0
votes
2answers
45 views

Integer roots to cubic equation

If I have a cubic equation $x^3 + ax^2 + bx + c = 0$, what constraints exist on $a,b,c$ when we have three integer solutions? How do I choose $a,b,c$ to force integer solutions?
4
votes
1answer
40 views

Is it possible to calculate the roots of the difference between successive terms of this polynomial series $\rm{P}_n (x)=x\rm{P}_{n-1}-r\rm{P}_{n-2}$

Consider the polynomial series defined by the following recursion formula: $$ \begin{align} &\mathrm{P}_0 = 1 \\ &\mathrm{P}_1 = x-r \\ &\mathrm{P}_n = x\mathrm{P}_{n-1} - ...
3
votes
3answers
184 views

Roots of polynomial of degree 6

I'm struggling to find the complex roots of $x^6-9x^3+8 = 0$. I've managed to find the real roots (1 and 2) by letting a variable, say $α = x^3$ and substituting where relevant, leading to a ...
1
vote
1answer
24 views

Sum of Non Real Roots of Bi Quadratic

Consider $$f(x)=8x^4-16x^3+16x^2-8x+k=0$$ where $k \in \mathbb{R}$,then find sum of non Real roots of f(x). My approach: we have $$f'(x)=32x^3-48x^2+32x-8=0$$ Also ...
0
votes
1answer
21 views

Projective roots of a homogeneous polynomial

Suppose that $f(X,Y)\in\mathbb C[X,Y]$ is a homogeneous polynomial of degree $n$, then we can consider it as a function on $\mathbb P^1_\mathbb C$. It has at most $n+1$ projective roots (points of ...
1
vote
0answers
23 views

Quartic Polynomial Manipulation

I have a quartic polynomial in $x$ (too long to write here) $f(x,c_1, c_2, c_2)$ where $c_1, c_2, c_3$ are constants which I have complete freedom over how to fix their values, as long as they are ...
0
votes
2answers
24 views

Understanding Multiplicities

I am having troubles understanding what 'multiplicities' mean. In example what does $-1/3(multiplicity 2)$ translate into?? To clarify this is for finding zero's in a polynomial function Any help ...
5
votes
1answer
75 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} - ...
1
vote
1answer
78 views

The number of zeros in a polynomial

Does anyone know of a relationship between the number of zeros (complex and real) and the degree of a polynomial? Specifically, if a polynomial has a double root, is it compensated with complex ...
0
votes
1answer
48 views

If one root is unsolvable, can there be a solvable root?

Suppose you have some polynomial $p(x)$ with rational coefficients in which at least one root is unsolvable by radicals, does this imply that all other roots of $p(x)$ are unsolvable by radicals? ...
1
vote
0answers
22 views

What is the condition for roots of conjugate reciprocal polynomials to be on the unit circle?

Given an Nth order complex polynomial $P(z) = \sum\limits_{n=0}^N a_nz^n$ such that $a_n = a^*_{N-n}$ i.e. conjugate reciprocal, Lakatos and Losonczi mention that a necessary and sufficient condition ...
1
vote
0answers
29 views

Version of Chevalley's Theorem proving $\mathbb{F}_q$ is a $C_1$-field for $\mathbb{Z}/p^n\mathbb{Z}$?

Let $k=\mathbb{F}_q$ Chevalley's theorem states that if $f(x_i)\in k[x_1,...,x_n]$ is such that $f(0,...,0)=0$ and $deg(f)<n$, then there is a non-trivial $a_i\in k^n$ such that $f(a_i)=0$. Is ...
1
vote
1answer
20 views

Relation between the roots of a cubic equation and the coefficients

$ax^3 +bx^2 + cx + d= 0$ If the roots are $\alpha$ $\beta$ and $\gamma$, Is there any relationship between the sum of the squares of the roots and the coefficients of the quadratic equation.. In ...
7
votes
1answer
80 views

Find the maximum value of $ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $

If $x\in\mathbb{R}$ find the maximum value of $$ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $$ I tried this: Let $$y= \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1}$$ For maxima ...
1
vote
1answer
46 views

Kantorovich Theorem example

I need to write in C a program that finds roots of a 6th order polynomial. I was thinking of using Kantorovich Theorem convergence of Newton's method to find when can I use Newton-Rhapson method. I'm ...
1
vote
1answer
56 views

I need help proving a statement about rational roots

I have no idea where to start...this is the statement: If a polynomial of degree not greater than 5 with rational coefficients has multiple roots, it has also a rational root, except in the case ...
1
vote
2answers
78 views

Most Efficient Method to Find Roots of Polynomial [duplicate]

I am designing a software that has to find the roots of polynomials. I have to write this software from scratch as opposed to using an already existing library due to company instructions. I currently ...
2
votes
1answer
40 views

Determine the coefficients of a polynomial knowing its roots

My prof. gave this problem as a bonus in an exam, and I couldn't figure out a solution. Some hints or a general method of solving it would be very nice. Given the following polynomial: ...
7
votes
4answers
111 views

Finding double root of $x^5-x+\alpha$

Given the polynomial $$x^5-x+\alpha$$ Find a value of $\alpha>0$ for which the above polynomial has a double root. Here's an animated plot of the roots as you change $\alpha$ from $0$ to $1$ I'm ...
0
votes
0answers
38 views

Extension of quadratic forms

A homogen multivariate polynomial with degree 2 is a quadratic form. It can be checked if the polynomial is positive for any non-zero vector by checking if the corresponding matrix A is positive ...
3
votes
1answer
52 views

What is the Most Efficient Way to Calculate the Internal Rate of Return?

I have built a program that prices financial assets and it does this in part by calculating the IRR. The problem is that it does not run as quickly as I would like it to. I currently use the ...
3
votes
2answers
32 views

Build field extension and solve equation

Build quadratic extension of field that contains $5$ elements. And solve $x^2+x+2=0$ in this field. As I understand we need to build $\mathbb{F}_{5^{2}}$. Field $\mathbb{F}_5$ contains ...
2
votes
1answer
54 views

Given roots (real and complex), find the polynomial

This is not a duplicate of theory of equations finding roots from given polynomial. Given that the roots (both real and complex) of a polynomial are $\frac{2}{3}$, $-1$, $3+\sqrt2i$, and $3+\sqrt2i$, ...
1
vote
1answer
24 views

Perturbation of complex polynomials

Let $f(z)=\sum\limits_{k=0}^N a_kz^k$ be a (monic) complex polynomial and $\{\xi_{k}\}_{k=1}^{N}$ be the roots of $f$ (with multiplicities). Let $\{\tilde{\xi_{k}}\}_{k=1}^{N}$ be the perturbed ...
1
vote
1answer
27 views

On the existence of polynomial roots

Assume $F$ is a field, and $f\in F[x]$ is polynomial. To see that $f$ has a root in some extension of $F$, without loss of generality we can assume $f$ is irreducible. Indeed any polynomial $f$ is ...
2
votes
1answer
57 views

Solving a cubic polynomial equation.

Overview I have tried finding a solution to this problem myself and I have flailed. Its just a challenge for me. could you please tell me how far am I in solving this question? My approach for ...
14
votes
2answers
356 views

Does this polynomial have all its roots both distinct and real?

Recently, I wondered about the following problem: let $n\geq 5$ and let $$ P_n(x)=(x-1)(x-2)\ldots (x-n)-1 $$ Is it true that $P_n(x)$ has $n$ distinct real roots for any $n\geq 5$ ? I checked it ...
3
votes
2answers
71 views

Root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$

Is there an analytic way to obtain the highest root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$ in terms of $K$ and $C$? The integer $K \ll x$ and the constant $C$ are known. The other way to ask ...
6
votes
2answers
186 views

Find the maximum possible value.

For all ordered triples $(p,q,r)$ define the polynomial $$f_{p,q,r}(x)=x^3-px^2+qx-r$$ Let $a_{1},a_{2},a_{3},b_{1},b_{2},b_{3},c_{1},c_{2},c_{3}$ be (not necessarily distinct) positive reals such ...
2
votes
2answers
48 views

Information about the roots of a polynomial without their calculation

Suppose I have a polynomial (of any order) and I'm not able to calculate the roots. Is there a way to get at least some information about the roots such as how many of them are complex, negative or ...
6
votes
3answers
80 views

Solving Equation of Degree n, where n is any value between 1 and 2

How does one solve an equation of the form: $$ax^n + bx + c = 0$$ where n is a non integer value between 1 and 2. Is there a formula to provide an analytic solution?
30
votes
4answers
858 views

$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x)$, then all the roots of $p_k(x)$ are real

$p_0(x)=a_mx^m+a_{m-1}x^{m-1}+\dotsb+a_1x+a_0(a_m,\dotsc,a_1,a_0\in\Bbb R)$ is a polynomial, and $$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x),\qquad n=1,2,\dotsc$$ then, there exist $N\in\Bbb N$, such ...
6
votes
4answers
113 views

Solve $x^{3}-3x=\sqrt{x+2}$

Solve for real $x$ $$x^{3}-3x=\sqrt{x+2}$$ By inspection, $x=2$ is a root of this equation. So, I squared both sides and divided the six degree polynomial obtained by $x-2$. Then I got a ...
5
votes
1answer
84 views

Find $\lfloor {\alpha}^6 \rfloor$

If $\alpha$ is a real root of the equation $$x^5-x^3+x-2=0$$ find the value of $\lfloor {\alpha}^6 \rfloor$. This one totally stumped me. We are asked to calculate $\lfloor {\alpha}^6 ...
0
votes
1answer
39 views

Using elementary polynomials to solve system of linear polynomials

Problem Statement I am given a finite set of monic polynomials in t, parameterized by $r_i$ $X_i = t - r_i$ where the $r_i$ are guaranteed unique. Neither $t$ nor $r_i$ are known, only $X_i$. I ...
7
votes
3answers
114 views

How to solve the following? $ x^3+1=2{(2x-1)}^{1/3} $.

Find all the real solutions of $$x^3+1=2{(2x-1)}^{1/3} $$ I tried to cube both sides but got messed up with a nine degree equation! Please help. Thanks in advance!
0
votes
1answer
18 views

prove that the next two affirmations are equivalent

prove that the next two affirmations are equivalent : 1) every non constant $f(x)\in \mathbb C[x]$ has all of its roots in $\mathbb C$ 2)every non constant $f(x)\in \mathbb C[x]$ has at least one ...
1
vote
2answers
25 views

how to find the roots of: $x^{3}+6x^{2}-24x+160$ if one root is $2-2(3)^{1/2}i$

how to find all the roots of the next two polinomials?: $x^{3}+6x^{2}-24x+160$ if one root is $2-2(3)^{1/2}i$ and $x^{5}-3x^{4}+4x^{3}-4x+4$ if $1+i$ is a double root I don´t know how to solve this, ...
0
votes
0answers
31 views

Let $f(x)\in K[x]$ ($K$ field). Prove that if $(f(x),f´(x))=1$ (greatest common divisor is 1) then $f(x)$ does not have multiple roots in $K$

Please I would like you to tell me if my proof is correct Let $f(x)\in K[x]$ ($K$ field). Prove that if $(f(x),f´(x))=1$ (greates common divisor is 1) then $f(x)$ does not have multiple roots in $K$ ...
1
vote
2answers
58 views

Finding zero of function which is a real number

Is there an easier way of finding or approximating the x-axis-intersect of this function: $$ 0=x^3-3x^2+x+3 $$ The approximate solution is: $$ x=-0.76929 $$ and the precise solution is: $$ x=1 - ...
0
votes
1answer
36 views

Application of Rouché's (Rouche's) Theorem to a Polynomial

Here is my question: State Rouché's theorem. How many roots of the polynomial $p(z) = z^8 + 3 z^7 + 6 z^2 + 1$ are contained in the annulus {$1 < |z| < 2$}? The statement is fine. I then ...
3
votes
2answers
69 views

Cubic polynomial - radical expression of roots

Let $f=X^3+X^2-2X-1$ be a polynomial with the three roots $x_1,x_2,x_3$ with $x_1=2\text{cos}(\frac{2 \pi}{7})$. We define $z:=(x_1-x_2)(x_1-x_3)(x_2-x_3)$. I want to find a radical expression for ...
0
votes
3answers
46 views

Find the rational roots of $x^{3}-{2x^{2}\over 3}+3x-2$

I need to find the rational roots of $$x^{3}-{2x^{2}\over 3}+3x-2$$ I thought about using descartes´ rational root theorem but I need to have integers as coefficients of my polynomial: can I work with ...
1
vote
1answer
42 views

Finding the scope of a parameter where a polynomial can have roots

I have this problem- lets say I have a polynomial which has real parameters as coefficients and I'm looking for the scope of the parameters where the polynomial can have real roots. e.g $x^2+kx+k$ we ...
1
vote
1answer
20 views

relation between the number of real roots of the derivative and the original polynomial

If the derivative of polynomial has n real roots then can we conclude that the original polynomial has to have n+1 real roots?
0
votes
1answer
37 views

conclusion about roots for positive derivative of a polynomial

If the derivative of a polynomial is always positive then what can we conclude about the number of real roots the original polynomial?
1
vote
3answers
71 views

Relation between the roots of $x^2+x+1$ and its derivative

If $f(x)$ is a polynomial in n degree and has $n$ real roots then is it necessary that $f'(x)$ has to have $n-1$ real roots? If this is so then $x^2+x+1$ has no real roots but the derivative of the ...
0
votes
2answers
55 views

$x^3+3x^2+4x+5=0$ and $x^3+2x^2+7x+3=0$, how many common roots they have?

My attempt, Equate both, at the end you will get $x^2-3x-2=0$ That means $x=-1$ and $x=2$. But what after that. Please provide solutions as well.
4
votes
2answers
46 views

Disk with root in center with no other roots in polynomial

Say we have a polynomial $p$ with roots $r_1,r_2...r_n$, I'm looking for a way to find a disk which, if placed on the center of any root, does not contain any other root (multiple roots considered as ...
1
vote
1answer
35 views

Finding a disk containing all roots of a complex polynomial

I'm trying to list all roots of a polynomial so I found this paper, in Part 9 on page 29 it gives a simple recipe to find all the roots. But there is this remark: We have assumed throughout the ...