c$, and having roots $\alpha$ and $\beta$. Find $\alpha^3+\beta^3$. Also find $\frac1\alpha^3+\frac1\beta^3$ I don't know how to proceed. ...
0
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1answer
54 views

Find a root of f(x) = 0, arccos & arcsin

Can someone please help me with this question? Let $f(x) = 2\arccos(\frac{x}{2}) + 6\arcsin(\frac{3}{2x}) - 2 \pi$ Find a root of $f(x) = 0$, that is a point x where $f(x) = 0$.
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0answers
39 views

Companion matrix of bivariate polynomial

A polynomial in one variable can be expressed as a companion matrix, of which the eigenvalues are the roots of the polynomial and which can be found by using e.g. QR decomposition or power iteration. ...
4
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1answer
102 views

root pattern of second degree polynomial

I'm considering the following 2nd degree polynomial for the case where the roots are complex conjugate. $ P(z) = z^2 + (f^2 + f q -2)z + (1 - f q) = (z - z_1) (z - z^*_1) $ where f and q are real ...