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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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
vote
2answers
37 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
vote
0answers
88 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 ...
0
votes
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
votes
0answers
58 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?
0
votes
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.
-2
votes
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
84 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
vote
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
735 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
vote
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
votes
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
vote
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
52 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
votes
0answers
26 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
vote
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 ...
0
votes
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 ...
-2
votes
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
441 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
215 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
79 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.
1
vote
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.
1
vote
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
64 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
125 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
75 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 ...
0
votes
3answers
67 views

Theorem 1.21 in Baby Rudin: How do we obtain $\sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a}$ for any $a>0$ and $m, n \in \mathbb{N}$?

Here is Theorem 1.21 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: For every real $x>0$ and every integer $n > 0$ there is one and only one positive real $y$ such ...
1
vote
1answer
28 views

Factorize polynomial in Matlab

I want to factor (break up) a polynomial $P(x)$ into first orders $(x + a_i)$ for real roots and second orders $(x^2 + b_ix + c_i)$ for complex roots. That is to say, $$P(x) = \prod (x + a_i) \prod ...
3
votes
1answer
34 views

Why are the factors of some solutions to a Pell equation also a solution?

I came across this observation while trying to answer this post using the Pell equation $x^2-2y^2=1$. Define, $$P(m) = \frac{ (3+2\sqrt{2})^m+(3-2\sqrt{2})^m}{2}$$ $$Q(m) = \frac{ ...
1
vote
1answer
12 views

Combining the rule of signs and index shifting.

When applying the rule of signs to a polynomial, one can determine possible positive and negative roots. But we can also apply a substitution or index shifting as follows:$$P(x)\to P(x\pm n)$$ Then ...
1
vote
9answers
295 views

Why is $x=2 \implies (x-2)(x-3)=0$ false?

Let $P(x)$ be the equation $x=2$ and $Q(x)$ be the equation $(x-2)(x-3)=0$ By definition of implication I see that $P(x)$ implies $Q(x)$... As I see it, any premise that is false can give any ...
1
vote
0answers
37 views

Zeros of a sum of real powers

Consider the complex function $$f(z)=\sum_{k=1}^n z^{\alpha_k}\,, \quad z\in\mathbb C,\;\Re z>0$$ where $\alpha_k$ are real numbers (assume positive without loss of generality). What can I say ...
1
vote
1answer
51 views

$x^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0=0$ has real coefficients which satisfy $0<a_0 \le a_1 \le \cdots \le a_{n-1} \le 1$ prove that $z$ is a root

Suppose that the coefficients of the equation $x^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0=0$ are real and satisfy $0<a_0 \le a_1 \le \cdots \le a_{n-1} \le 1$. Let $z$ be a complex root of the ...
4
votes
2answers
89 views

How can one find the zeroes of $f(x)=ae^{bx}+cx+d$?

A certain physics problem I have been working on has turned into a math problem. Particularly, I want to find the solutions of some equation of the form $$f(x)=ae^{bx}+cx+d = 0$$ where $a, b, c,$ ...
0
votes
4answers
47 views

The minimum number of non real roots of the equation $x^4-2x^3+2x^2-x=k$ is?

The minimum number of non real roots of the equation $x^4-2x^3+2x^2-x=k$ is? k is any real number. I plotted this on https://www.desmos.com/calculator/vpfpjwyxz8.It seems that the answer will ...