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.

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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 ...
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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 ...
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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 $x-1$?
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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 ...
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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 ...
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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 $C_n$.
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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 ...
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4answers
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Expansion of $x^n-y^n$

Studying polynomials I couldn't find a way to expand $x^n-y^n$ as a product of other polynomials. Now of course we know that $$x^4-y^4=(x^2+y^2)(x^2-y^2)=(x^2+y^2)(x+y)(x-y)$$ and I came up with this: ...
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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 ...
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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 ...
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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 $k[y_1,...,y_n]$. ...
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1answer
37 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 ...
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5answers
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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 ...
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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 = ...
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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 ...
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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. ...
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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 ...
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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 ...
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0answers
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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 ...
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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 ...
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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) ...
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3answers
678 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 ...
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1answer
60 views

Show that $\prod _{i<j}(x_i-x_j)$ can be divided wihout remainder in $\prod_{i<j}(i-j)$ [duplicate]

Let $x_1,...,x_n$ be a natural numbers, show that $\prod _{i<j}(x_i-x_j)$ can be divided wihout remainder in $\prod_{i<j}(i-j)$ I know $\prod \left(x_i-x_j\right)$ is the result of ...
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3answers
289 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.
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4answers
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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6answers
109 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 ...
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1answer
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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 ...
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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 ...
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2answers
51 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 ...
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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 ...
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1answer
54 views

The existence of a polynomial factor

Given two polynomials $p_1(x_1,\dots, x_m)$ and $p_2(x_1,\dots, x_n)$ over reals, where $m > n$, and we know that $p_2(x_1,\dots, x_n)=0 \implies p_1(x_1,\dots, x_m) =0$. My question is: ...
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2answers
49 views

Polynomials ( Bezout's Theorem and Vieta's Formula) [closed]

Another question for faculty entrance exam (workbook tasks). Equation $x^3 + x^2 + ax + b = 0$ (a,b $\in R$) have solutions $1-\sqrt{2} $ and $1+ \sqrt{2}$. Product of all the available solutions of ...
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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 ...
2
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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 ...
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1answer
36 views

A question on polynomial congruence

If $m\ne 4$ is a composite number, then does there exist two integers $a$ and $b$ which, divided by $m$, give a remainder different from zero and such that if $f(x)$ is a polynomial with integral ...
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0answers
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Why has $\Phi(X,Y)$ integral coefficients?

Let $j$ be Klein's $j$-invariant and let $M$ be a the set of integral $2\times 2$ matrices with determinant $n$, $A$ and $-A$ identified. Why has the polynomial $\Phi(X,Y)\in\mathbb C[X,Y]$ with ...
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21 views

Finding the Inverse of Polynomial Equations (Approximatly)

Assume one is given a set of two equations of the form: $$x(u,v) = u + a_1 u^2 + b_1 u v + c_1 v^2$$ $$y(u,v) = v + a_2 u^2 + b_2 u v + c_2 v^2$$ And one would like to find the inverse functions, ...
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1answer
32 views

Proving an inequality involving integer polynomial

So we've got an integer polynomial $P$, and all we know about it is that $P(1) = 1$, $P(2) = 2$, and also $P(100) = -k$, where $k \in \mathbb{Z},\, k \geqslant 0$ - some unknown constant, which will ...
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1answer
58 views

Product of $n(n-1)/2$ polynomials of the same degree is symmetric

I am trying to prove a simple fact about polynomials in the multivariate polynomial ring $\mathbb{C}[x_1,x_2,...x_n]$, for $n \gt 3$ but I've been getting stuck. EDIT: After a comment by Tad I ...
3
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1answer
44 views

Looking for group of polynomials with only real roots

Assume $P_\mathbb R$ is the set of all polynomials which have only real coefficients and only real roots. Define $0$ as a polynomial with infinitely many real roots and all other constant polynomials ...
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1answer
18 views

Prove that the sum of the Lagrange (interpolation) coefficients is equal to 1

Prove that the sum of the Lagrange (interpolation) coefficients is equal to 1. Please suggest me a book-reference or give a solution for me. Thanks a lot in advance. If $f = ...
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1answer
46 views

Position of roots of $(x-a)(x-b)+2(x-c)(x-d)=0.$

Problem: If $a<b<c<d$, then consider the equation $(x-a)(x-b)+2(x-c)(x-d)=0.$ Does the equation have both roots in $[a,b]$, both roots in $[c,d]$ or have one root in $(a,b)$ and the other ...
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1answer
32 views

Is there a relationship between these polynomial concepts?

I'm currently doing a bit of reading on abstract algebra (more specifically Polynomial Theory), and noticed something that may have some sort of significance perhaps? The section I'm reading on at ...
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1answer
30 views

Show $f(x)\geq 0$ for all $x$ if and only if it has even degree and is of the form $f(x)=r(x)^2(x^2+a_1x+b_1)(x^2+a_2x+b_2) \cdots (x^2+a_nx+b_n)$

Suppose $f(x)$ is a polynomial with real coefficients. Show that $f(x)$ is nonnegative for all real numbers $x$ if and only if it has even degree and is of the form ...
2
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1answer
36 views

Connected component identification?

Suppose I give a random 2 variable polynomial relation such as: $$x^3+y^3=10$$ $$x^2 + 7yx^4 + x^2-15=0$$ Etc... How do I determine how many individual pieces there are to the graph?