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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1answer
103 views

Three phase voltage system of polynomial equations

I'm working with the development of a product in the company where I work. This product measures three phase voltages and currents. I cannot change the circuit because it has been sold for a long time ...
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0answers
12 views

Differential Formula Simplification

Define operators $x,D,1$ by $xf=xf$, $Df=\frac{d}{dx}f=f'$, and $1f=f$. Notice, then that $$(x+D)^nf=\sum_{k=0}^np_k(x)D^kf,\ \ \ \ \ \ \ f\in R[x],$$ for some sequence of polynomials ...
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0answers
25 views

2nd Order Polynomial Trendline

Hi apologies in advance if this is very trivial or I am out of my depth. I am working on a 2nd Order Polynomial Trendline How do I solve for x here? $$y = (c_2 \times x^2) + (c_1 \times x ^1) + ...
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0answers
55 views

Question on Primitive Polynomials

How can we show that any $g(x)\in \mathbb{Q}[x]$ can be uniquely written as $g(x)=cf(x)$ where c is rational and $f\in\mathbb{Z}[x]$ is primtive? This property seems intuitive but I am unable to ...
0
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1answer
23 views

numerical algorithms for determining least common multiple of polynomials

I have a pair of rational polynomial fractions $\frac{A(x)}{B(x)} + \frac{C(x)}{D(x)}$ where A, B, C, and D are all polynomials in x, and I have their coefficients as an array of numbers. I would ...
0
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1answer
30 views

What do I do wrong with Möbius method of inversion?

I use the Möbius inversion with polynomials as e.g. in the well-known inversion formula of the cyclotomic polynomials. So I have $$p_{2n}(x)=\prod_{d|n}(2q_d(x))^{\mu(\frac{n}{d})}$$ Now I get the ...
2
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1answer
40 views

Efficient Extended GCD Algorithm for Polynomials

For computing the GCD of two multivariate polynomials we have the Euclidian algorithm. However, it's well known that the Euclidian algorithm is not very efficient (because of intermediate expression ...
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4answers
83 views

Sums of solutions to $z^n-1 = 0$ that equal 0

Consider the solutions of the equation $z^n - 1 = 0$, where $z$ is a complex number: ${z_1,z_2...z_n}$. What are ALL the possible sums $\sum_{i=1}^n a_iz_i$ over these n solutions, where $a_i$ are ...
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1answer
34 views

Complex conjugate root theorem question

From the Complex conjugate root theorem we get that if a polynomial in one varaible with real coefficients has as solution $a + bi$ , than $a-bi$ must also be a solution...however, what happens if ...
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1answer
43 views

Number of monic irreducible polynomials over a finite field

Let $\mathbb{K}=\mathbb{F}_q$ and $\nu_n$ denote the number of monic irreducible polynomials over $\mathbb{K}$. It holds $$\nu_n=\frac{1}{n}\sum_{d\mid n}\mu\left(\frac{n}{d}\right)q^d$$ What I need ...
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5answers
69 views

Graphing polynomials

Sketch a graph of the polynomial $P(x)=(x-2)^2(x+1)^3$. You must plot and label the x and y intercepts and these should be the only points you plot. How do I sketch the graph of a polynomial?
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Polynomial representation of binary

It is well known that we can represent binary using polynomial. For example, $11$ can be represented as $x+1$. So when we compute $11\times11$, we should obtain $1001$, which is equal to $9$ in ...
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4answers
71 views

Prove that the characteristic polynomial of a nilpotent matrix is $x^n$

How can I prove that the char.pol. of a nilpotent matrix is of the form $x^k$? I'm trying to do it by contradiction but assuming that $p_{xA}=a_0+a_1x+\dots+a_mx^m+\dots+a_nx^n$ seems not giving any ...
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2answers
17 views

Dimension of quotient construction

If I have an irreducible polynomial, $f$ with $deg(f) = n$ and I look at the quotient: $$R = \frac{\mathbb{Q}[x]}{(f)}$$ How can we show that the dimension of $R$ as a $\mathbb{Q}$ vector space is ...
4
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1answer
33 views

Find discriminant of polynomial

$f(x)=x^{n-1}+x^{n-2}+...+1$ I solved this issue and get $-1^{\frac{(n-1)(n-2)}{2}}*n^{n-2}$ as solution. I did it in a way of substitution n=1,2,3.. then i get my answer. Now i want to solve it ...
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0answers
28 views

Facts about quotient rings - example

I have three quotient rings: $R_1 = \frac{\mathbb{Q}[x]}{(x^2 -1)}$ $R_2 = \frac{\mathbb{Q}[x]}{(x^2 +1)}$ $R_3 = \frac{\mathbb{Q}[x]}{((x -1)^2)}$ I am trying to decide whether these are integral ...
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2answers
57 views

Basic irreducible polynomial

I'm studying cyclic codes over a ring $R$. It is well known that a cyclic code over $R$ of length $n$ is an ideal of $R\left[ x \right]/\left( {{x^n} - 1} \right)$. Hence the factorization of ...
0
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1answer
54 views

Factoring $x^4 -8a^2x^2 -48a^4 -8bx^3 - 32a^2 bx +16b^2x^2 +64a^2b^2$

The subject line pretty much says it all. In my geometry class today, the following equation came up: $$x^4 -8a^2x^2 -48a^4 -8bx^3 - 32a^2 bx +16b^2x^2 +64a^2b^2 = 0$$ Specifically, it was in the ...
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3answers
41 views

How to establish these two facts about polynomials?

Let $f(x) := \sum_{k=0}^n c_k x^k $ be a polynomial of degree $n\geq 0$ with real coefficeints such that $f(x) = 0$ for $n+1$ distinct real values of $x$. Then how to prove that each $c_k = 0$ and ...
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1answer
83 views

What do we know about $\displaystyle \frac{f}{\gcd(f,f')}$ if $f\in\mathbb{F}_{p^d}[X]$?

Let $\mathbb{K}=\mathbb{F}_{p^d}$ and $f\in\mathbb{K}[X]$ be a non-constant polynomial with the factorization $$f=\prod_{i=1}^nf_i^{k_i}$$ where $f_i\in\mathbb{K}[X]$ is irreducible and ...
0
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0answers
48 views

Proof that $\mathbb{Z}[\sqrt{-5}]$ is integrally closed

There are demonstrations on the Internet saying that the polynomial $$\left(x-\frac{a}{c}-\frac{b}{d}\sqrt{-5}\right)\left(x-\frac{a}{c}+\frac{b}{d}\sqrt{-5}\right)$$ is monic if and only if ...
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1answer
23 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 ...
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1answer
16 views

Determining if any of these three are an ideal of $\mathbb{R}[x]$

$\mathbb{R}[x]$ denotes the ring of polynomials in $x$ with real coefficients. Let $I \subset \mathbb{R}[x]$ be the subset of those polynomials with constant coefficient $0$, and let $J \subset ...
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0answers
36 views

Finding the general form of an element in $\frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$

I'm trying to find the general form of elements in the quotient ring: $$R = \frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$$ Now my initial thoughts are to take a general element $f \in R$ so that $f = g + (x^2 ...
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2answers
47 views

roots of cubic polynomial

On page 26 of Milne's Elliptic Curves (http://www.jmilne.org/math/Books/ectext5.pdf), he states the following: "... a cubic polynomial $h(x) \in k[x]$ with two roots in $k$ has all of its roots in ...
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1answer
23 views

A sum of Laguerre polynomials

I'm looking to find a closed-form expression for the sum $$S = \sum_{n=0}^N e^{-x/2} L_n^{0}(x),$$ where $L_n^{0}$ is the $n$th Laguerre polynomial. Using the formula $$L_n^{\alpha}(x) = \sum_{m=0}^n ...
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2answers
47 views

how to solve this question of polynomials

Given the polynmial is exactly divided by $x+1$, when it is divided by $3x-1$, the remainder is $4$. The polynomial leaves remainder $hx+k$ when divided by $3x^2+2x-1$. Find $h$ and $k$. This ...
0
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1answer
42 views

Proving Newton's identities

Assume $F$ is a field of zero characteristic. Denote the elementary symmetric polynomials of $n$ variables by $e_k$, $\quad k=\overline{1,n}$. Let the symbol $\sum ax_1^{i_1}\dots x_n^{i_n}$ denote ...
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0answers
79 views

Approximating polynomials over finite fields

Consider a binary finite field $F = GF[2^{n}]$ with addition and multiplication denoted by $\oplus$ and $*$, respectively. Let me represent the elements of $F$ by $n$-bit strings, which means that ...
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2answers
114 views

Polynomial multiplication modulo polynomial

Suppose we are working on finite field $F_{16}$ and have pritimive polynomial $z^4+z+1$. I stuck at how to compute polynomial modulo. For example, we have $z^5+z+1$ mod $z^4+z+1$. I use the usual ...
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0answers
280 views

A problem with sign of coefficients of a polynomial expression

Let $f$ be a real coefficient homogeneous polynomial in $n$ undeterminates, such that $f(x_1,\cdots,x_n)>0$ whenever $x_1,...,x_n$ are non-negative real numbers, not all $0$. Then how to show ...
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2answers
45 views

Prove an inequality (Using Taylor expansion)

Prove: $\frac{x}{1+x} < \ln(1+x) < x$. I thought a good practice would be to prove it using Taylor Expansion. Here's my try: $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3}...$$ The n=1 ...
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4answers
43 views

find f(x) polynomial with rational coefficients such that $f(x)^{2} = g(x)^{2}(x^{2}+1)$

g(x) is a polynomial with rational coefficients that is not 0 . I need to find f(x) polynomial with rational coefficients such that: $f(x)^{2} = g(x)^{2}(x^{2}+1)$ or prove such polynomial does not ...
4
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1answer
42 views

If $f(n) \in \mathbb{Z}$ for an infinite number of $n \in \mathbb{Z}$, then $f \in \mathbb{Q}[x]$.

Do you think that the following statement is true? Do you have any idea about the proof? Let $\; f(x) \in \mathbb{C}[x]$ be a polynomial. If $f(n) \in \mathbb{Z}$ for an infinite number of $n \in ...
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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 ...
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1answer
55 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 ...
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3answers
41 views

On Bezout's identity

Assume $F$ is a field. Then for each $f,g\in F[x]$ with greatest common divisor $(f,g)=d$ by Bezout's identity $uf+vg=d$ for some $u,v\in F[x]$. How can we see that $(u,v)=1$? Furthermore, if for ...
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2answers
47 views

how to find the remainder when a polynomial $p(x)$ is divided my another polynomial $q(x)$

i was solving the question from the book IIT FOUNDATION AND OLYMPIAD - X and i was solving the problems of polynomials-III. so ...
2
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1answer
57 views

Polynomial equal to the ceiling of x

For a few days, I've been looking for a polynomial who's value is equal to the ceiling function of the only variable it contains. I thought about it for while and I haven't got a clue.
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1answer
14 views

The degree of a map between complex projective lines

Let $P$ and $Q$ be complex polynomials such that $\deg P=p$, $\deg Q=q$ and $\gcd(P,Q)=1$. How can I: show that $F(z)=\frac{P(z)}{Q(z)}$ defines a smooth map $\mathbb{C}P^1\to\mathbb{C}P^1$? ...
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1answer
39 views

prove that if T is invertible transformation there is polynomial $p$ such that $T^{-1} = p(T) $

I know how to prove this using Hamilton.C but something doesn't make sense to me. if I assume that there is such polynomial p(x), so p(T)T = I . then looking at these polynomials I get: p(x)x = 1 so ...
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2answers
29 views

Does Property of Division of Polynomials apply to Constant functions in the Numerator and Denominator

My text book states that "if $p$ and $q$ are polynomials, with $q \ne 0$, then there exist polynomials $G$ and $R$ such that $p/q = G + R/q$, and $\deg R < deg q$ or $R=0$" So if $p(x) = 1$ and ...
0
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1answer
20 views

Question about calculating exponent of polynomial

$V=R_{3}[X] $ and $T:V->V$ is a linear transformation : $T(p(x)) = p(x) + xp'(x)$ I need to find $e^{T(1+x+x^{2}-x^{3})}$ I don't understand how to do it? what does it mean to calculate exponent ...
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4answers
74 views

Is this polynomial irreducible in $\mathbb{Q}$?

this is a really easy question but I cant find an answer; In need to see if $x^4+x^2+x+1$ is an irreducible polynomial over $\mathbb{Q}$
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0answers
35 views

Maximum of $P$ in the disk $|z|=1$ depending on co-efficients

Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m$ be a polynomial with complex coefficients such that $a_m\neq 0, a_n\neq 0$ and $n>m$. Prove that ...
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1answer
23 views

What's the complexity of expanding a general polynomial?

Suppose I have a polynomial in the form $(a_1 x_1+ a_2 x_2+...+ a_m x_m)^n$, where $x_1,...,x_m$ are the independent variables. I want to expand it to the form of sum of products. What is the ...
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40 views

Matrix Polynomials

Let $A$ and $B$ be $n × n$ matrices such that rank $(AB − BA) ≤ 1$. Show that $A$ and $B$ have a common eigenvector. Find a common eigenvector $($probably belonging to different eigenvalues$)$ for ...
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35 views

How can I find a tranformation matrix/Mathematical relation between two 5th degree polynomial curves in space?

I have the equation of two 5th degree polynomials which they don`t intersect with each other .Each curve is made of 100 points and these two curves looks similar but there are small differences .I am ...
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1answer
23 views

solving a simple polynomial equation?

I am looking for a solution to an equation of the form $a x^4 + b x + c = 0$. (I need the positive solutions only, but I can filter the negative ones out, if I get the four solutions for the above.) ...
2
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1answer
60 views

Find the remainder when $x^{100}$ is divided by $x^2-3x+2$

We have to find the remainder when $x^{100}$ is divided by $x^2-3x+2$.I tried to use the remainder theorem but am not just able to solve it.please help.