Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, polynomial long division, factoring and solving for roots.

learn more… | top users | synonyms

3
votes
3answers
36 views

Abstract Algebra, a question on polynomials .

Theory : Every polynomial $P_n$ , of n-th power over the field $R$- real numbers, can be written as a product of polynomials of the power $<3$, who's coefficients are real numbers. Proof: ...
3
votes
0answers
63 views

In the ring of polynomials $F[t]$, every ideal is principal

Let $$\langle a \rangle=\bigcap_{ a \in I} I$$ $$ \langle a \rangle =\{ Ira + na, r\in R,n \in Z\}$$ What is unclear is why$$ I=\langle 0 \rangle$$ proves that I is a principal ideal. My definition ...
3
votes
1answer
19 views

Abstract algebra. Proof of: Let $F$ be a finite field and $P$ an irreducible polynomial upon $F$. Then $(F[t]|_{\equiv_P}, + , \ast)$ is a field.

Division of polynomials I put what is unclear to me in between three asterisks bounding the unclear lines... $\equiv_{P}$ is defined as ($\forall Q, S \in F[t]$) $Q\equiv_{P} S \iff ...
2
votes
1answer
58 views

Solve $x+\frac{2}{y}=3,y+\frac{2}{z}=3,z+\frac{2}{x}=3 $ in reals

Find answers of this system of equations in real numbers$$ \left\{ \begin{array}{c} x+\frac{2}{y}=3 \\ y+\frac{2}{z}=3 \\ z+\frac{2}{x}=3 \end{array} \right. $$ Things I have done: first I ...
1
vote
1answer
31 views

How does this relate to Vieta's formula?

I was reading this PDF: http://diendantoanhoc.net/forum/index.php?app=core&module=attach&section=attach&attach_id=219 On Page 2, the author mentions Vieta's Formula. Now I am familiar ...
0
votes
1answer
19 views

Abstract algebra, polynomials , division.

It says here that $S,T$ are polynomials and $S=0, T \neq 0$ then $GCD(S,T)=a^{-1}T$ where $a$is the leading coefficient of polynomial $T$. Why is this?
0
votes
1answer
14 views

Finding a value of a constant coefficient to get a specific zero in polynomials

Find the value of the constant a for which the polynomial $x^3 + ax^2 − 1$ will have $−1$ as $a$ zero.
1
vote
2answers
566 views

Can you help me reverse the Minimum Curvature Method?

The minimum curvature method is used in oil drilling to calculate positional data from directional data. A survey is a reading at a certain depth down the borehole that contains measured depth, ...
5
votes
3answers
679 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
61 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 ...
0
votes
0answers
19 views

approximation of function by polynomials

Given a function $f \in L^2[a,b]$, it can be written as $f(x)=\sum_{n=0}^\infty c_nL_n(x)$. where $L_n(x)$ is shifted Legendre polynomial. I am taking the finite sum to approximate. If I take some ...
2
votes
6answers
110 views

Proof that a degree 4 polynomial has at least two roots

Let $$P(x) = x^4+a_3x^3+a_2x^2+a_1x+a_0$$ $$P(x_0) = 0$$ $$P'(x_0) \not= 0$$ with $x_0$ and each $a_i$ real. Prove that $P(x)$ has a at least two real roots. I can't figure why this is ...
2
votes
0answers
40 views

Prove that the 4 degree polynomial has at least two roots [duplicate]

In my assignment I have to prove that: Let $P(x)=x^4+a_{3}x^3+a_{2}x^2+a_{1}x+a_{0}$. Prove that if P has a root in $x_{0}$ and $P'(x_{0})\ne0$ then P has a least two roots. My solution has ...
0
votes
2answers
26 views

$p(x)=x^3+6x^2+wx-4$ have the same remainder when it is divided by $x+2$ and $x-1$.

For what value of $w$ will the polynomial $p(x)=x^3+6x^2+wx-4$ have the same remainder when it is divided by $x+2$ and $x-1$?
0
votes
0answers
31 views

How to prove that polynomials are dense in the set of bounded analytic functions

Let $H^\infty$ denote the set of all functions holomorphic and bounded in the open unit disk $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$, i.e. $\|f\|_{H^\infty} = \sup_{0<r<1} \sup_{\varphi ...
3
votes
4answers
39 views

Quartic Solution on Wikipedia special cases problem $S=0$ how to “change the choice of cubic root”?

So, I've posted a question regarding Wikipedia's quartic page. This was from the first question. I'm trying to implement the general quartic solution for use in a ray tracer, but I'm having some ...
0
votes
0answers
14 views

Invertible polynomials and polynomial norms

I am interested in normed rings, and I got to thinking about polynomial rings. In particular, if $R=k[x]$ is the ring of polynomials in one variable over a field $k$ (say characteristic 0), then the ...
1
vote
1answer
20 views

Deriving chromatic polynomials [duplicate]

How to derive the chromatic polynomial from a Cycle? I derived the chromatic polynomial for a triangle $ K_3$ it's: $t(t-1)(t-2)$ But I don't understand how to get it for Cycles $C_n$.
2
votes
1answer
42 views

A number root of two irreducible polynomials?

I woke up today doing me a question: is there a complex number that is root of two different irreducible polynomials of $\mathbb{Q} [x]$? I think not but I'm not sure and I am trying to prove. Some ...
2
votes
1answer
41 views

Show that $\alpha_A^{-1}(I'+J')=\alpha_A^{-1}(I')+\alpha_A^{-1}(J')$, where $I',J'$ are ideals and $\alpha_A$ is a surjective ring homomorphism.

Let $\alpha_A: k[x_1,...,x_m]\rightarrow k[y_1,...,y_n]$ be a map defined by $\alpha_A(f)(y)=f(Ay)$ where $A$ is an $m\times n$ constant matrix. Let $I',J'$ be ideals in $k[y_1,...,y_n]$. ...
0
votes
1answer
22 views

A formula about polynomials: $x^{n^i-1}-1=(x^s-1)\prod_{j=1}^r(x^{2^{j-1}s}+1)$, where $n^i-1=2^rs$

$x^{n^i-1}-1=(x^s-1)\prod_{j=1}^r(x^{2^{j-1}s}+1)$, where $n^i-1=2^rs$. Also, the factors on the right side are relatively prime polynomials. I found this formula on Grantham's "Frobenius ...
2
votes
1answer
35 views

Show that $G$ is a Groebner bases of $I$ if division of $f$ on $G$ is zero for all $f\in I$.

Let $I=\langle g_1,\dots, g_t\rangle$ be an ideal in $k[x_1,\dots,x_n]$ with $k$ a field. Let $G=\{g_1,\dots,g_t\}$. Show that if the remainder of $f$ on division by $G$ is $0$ for all $f\in I$, ...
2
votes
1answer
27 views

Show that $\left\langle\alpha_A(I\cap J)\right\rangle \subset \left\langle\alpha_A(I)\right\rangle \cap \left\langle\alpha_A(J)\right\rangle $.

Let $\alpha_A: k[x_1,...,x_m]\rightarrow k[y_1,...,y_n]$ be a map defined by $\alpha_A(f)(y)=f(Ay)$ where $A$ is an $m\times n$ matrix. Show that $\left\langle\alpha_A(I\cap J)\right\rangle \subset ...
2
votes
0answers
33 views

Prove that $I_1^m\cap I_2^m \cap \dots \cap I_r^m=I_1^m\cdots I_r^m$, where $I_1,…,I_r$ are ideals in $k[x_1,…,x_n]$ and are comaximal.

This is an exercise from Ideals, Varieties and Algorithms by Cox, etc. If $I_i$ and $J_i=\cap_{j\ne i}I_j$ are comaximal for all $i$, where $I_1,...,I_r$ are ideals in $k[x_1,...,x_n]$, prove that ...
1
vote
0answers
15 views

Rook Polynomials with Symmetrical Overlap (Count Permutations Restricted by Distance)

Consider the cardinality $P(n,d)$ of permutations where elements can move up to distance $d$; for example, the permutation $\binom{012}{102}$ with $d = 1$ would be valid but $\binom{012}{201}$ would ...
3
votes
1answer
38 views

Solving characteristic equation to find eigenvalue.

I came across the following question: The characteristic polynomial of a $3 \times 3$ matrix $A$ is $|\lambda I -A| = \lambda^3 + 3 \lambda^2+4 \lambda +3$. Find $trace(A)$ and $det(A)$. I know ...
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 ...
1
vote
3answers
49 views

Divisibility of a polynomial by another polynomial

I have this question: Find all numbers $n\geq 1$ for which the polynomial $x^{n+1}+x^n+1$ is divisible by $x^2-x+1$. How do I even begin? So far I have that $x^{n+1}+x^n+1 = ...
3
votes
1answer
57 views

Discriminant of Polynomials (Galois Theory)

So I'm reading Dummit and Foote and they define the discriminant of $x_{1},...,x_{n}$ by $$D=\prod_{i<j}(x_{i}-x_{j})^2$$ and the discriminant of a polynomial to be the discriminant of the roots. ...
5
votes
2answers
237 views

Artin Chapter 11, Exercise 9.12, polynomials without common zeroes [closed]

How do I show that the three polynomials $f_1 = t^2 + x^2 - 2$, $f_2 = tx - 1$, $f_3 = t^3 + 5tx^3 + 1$ generate the unit ideal in $\mathbb{C}[t, x]$? Artin mentions two approaches: by showing ...
1
vote
1answer
41 views

How to calculate the gcd of two polynomials $\mod 7$

I need to find gcd of $x^4-3x^3-2x+6$ and $x^3-5x^2+6x+7$ in $\mathbb Z/7 \mathbb Z[x]$, the integer polynomials mod $7$. Please any help will be appreciated.
8
votes
3answers
390 views

A Curious Binomial Sum Identity without Calculus of Finite Differences

Let $f$ be a polynomial of degree $m$ in $t$. The following curious identity holds for $n \geq m$, \begin{align} \binom{t}{n+1} \sum_{j = 0}^{n} (-1)^{j} \binom{n}{j} \frac{f(j)}{t - j} = (-1)^{n} ...
4
votes
2answers
20 views

Proving a property of two polynomials when one of them divides the another

Suppose $P(x)$ is a polynomial which can be factored into a product of different linear terms, that is $P(x)=(x-\alpha_1)(x-\alpha_2)\cdots(x-\alpha_k)$ and suppose $Q(x)$ divides $P(x)$ (i.e. $Q \mid ...
2
votes
1answer
37 views

finding two conditions so that for two polynomials, there exists exactly one matrix

Let $K$ be a field, and $f, g \in K[t]$, the ring of polynomials over $K$. I want to find a necessary and a sufficient condition for $f$ and $g$, so that there exists an (except for similar matrices) ...
0
votes
1answer
25 views

Finding coefficients of a function, given a list of points on the function

Given $f(x) = ax^n + bx^{n-1} + ... + cx + d$, a list of points, and a specification of a tangent line (point $p_t$ and equation) find $a, b, ..., c, d$ s.t. $f(x)$ passes through each point the ...
1
vote
0answers
11 views

Laurent polynomial regression?

Polynomial regression is a common way of doing curvilinear regression. It is common to also use the inverse transform x^-1 (http://pareonline.net/getvn.asp?v=8&n=6). One can extend the concept ...
1
vote
3answers
291 views

One more confusing factoring question.

The question is: $x^6 + 5x^3 + 8$ Please can someone help me in factorising this. I saw some solutions but they are not meant for a IX grade student. Thanks for the help.
2
votes
6answers
164 views

Given $x^2 + 4x + 6$ as factor of $x^4 + ax^2 + b$, then $a + b$ is [closed]

I got this task two days ago, quite difficult for me, since I have not done applications of Vieta's formulas and Bezout's Theorem for a while. If can someone solve this and add exactly how I am ...
0
votes
4answers
46 views

Factorisation question. [closed]

I was going through a math worksheet and I am stuck at this question: Factors of $(x^4 + 4)$: Can someone tell me how to factorise this? Thanks for your help.
-1
votes
1answer
40 views

Prove the polynomial is irreducible [duplicate]

I tried this problem for a while, but didn't see the application of Eisenstein's irreducibility criterion here. All the coefficients, including the leading coefficient, are equal to 1. p is a prime ...
2
votes
1answer
48 views

How can I apply Newton's sums to solve this problem?

Given $x_1,x_2,x_3,x_4$ real numbers such that $x_1+x_2+x_3+x_4 = 0$ and $x_1^7+x_2^7+x_3^7+x_4^7 = 0,$ how can I use symmetric functions and Newton's sums to prove that ...
0
votes
1answer
22 views

Proving that a binomial coefficient involving a power of $2$ is even

In the process of proving that the polynomial $x^{2^n} + 1$ is irreducible in $\mathbb{Z}[x]$, I am getting stuck on proving an intermediate result: Denote $f(X)=X^{2^n}+1.$ By a linear change of ...
1
vote
0answers
73 views

Determining the cycle type of complex conjugation

This arose recently in an online discussion about roots and irreducibility. Let $f(X) = X^4 - 4X + 2$. $f(X)$ has two real roots and two complex roots, which means that complex conjugation $\sigma$ ...
11
votes
1answer
282 views

How to solve $4x^3-3z^2=y^6$ in positive integers?

Solve in positive integers $$4x^3-3z^2=y^6$$ We are given that $\gcd (x,y) = \gcd (y,z) = \gcd (x,z) = \gcd (x,y,z) = 1$. I do not have the slightest idea how to even begin this question. ...
1
vote
1answer
25 views

Trouble Understanding general Quartic soulution from the Wiki what is $\Delta_1^2$ and $\Delta_0^3$

I'm trying to implement the general quartic solution for use in a ray tracer, but I'm having some trouble. The solvers I've found do cause some strange false negatives leaving holes in the tori I'm ...
1
vote
2answers
23 views

Higher Order Polynomial Function Solver

I have a 5th order, uni-variable, polynomial :( As I understand the only way to solve this is to guess? Since this is a real world equation, rather than something from a textbook, there really isn't ...
0
votes
1answer
17 views

Content of Polynomials and Gauss's Lemma

I am getting stuck on a little part of a proof: Let $R$ be a PID and let $K=$Frac$(R)$. If $f\in R[x]$ and $f=gh$ with $g\in R[x]$ of content 1, show that $h\in R[x]$. We can clear the denominators ...
0
votes
1answer
74 views

Find all polynomials $P(x)$ such that $P(x^2+1)=P(x)^2+1$ and $P(0)=0$

Find all polynomials $P(x)$ such that $P(x^2+1)=P(x)^2+1$ and $P(0)=0$. It's almost the same question as this: Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$. But here I have the ...
1
vote
5answers
61 views

System of equations $x + xy + y = 11$ and $yx^2 + xy^2 = 30$

I have problem with solving this one. Total number of solutions from system of equations? \begin{cases} x + xy + y = 11 \\ x^2y + y^2x = 30 \end{cases} There is a system of equation and I have ...
0
votes
2answers
53 views

finding maxima and minima of a quadratic equation

I'm dealing with a quadratic equation(with 2 independent variable) which looks like: $$f(x,y) = 15.390x^2 - 0.001y^2 - 0.003xy - 69.985x + 0.263y + 58.740 $$ But I'm not being able to determine the ...