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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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, ...
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17 views

Composition of polynomial and multiplicative is multiplicative .

I made the following problem a while ago but I can't solve it (also I don't think it's extremely hard ) : Let $f$ be a non-constant completely multiplicative function over $\mathbb{Z}$ . Assume ...
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36 views

Why does $\frac{x^n}{n^x}$ stop growing at the approximate value of $\pi (n)$?

I noticed while playing around with these functions that $n^x$ will start slow and then speed up really fast in its growth rate. While $x^n$ grows more slowly, but faster than $n^x$ at the start. ...
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230 views

Why is the (-1)-th coefficient of $f^n f'$ equal to 0, without dividing by $n+1$?

Let $R$ be a commutative ring, and $n$ be a nonnegative integer. Let $f\in R\left[t,t^{-1}\right]$ be a Laurent polynomial in one variable $t$ over $R$ (this means a formal $R$-linear combination of ...
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2answers
58 views

a matrix of rank $r$ satisfies a polynomial of degree $r+1$.

Let $M$ be an $n\times n$ matrix with coefficients in $\mathbb C$. Suppose $M$ has rank $r$ with $r<n$. Prove there is a polynomial $P(x)$ with degree $r+1$ and coefficients in $\mathbb C$ such ...
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1answer
161 views

Show that $\sqrt{2}$ is irrational using the integer root theorem

Show that $\sqrt{2}$ is irrational using integer root theorem. Let $P(x)=x^2-2$. Since $\sqrt{2}$ is a root of this polynomial, had it been a rational (suppose $\sqrt{2}=\frac{p}{q}$) no, by ...
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44 views

Finding coefficients of $x^n$ and $x^{n+r}$ in an expansion

I have to find the coefficients of $x^n$ and $x^{n+r}$ $(1 < r < n)$ in the expansion of: $$(1 + x)^{2n} + x(1 + x)^{2n - 1} + x^2(1 + x)^{2n - 2} + ... + x^n(1 + x)^n$$ How do I solve it?
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1answer
43 views

Conditions for a unique root of a fifth degree polynomial

Fifth degree polynomials cannot generally be solved analytically, but at least one solution always exists. Given the normal form $$ax^5+bx^4+cx^3+dx^2+ex+f=0,$$ is it possible to find sufficient ...
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2answers
41 views

How to factor $ s^2LC + sRC + 2$

or $$ s^2+s\frac{R}{L}+\frac{2}{LC}=0 $$ Is there any way? I can't find out. Thanks in advance.
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1answer
45 views

Sum of digits modulo a polynomial

I made the following problems a while ago but I can't solve them (though I don't think it's too hard) 1.Let $s(n)$ be the digits sum of $n$. Let also $f(n)$, $g(n)$ $\in Z[X]$ . Assume that: ...
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0answers
35 views

Legendre symbol identity

I try to solve the following problems ($p$ is an odd prime) Find the sum $$\sum_{a=1}^{p-1}a \cdot \left (\frac{a}{p} \right)$$ Find the sum $$\sum_{a=1}^{p-1} 2^a \cdot \left (\frac{a}{p} \right)$$ ...
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3answers
86 views

How do I solve an equation like this?

How do I solve following equation for $X$: $$ AX^n + BX^{n-1} + CX^{n-2} + \dotsb + YX + Z = 0, $$ where $A,B,C,\dotsc,Z,n$ are known?
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2answers
6k views

Reed Solomon Polynomial Generator

I am developing a sample program to generate a 2D Barcode. And i am using reed solomon error correction code. By Going through this article i am developing the program. But i couldn't understand how ...
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2answers
292 views

Finding characteristic polynomial of adjacency matrix

Short question im having a tad difficulty with. I'm trying to find the characteristic polynomial of a graph that is just a circle with n vertices and n edges. I think the adjacency matrix should ...
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1answer
17 views

Durand-Kerner with derivative in denominator

The correction term for Durand-Kerner root finding method is $w_k = -\frac{f(z_k)}{\prod_{j\not=k}(z_k - z_j)}$ Wikipedia Talk page mentions that it is also possible to use derivative in ...
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2answers
43 views

Contest problem in functional equations.

Let n be a positive integer with $f(n)= 1! +2! +3!+... +n!$ and P(x), Q(x) be polynomials in $x$ such that $f(n+2)=P(n)f(n+1)+Q(n)f(n)$ for all $n \geq 1$, then which of the options is/are ...
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1answer
174 views

The asymptotic of the number of integers that are sums of three nonnegative cubes

Let $c(n) $ be the number of distinct integers between $0 $ and $n $ of the form $ a^3 + b^3 + c^3$, meaning the sum of $3$ nonnegative cubes. $C(n) = O( n \space \ln(n)^x ) $ Find and prove the ...
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1answer
399 views

Working with casus irreducibilis

I read about casus irreducibilis here. As an example of casus irreducibilis, it says we can factor $x^3 - 15x - 4$ to find $4$ as a root and it also has two other real roots. Using Cardano's method we ...
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4answers
174 views

Express roots in polynomials of equation $x^3+x^2-2x-1=0$

If $\alpha$ is a root of equation $x^3+x^2-2x-1=0$, then find the other two roots in polynomials of $\alpha$, with rational coefficients. I've seen some other examples [1] that other roots were ...
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0answers
8 views

Reference for a Dickson Determinant Polynomial

For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k] \end{equation} ...
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2answers
182 views

Divisors of zero in polynomial ring

I have the following theorem: McCoy: Let $R$ be a commutative ring with identity. If $f=\sum_{i=0}^na_iX^i$ is a zero divisor in $R[X]$, then there exists a nonzero $c$ in $R$ such that $cf=0$. ...
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1answer
100 views

Reference request: Newton-Kantorovich hypothesis for polynomials of integral coefficients

Kantorovich's theorem states that the Newton method for finding the roots of a nonlinear function is guaranteed to converge if a parameter $h$, determined by the values of the function and its ...
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1answer
39 views

Proof for the coefficient of $x^n$ in $(x^0 + x^1 + \dots + x^n)^n$

Theorem: The coefficient of $x^n$ in $(x^0 + x^1 + \dots + x^n)^n$ is $\binom{2n-1}{n-1}$. How to prove this? Multinomial theorem produces the following $$ \left(\sum_{k=0}^{n} x^k \right)^n = ...
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1answer
49 views

Product of a Finite Number of Matrices with a Cosine Entry

Does any one know how to prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} 2\cos\frac{2j\pi}{n} & a \\ b & 0 \end{pmatrix}\right)=2 $$ when $n$ is ...
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1answer
69 views

Simplifying a sum of products related to Vandermonde determinant

How to show this equality? $$ 1=(-1)^n\sum_{k=0}^n\frac{x_k^n}{\prod_{\substack{l=0 \\ l \neq k}}^n(x_l-x_k)} $$ This is part of a proof to show the value of the determinant of the Vandermonde matrix ...
2
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2answers
46 views

homogeneous polynomials over finite fields

Let $F$ be a finite field and $p(X_1,\dots,X_n)\neq 0$ an homogeneous polynomial with coefficients in $F$. Is it possible that $p(x_1,\dots,x_n)=0$ for every $(x_1,\dots, x_n)\in F^n$?
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1answer
35 views

Why is $Y_n$ of that form?

Let $R$ be any integral domain of characteristic zero. We consider the Pell equation $$X^2-(T^2-1)Y^2=1\tag 1$$ over $R[T]$. Let $U$ be an element in the algebraic closure of $R[T]$ satisfying ...
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25 views

Solving a system of polynomial equations

How can I solve a system of polynomial equations like this one Maybe I'm missing a very basic trick... Can anybody suggest me an approach?
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Exercise in arithmetic of a finite field

I am in difficult in resolving this exercise in Galois Theory : "in $GF(2^5)$ calculates the product $(1,1,1,0,1)(0,1,0,1,0)$ , generator of $GF(2^5)^*$ ". I don't know how to proceed.. thank you
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1answer
66 views

Polynomial Interpolation and Security

Let polynomial $P$ be $P(x)=g(x).(x−β)$, where $g$ is a polynomial and $\beta \leftarrow \mathbb{F}_p$. We evaluate $P$ at some $\textbf{x}=(x_1,..,x_n)$. This gives us $\textbf{y}=(y_1,..,y_n)$. ...
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1answer
31 views

Tricks for a Specific System of Polynomial Equations

I'm looking for all the complex solutions to the following 3 equations (and for this consider $a$ to be some given constant, so that there are really just 3 unknowns in solving): ...
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A question in perturbation of $P(\lambda )$

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
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1answer
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How to estimate the error on the position of the point x where y is maximal in quadratic relationships?

I would like to estimate the elevation at which species richness is expected to be maximal. The relationships between species richness ($y$) and elevation ($x$) follows a second order polynomial ...
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34 views

Is there a linear decomposition of the Hadamard inverse of the sum of two matrices?

Let the matrix $$\Gamma = \alpha A + (1-\alpha)B$$ where $B$ is a square symmetric matrix, $A = c\ ee'$, where $e$ is a vector of ones, and $c$ a positive constant and $0 < \alpha < 1$. The ...
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4answers
246 views

Why do we choose cubic polynomials when we make a spline?

Good morning, I want to learn more about cubic splines but unfortunately my class goes pretty quickly and we really only get the high level overview of why they're important and why they work. To me ...
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28 views

Polynomial Interpolating; When $y_i$'s are Changed

This is a comlpementry question to the one posted in: Polynomial Interpolation And polynomial Roots Given $\{(x_1,y_1),...,(x_n,y_n)\}$, we can interpolate a polynomial $P$. Assume polynomial $P$ has ...
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79 views

Polynomial Interpolation And polynomial Roots

Given $\{(x_1,y_1),...,(x_n,y_n)\}$, we can interpolate a polynomial $P$. Assume polynomial $P$ has some roots including an specific root $\beta$. Consider we change one of $y_i$ to $y'_i$. Given ...
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Need For group theory [closed]

How are permutation groups used to find the relationship between roots of a $n$-th degree polynomial? What is the geometrical significance of stabilizers and orbits?
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35 views

Linear Algebra Change of Basis problem

So, $\mathbb{P}_2$ is the vector space of all polynomials with degree less than or equal to 2 and that $E=\{1,t,t^2\}$ is a basis for $\mathbb{P}_2$ We define $p_1(t)=1+2t$ $p_2(t)=t-t^2$ ...
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closed-form expression for roots of a polynomial

It is often said colloquially that the roots of a general polynomial of degree $5$ or higher have "no closed-form formula," but the Abel-Ruffini theorem only proves nonexistence of algebraic ...
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4answers
64 views

Factorization of a polynomials in complex number.

Factorize this expression: $$a^2+b^2+c^2-ab-bc-ca.$$ The result is $$(a+b\Omega+c\Omega^2)(a+b\Omega^2+c\Omega)$$ How I can get $\Omega$ here?What's the approach?
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4answers
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Simplifying $\sum_{i=1}^{n-2}i(n-1-i)$

I have been trying to simplify $\sum_{i=1}^{n-2}i(n-1-i)$ i.e - remove the summation, put it in polynomial form Since $i$ is the changing variable, I don't think this is possible. I also know that ...
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1answer
79 views

Find a polynomial that has two algebraic numbers as a root

If you have two algebraic numbers $\alpha$ and $\beta$ with two polynomials $u(x)$ and $v(x)$ such that $u(\alpha)=0$ and $v(\beta)=0$ you can, for example, find out a polynomial $q(x)$ that has ...
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How to solve this polynomial problem

$(3x-1)^4=a_4x^4+a_3x^3+a_2x^2+a_1x+a_0$ Value required to be found :- $a_4+3a_3+9a_2+27a_1+81a_0$ I can find the value of $a_4+a_3+a_2+a_1+a_0$.Then I don't know how to continue.Please help.
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1answer
170 views

Maximum volume of a box with a lid that can be made out of a square

Snacks will be provided in a box with a lid (made by removing squares from each corner of a rectangular piece of card and then folding up the sides) You have a piece of cardboard that is 40cm by 40 ...
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1answer
51 views

Why does this equivalence stand?

I am reading the proof of the following theorem: THEOREM A. Let $R$ be an integral domain of characteristic zero; then the diophantine problem for $R[T]$ with coefficients in ...
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19 views

Construct bivariate symmetric (polynomial) nonnegative functions (distributions) over the unit square with certain properties

Construct bivariate symmetric polynomials $f(x,y) = f(y,x) \ge 0$ over $[0,1]^2$, with $f(1,y) = f(x,1)=0$, such that the univariate marginal distributions are both proportional to $$(1-u^2)^4$$, ...
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2answers
38 views

Polynomial must be monotone between its extrema

Suppose that the polynomial function $f(x)=x^n+a_{n-1}x^{n-1}+\cdots +a_0$ has $k_1$ local maximum points and $k_2$ local minimum points. Show that $k_2=k_1+1$ if $n$ is even, and $k_2=k_1$ if $n$ is ...
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33 views

Existence of a splitting ring

Let $R$ be a commutative ring and $f\in R[X]$ be a monic non-constant polynomial. How can one show that there exists a commutative ring $S$ so that $R$ is a subring of $S$ and $f$ can be written as a ...
1
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1answer
17 views

Transform to flatten a parametric curve (polynomial)

Given a polynomial parametrized by $p(t)=(x(t),y(t))$ such that $y(t)=p(t)$, $x(t)=t$, and $p(t)= \sum_{i=0}^na_it^i$, for real coefficients $a_i$, is there some transformation I can take such that ...