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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3answers
50 views

Cubic equation (polynomial)

A cubic polynomial with real coefficients, $a x^3 + b x^2 + c x + d$, has either three real roots, or one real root and a pair of complex conjugate ones. If the latter happens, what is the explicit ...
0
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1answer
18 views

How to guess initial intervals for bisection method in order to reduce the no. of iterations?

SO, A function $f(x)$ is given to me and but the initial intervals are not given. I need to find the root of the equation using Bisection method. Sometimes when I randomly guess the initial interval ...
1
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0answers
35 views

Analytic closed-form solution

I have the following equation: $$\left(\frac{x}{\cosh(x)}\right)^2-x\tanh(x)+\ln\cosh(x)=0$$ and I would like to know if there is some analytic closed form solution. WA gives me two non-zero ...
0
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2answers
61 views

Calculating $a_1^4+a_2^4+a_3^4$ of the roots of a polynomial

We have a polynomial $f=X^3+19X^2+12X+3\in\mathbb{C}[X]$ with roots $a_1,a_2,a_3$. What is $a_1^4+a_2^4+a_3^4$? And how do I know that these roots are all different? Edit: How can I show that ...
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3answers
84 views

Functions Mapping Integers to Zero?

I am looking for functions such that: $z∈$ Z ⇔ $f(z)=0$ That is to say, functions that map from Z to the zero set. One example is $f(z)=\sin(πz)$. EDIT: To narrow the possible group of ...
2
votes
0answers
30 views

Finding the roots of a polynomial with limited information

Let $\ f(x) \in \mathbb{R}[x]$ be a 7-th degree polynomial, such that. $$ f(0)=0 \land f(i)=-3i $$ $$ f'(0)=0 \land f'(i)=-21 $$ Find all the complex roots of $ f(x)-3x^7$. Find all possible ...
3
votes
1answer
35 views

Let $a,b,c,d$ be distinct integers such that the equation $(x-a)(x-b)(x-c)(x-d)-9=0$ has an integer root $r$,then find the value of $a+b+c+d-4r.$

Let $a,b,c,d$ be distinct integers such that the equation $(x-a)(x-b)(x-c)(x-d)-9=0$ has an integer root $r$,then find the value of $a+b+c+d-4r.$ As $r$ is the integer root of the equation ...
2
votes
1answer
57 views

How do to find all the roots of $x^4-2yx^3+3y^3x-2y^4=0$?

The main question is how do we factorize $x^4-2yx^3+3y^3x-2y^4=0$ where $y$ is a parameter. I thought we could use Vietta's formula and solve the following system: $$\begin{cases} \begin{split} ...
1
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2answers
63 views

Find all complex roots of $T^4-{1/2}T^2-\sqrt{15}T+{69/16}$

I want to find all complex roots of $T^4-{1/2}T^2-\sqrt{15}T+{69/16}$. The only way I can think to do it is to find 1 complex root, $\alpha$, by inspection, so we can rearrange the polynomial to ...
2
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1answer
27 views

Solution to Sextic Polynomial with Two Real Roots

I have the polynomial $$f(x;a)=3ax^6+6x^5-9ax^4-4x^3+9ax^2+6x-3a$$ where the variable $a$ is a random variable from the uniform distribution in the range $[0,1)$. When I analyze this function using ...
1
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2answers
38 views

Polynomials and common roots

When dividing $f(x)$ by $g(x)$: $f(x)=g(x)Q(x)+R(x)$. How to find the quotient $Q(x)$ and the remainder $R(x)$? For example: $f(x)=\ 2x^4+13x^3+18x^2+x-4 \ $ , $g(x)=\ x^2+5x+2 \ $ At first $g(x)= ...
1
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0answers
89 views

Finding a solution to $\sum _{n=1}^{n=k} \frac{1}{n^x}+\sum _{n=1}^{n=k} \frac{1}{n^y}=0$

Scroll down to the update to see what I am meaning. The Mathematica program below finds a solution to the equation: $$\sum _{n=1}^5 \frac{1}{n^x}+\sum _{n=1}^5 \frac{1}{n^y}=0$$ My question is if you ...
0
votes
2answers
38 views

Show that sum of roots is rational.

If $f(X)$ in $\mathbb{Q}[X]$ is an irreducible polynomial polynomial of degree $n \geq 2,$ with roots $\alpha_1, \alpha_2,\ldots,\alpha_n$ in $\mathbb{C},$ show that $\displaystyle ...
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0answers
13 views

Count the number of positive real solutions of a polynomial of arbitrary degree?

Let $$P(x) = a_0 + a_1 x + \dots + a_n x^n$$ be a generic polynomial of order $n$. I need to know the number of positive real solutions to the equation $P(x)=0$. Specifically, I need to determine if ...
5
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0answers
59 views

Geometry of the zeros of a power series.

This is probably a basic question that is easily googlable, but it seems that I dont have the right keywords. So my question is, having some power series $$ f(z)=\sum_{k=0}^{\infty}C_{k}z^{k}, ...
-1
votes
1answer
40 views

Is there a general formula the solutions of a polynomial equation of the form $Ax^n + Bx^{n-1} + C = 0$?

Is there a general formula for the solutions of a polynomial equation of the form $$Ax^n + Bx^{n-1} + C = 0,$$ where $A$, $B$, $C$, and $n$ are constants?
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2answers
39 views

How would you find the roots of the following equation?

How would you find the roots of the following equation? $$x^{13} + 1 = 0.$$ I am absolutely new at this and have no clue.
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1answer
52 views

Can someone explain to me how to find zeros of a function? $10x^2+20x+19x+97^1$

I got this function right here and my teacher wants me to find all real number zeros $$10x^2+20x+19x+97^1.$$ I looked up this video on how to find it and they were using the $P/Q$ and I found ...
2
votes
0answers
39 views

Does $y=\lim_{n\to\infty}x\uparrow\uparrow n-f(x,n)$ have a root at $2$?

While messing around on Desmos calculator, I found an interesting factorial/tetration function, where we are using Nutch arrow notation. $$y=\lim_{n\to\infty}x\uparrow\uparrow ...
2
votes
2answers
27 views

prove that $x_{n+1}=\frac{x_n(x_n^2+15)}{3x_n^2+5}$ is cubic order of convergence near $x_0=\sqrt{5}$

To solve the equation $$x^2-5=0$$ There exitsts a iteration method $$x_{n+1}=\frac{x_n(x_n^2+15)}{3x_n^2+5}$$ I know that it is cubic convergence but I don't know how to prove it. I have tried the ...
3
votes
4answers
87 views

Minimal polynomial over $\mathbb Q(\sqrt{-2})$

Find the minimal polynomial for $\sqrt[3]{25} - \sqrt[3]{5} $ over $\mathbb Q$ and $\mathbb Q(\sqrt{-2})$. I have done the first part of this, over $ Q$, and have a polynomial. But I do not know ...
0
votes
1answer
29 views

Describe the rational solutions to this quartic.

Consider the quartic $-(a^5 + b^5)x^4 + (4a^5 - b^5)x^3 - (6a^5 + b^5)x^2 + (4a^5 + b^5)x - (a^5 + b^5) = 0$ where $a$ ...
2
votes
0answers
37 views

Find all such $a$ that $x+2\lvert x-3 \rvert = 7\lvert x-a \rvert + 3 \lvert x-a-4|$ has at least one root.

In the equation, $a$ is a parameter and $x$ is a variable: $$x+2\lvert x-3 \rvert = 7\lvert x-a \rvert + 3 \lvert x-a-4|.$$ I want to find all values of $a$ that make the equation have at least one ...
1
vote
1answer
40 views

Is this true of all linear recurrences?

Is it true that any linear recurrence $f_n$ can be written as: $$f_n = \sum_{i=1}^{k} \alpha_i r_i^n$$ where $f_n$ is a linear recurrence of degree $k$ and $r_i$ represents a root of the ...
1
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1answer
33 views

Monic polynomal $f \in \mathbb{Z}[X]$ such that $f(2)=13$ have at most three distinct roots in $\mathbb{Q}$?

how can I show that given a monic polynomial $f \in \mathbb{Z}[X]$ such that $f(2)=13$ have at most three distinct roots in $\mathbb{Q}$? I know that these roots has to be in $\mathbb{Z}$, but I ...
0
votes
1answer
49 views

Find $a$ such that $p(x)\geq 0$

The problem is: Let $p(x)=x^4-2x^3+ax^2-2x+1$, let a and x be real numbers, find a such that $p(x)\ge0$. My intent to solve it: We see that $(x^2-x+1)^2-3x^2+ax^2\ge0$ then ...
4
votes
4answers
736 views

How to find root of derivative of any polynomial/equation?

Let $$f(x) = (x-1)(x-2)(x-3)(x-4)(x-5),\quad -\infty< x<\infty.$$ The number of distinct roots of equation $$\frac{d}{dx}f(x) = 0$$ is exactly ? Source. The only method that I know ...
4
votes
2answers
53 views

Polynomial whose roots are not integers but almost so

Let $\varepsilon \in (0,\frac{1}{2})$. Say that a real number $x$ is an $\varepsilon$-pseudointeger if it is not an integer but at distance at most $\varepsilon$ from some integer (thus $|x-i|\leq ...
1
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0answers
54 views

How can I apply Rouche's Theorem here?

How many solutions lie in the left half-plane? $$f(z) = z^3+2z^2-z-2+e^z=0$$ My work so far: Factoring the polynomial, moving the exponential term over to the RHS, and taking the modulus of both ...
2
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1answer
122 views

how to find the roots of the following floor-equation: [closed]

How to find the roots of $$\lfloor x\rfloor+\lfloor 2x\rfloor+\lfloor 3x\rfloor=6$$
1
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3answers
28 views

Showing that $1$ is the only root in $f=x^5 -1 \in \mathbb{F}_p[x]$ if $p-1$ is not divisible by $5$

I am trying to show that if $p-1$ is not divisible by $5$ then $1$ is the only root in $f=x^5 -1 \in \mathbb{F}_p[x]$, where $p$ is a prime. I can see that $f = (x-1)(x^4 + x^3 + x^2 + x + 1)$ so it ...
3
votes
0answers
54 views

Why does Ridders' method work as well as it does?

I've just read section 9.2.1 in Numerical Recipes Ed. 3 (Press et al. 2007), which describes Ridders' method of root finding. I understand that allowing for some curvature of the function by ...
0
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0answers
27 views

Upper bounds for the number of roots of polynomials, over finite fields, lying in given extensions

Let $F$ be the finite field with $q$ elements, where $q$ is a power of a prime, and let $E$ be its degree $n \geq 2$ extension. Let $f(x) \in F[x]$ such that $f(E) = F$. Clearly the number of distinct ...
1
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3answers
39 views

If $x=2+i$, $gcd(a,b,c)=1$, and $ax^4+bx^3+cx^2+bx+a=0$, then what is $|c|$?

Suppose $$a(2+i)^4 + b(2+i)^3 + c(2+i)^2 + b(2+i) + a = 0,$$ where $a,b,c$ are integers whose greatest common divisor is $1$. Determine $|c|$. So I first simplified the exponents and ...
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2answers
39 views

Which equation has roots -2c, 2c, and 2?

This is a multiple choice question $$-4c^2 -2c=0$$ $$-4c^2+2c=0$$ $$x^3 - 2x^2-4x+8=0$$ $$x^3 - 2x^2-4c^2x +8c^2=0$$ I know roots mean solutions, so do I plug in the given roots and see if they ...
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1answer
39 views

How do I find the roots from this? [closed]

Problem: We are give that $2+7i$ is a root of $x^4 - 8x^3 + 73x^2 - 228x + 212=0$ I got stuck on this one: how do I find all the roots? $\left(i=\sqrt{-1}\right)$
4
votes
4answers
444 views

Prove the roots of a complex polynomial are imaginary

For an equation $f(z) = z^5 - 6z^4 + 15z^3 - 34z^2 +36z -48$ show that roots $f(z) = 0$ of this equation include 2 purely imaginary roots, and find them. I thought to substitute in $z=x+iy$ to show ...
2
votes
1answer
49 views

How many roots lie inside the disk |z|<1

How many roots of $f(z) = z^3 +cz^2 + z + 1$ lie inside $|z|<1$ if Part 1) $c>3$ Part 2) $3\ge c >2$ Part 3) $2\ge c >1$ Part 4) $c=1$ I am able to solve Parts (1) and (4) by Rouche's ...
1
vote
2answers
77 views

Is the complex square root of $z^2 = \pm z$?

Is $\sqrt{z^2} = \pm z$, for $z$ complex? I think it is, since either $-z$ or $+z$ satisfies the definition $\sqrt{z^2}= e^{\large \frac{1}{2}\log(z)^2}$ but I just wanted to make sure. It's a bit ...
5
votes
3answers
217 views

$\sqrt[31]{12} +\sqrt[12]{31}$ is irrational

Prove that $\sqrt[31]{12} +\sqrt[12]{31}$ is irrational. I would assume that $\sqrt[31]{12} +\sqrt[12]{31}$ is rational and try to find a contradiction. However, I don't know where to start. Can ...
1
vote
1answer
80 views

Prove that all the roots lie inside a little circle |z|=r, for n large enough

$$f(z)= 1 + \frac{1}{z} + \frac{1}{2!z^2} + ... + \frac{1}{n!z^n}$$ I can't seem to apply Rouche's Theorem here. Another idea would be to use perhaps Big Picard's theorem, since f(z) will converge ...
6
votes
3answers
92 views

How do I find the roots of this polynomial of degree $4$?

I am studying for finals and in the review packet is shown this problem: $$P(x)=2x^4 + 5x^3 + 5x^2 + 20x - 12$$ I don't know what to do, I have already tried looking in the textbook and Khan academy.
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2answers
28 views

If the zeros of a cubic form an arithmetic sequence, its point of inflection lies on the $x$-axis [closed]

If the zeros of the cubic $$y=x^3+ax^2+bx+c $$ form an arithmetic sequence, then show that the point of inflexion lies on the $x$-axis.
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3answers
65 views

Find the range of values for k such that ${kx^2 + 3x + 9k = 0}$ has real roots

I am asked the question: Find the range of values for ${k}$ such that ${kx^2 + 3x + 9k = 0}$ has real roots. So from my understanding, there are distinct roots if ${b^2 - 4ac\ge 0}$ My first step ...
4
votes
2answers
66 views

Geometric solution of quintic equations

There are many method of geometric solution of quadratic equations (for example Carlyle Circle). Does there exist similar method for the quintic equation?
0
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3answers
32 views

How to find no. of complex root of any polynomial?

For $f(x)=x^5+x^3-2x+1$, we know there should be $5$ roots . We can also find real roots of this polynomial. Can we conclude that: No of complex roots = No of total roots - No of real roots . Do ...
1
vote
1answer
131 views

Algebra: Rational roots of a polynomial of degree $4$

Consider the following polynomial with real coefficients: $x^4(t^5 - 1) - x^3(1+4t^5) + x^2(6t^5 -1) - x(1+4t^5) + (t^5 -1 ) = 0$, where $x$ and $t$ are both rational and $t$ is fixed. By simple ...
5
votes
2answers
131 views

The general solution of $x^a = a^x$ for real $a >0$

What are the roots of $$f(x) = x^a - a^x$$ for real $a > 0$? Case 1: For $0 < a < 1$ there is 1 solution, $x=a$. Case 2: For $1\le a < e$ there are 2 solutions: $x=a$ and $[x>a]$. ...
0
votes
1answer
35 views

Show that the roots of P' lie in the same half plane as the roots of P

The problem statement is: Part(a) Assume P(z) is a non-constant polynomial with all of its roots in some half plane H. Show that the derivative P'(z) must also have all of its roots in H. Part (b) ...
2
votes
3answers
76 views

Applying trigonometry in solving quintic polynomials?

So I came across the unsolvable quintic polynomial noticing that solutions can be found by connections with ellipses and such here. But more importantly, I was considering methods we use (or at least ...