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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Divisibility !?? wihout mathematical induction if possible PLZ …

Hello I need to prove if $n\, |\, (x-a)$ and $n \, | \, f(x)$ then $n \, | \, f(a)$. It is true if $\operatorname{deg}(f)=1$ or $2$, but what for greater degree of $f(x)$? I don't know how to ...
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To make a polynomial with coefficients in a finite field uniform at random

We define the polynomial ring $R[x]$ consist of all polynomial with coefficients from $\mathbb Z_p$. Let $P_1$ be a polynomial such that $P_1 \in R[x]$. The aim is to make $P_1$ uniformly at random. ...
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What does a complex root signify?

What does it tell me when I find that a polynomial has complex roots, except for the obvious fact that it crosses zero for these values? To me it seems that the existance of complex roots must have ...
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561 views

Relative Maxima/Minima of polynomial functions

I am taking the Pre Calculus 12 course online. I came across this concept that the online material teaches in 3 different ways, and each one contradicts the other. I find this extremely frustrating. ...
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Is there a general formula for solving 4th degree equations?

There is a general formula for solving quadratic equations, namely the Quadratic Formula. For third degree equations of the form $ax^3+bx^2+cx+d=0$, there is a set of thee equations: one for each ...
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Increasing Function or Polynomial with Prescribed Values

Consider $n$ points $(a_1,b_1), (a_2,b_2),\cdots, (a_n,b_n)$ in Euclidean plane with $a_1<a_2<\cdots < a_n$ and $b_1<b_2<\cdots < b_n$. It is easy to construct a polynomial of degree ...
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Play with a,b,c polynomial [on hold]

I want to get $a^4-b^4-c^4 +2(ab)^2 + 2(ac)^2 +2(bc)^2 = (...)*(...)$ Thank you in Advance !
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Roots of the Lagrange polynomials

This question follows my previous one Coefficients of Lagrange polynomials. Notations : $ n\in\mathbb{N}^*$ $[|1,n|]=\{1,2,\dots,n\}$ $A=(a_1,\dots,a_n)\in\mathbb{K}[X]^n$ all different numbers ...
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35 views

Special representation of polynomial

How to prove that for natural $n$ the polynomial $(x^4-6x^2+1)^n$ can't be represented in such a way $$ (x^4-6x^2+1)^n=f(x)^2+1, (x^4-6x^2+1)^n=g(x)^3-1, $$ where$f(x), g(x)$ are polynomials. ...
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34 views

Using the pseudoinverse to find the linear combination of functions?

I'm working out this problem with a friend of mine on a group project and we are both stuck Our professor insists that we do all of our work in Maple. I like Maple, but it's not as great as ...
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61 views

Prove that the two polynomials intersect each other only at a single point

Here are the polynomials: $$D^K_1(\theta)=\sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}\theta^i(1-\theta)^{K-i}$$ and $$D^K_2(\theta)=\frac{1}{2}\sum_{i=\lceil{K/2}\rceil}^K ...
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Polynomial Irreducibility Test [on hold]

Is there any polynomial with constant c=8 that make this polynomial reducible over field Q but Eisensten Irreducibility Test does not apply?
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38 views

Uniformly at random polynomial

We have a polynomial of degree $d$, and multiply it by a polynomial whose coefficients are chosen uniformly at random and its degree is equal to or less than $d$. My question is whether the result is ...
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320 views

Approximate a polynomial function using a sum of sine waves

I have a polynomial function which I need to approximate by a sum of sine waves with constant amplitude along a given domain. From what I hear, this might be a good time to make use of Fourier ...
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193 views

Coefficients of Lagrange polynomials

Let $n\in\mathbb{N}^*,A=(a_1,\dots,a_n)\in\mathbb{K}[X]^n$ all different numbers and $B=(b_1,...,b_n)\in\mathbb{K}[X]^n$ all different numbers. Let $L_{A,B}$ be the polynomial of degree $n-1$ ...
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271 views

Krylov-like method for solving systems of polynomials?

To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods ...
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551 views

Irreducibility of Polynomials in $k[x,y]$

I'm working through some Hartshorne problems and have noticed that in order to do certain problems properly one must prove a given polynomial $f\in k[x,y]$ is irreducible. For example, in problem ...
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dimension of quotient space

Let $f(x)=x^4+3x^3-x^2-4x-3$ and $g(x)=3x^3+10x^2+2x-3$ and $U = \{u(x)f(x)+v(x)g(x) | u(x),v(x) \in \mathbb{F}[x]\}$, find the dimension of quotient space $\mathbb{F}[x]/U$ If $V$ is a finite ...
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Is the identification between symmetric tensors and homogeneous polynomials useful?

The general question: Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification $$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$ between the space of symmetric order ...
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Location of Roots Symmetric Polynomial

I'm trying to prove (or disprove) that the roots of an even degree real symmetric coefficient polynomial are all on the unit circle. If it is not true, I will then try to find the conditions such that ...
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proportion of primes in a polynomial sequence

It is conjectured (Bunyakovsky) that when $P(x)$ is a polynomial from $\mathbb{Z}[X]$, irreducible, with positive leading coefficient and so that the integers $P(n)$ , $n\gt0$ do not share a common ...
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41 views

Roots of $x^{4} -28 x^{2}+49$ with Horner

I am studying Horner's algorithm and I got a problem I can't solve. The polynomal is $x^{4} -28 x^{2}+49$. After trying $\pm 1, \pm 7, \pm49$ with Horner I couldn't find any solution. Wolfram alpha ...
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Number of (distinct) roots of derivative of polynomial

Let $f(x) = (x-a)(x-b)^3(x-c)^5(x-d)^7$, where $a,b,c,d \in \mathbb{R}$ and $a<b<c<d$. Thus $f(x)$ has 16 real roots counting multiplicities and among them 4 are distinct from each other. ...
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Find the size of squares cut from a box.?

This has been taking me days to do and I really want to do it for test practice. I actually have absolutely no idea how to even start this, so if I can get a hint, advice, or something to start me ...
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Solution to Quartic, Pentic, Hexic and Sietic Polynomials? [on hold]

Is there a mistake in this article: Solution to Quartic, Pentic, Hexic and Sietic Polynomials and isn't it in contradiction with Galois theory?
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SAGE: Is it possible to extract the irreducible factor of a polynomial for the purpose of constructing a Number Field?

I'm in the middle of making a program that tests a certain fact for many number fields. At this current step I get say a hundred polynomials, which are reducible. I want to factor them (over Q), take ...
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140 views

Polynomial satisfying $ P (P (x))=P (x)+ P(x*x)$

If $P(x)$ is a polynomial with integer coefficients such that for all integer $x$, $$P (P (x)) = P (x)+P (x*x).$$ I've tried solving it putting it as a function instead. Not much though. How do you ...
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37 views

If $\deg(f) > p^k$ then $f$ as an irreducible divisor of degree $> k$

Let $p$ be prime and let $f \in \mathbb{F}_p[X]$ with no repeated roots. Let $k \in \mathbb{N}^*$ such that $\deg(f) > p^k$. Show that $f$ has an irreducible divisor of degree $> k$. My ...
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Basis-free and noncommutative versions of the two-polynomials-over-ring problem (McCoy theorem etc.)

There is a rather canonical bunch of exercises in commutative algebra which tend to come up time and again on math.stackexchange: recently in #948010 and #83121, formerly in #227787 and #413788, and ...
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Factorizing Cubic Equations.

Factorization of Cubic Equations has always obstructed my way to the solution to a problem. Is there any simple technique to factorize them?
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Zero divisor in $R[x]$

Let $R$ be commutative ring with no (nonzero) nilpotent elements. If $f(x) = a_0+a_1x+\cdots+a_nx^n$ in $R[x]$ is a zero divisor, how do I show there's an element $b \ne 0$ in $R$ such that ...
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How can I simplify the polynomial $x^4+1$ into quadratic factors? [on hold]

The teacher gave us a hint that this polynomial expression can be written as the multiplication or sum of quadratic factors at the most. How can I do this?
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Show that there exists no integer b such that f(b) is 1993.

We are given a polynomial $f$ with integer coefficients such that for 4 distinct integers $a_1,a_2,a_3$ and $ a_4$, $f(a_1)=f(a_2)=f(a_3)=f(a_4)=1991$. Show that there exists no integer $b$ such that ...
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A Polynomial that Passes through the following four points?

I'm trying to do this for practice but I'm just going nowhere with it, I'd love to see some work and answers on it. Thanks :) Find a polynomial that passes through the points (-2,-1), (-1,7), ...
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257 views

Wolfram Mathematica - Newton Backward Interpolation?

I have the following task: Create a function (in Wolfram Mathematica), called $\mathrm{NewtonBackward}$[n_,x0_,h_,f_] which interpolates backwards the function $f(x)$ with nodes {x_i = x_0 + ...
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Minimal polynomial of f restricted to its image

Let $f:V\to V$ be a $F$-linear map, $V$ an $n$-dimensional vector space over $F$, $\operatorname{rank} E=r$, $W=\operatorname{Im} f$, $\tilde f:=f|_W:W\to W$. Let $\mu$ be the minimal polynomial of ...
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Secure Computation and polynomial evaluation

Consider we evaluate a polynomial P of degree d on some points (say 2d+1 points) to obtain Y's. My questions are: A) Given 2d+1 (or more) Y's can anybody 1) recover the original polynomial 2) ...
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Finding roots of characteristic polynomial of 3x3 matrix

I have never learned how to solve cubic equations and unfortunately need to do it in an upcoming exam for finding eigenvalues. I have been searching on the web for good resources, but whenever I find ...
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Irreducible polynomial over an algebraically closed field

Suppose $k$ is an algebraically closed field and $p(x,y)\in k[x,y]$ is an irreducible polynomial. Prove that there are only finite many $a\in k$ such that $p(x,y)+a$ is reducible, i.e. the set ...
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The min degree of polynomials of two variables with a special form

Let $f(x)$, $u(x,y)$ and $v(x,y)$ be non-constant polynomials over complex number field $\mathbb{C}$. Assume that $u(x,y)$ is not a polynomial only on $y$, and $v(x,y)$ is not a polynomials of only ...
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How do generating function created from solution of system of polynomials

What are the examples of generating function derived from solution of system of polynomials? how to count the number of points in varieties which are solution of system of polynomials? From Wiki, ...
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Question about multiple solutions to a polynomial

Assume that $f(X,Y,Z,V,W)\in \mathbb{Z}[X,Y,Z,V,W]$ is some polynomial and assume that $f(x,y,z,v,w)=0$. I would like to know if there is some way to figure out if there are non-trivial constants in ...
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How to solve an nth degree polynomial equation

The typical approach of solving a quadratic equation is to solve for the roots $$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$$ Here, the degree of x is given to be 2 However, I was wondering on how to solve ...
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$P(x)=x^5+a_4x^4+\cdots+a_0$ has roots $1,2,3,4$ and $k$. Find $P(5) -P(0)$.

A polynomial $P(x)$ with leading coefficient $1$ is of degree $5$, and its distinct roots are $1, 2, 3, 4$ and $k$. Find the value of $P(5) -P(0)$. I have no clue on what my initial steps should be.
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0-1 roots in a free algebra

Let $\mathbb{F}$ be a field and consider the free algebra $\mathbb{F}\langle x_1,\ldots, x_n \rangle $, that is, the algebra of non-commutative polynomials with coefficients from $\mathbb{F}$. Let ...
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46 views

How to solve the quadratic matrix equation

Given $\mathbf{A}$ and $\mathbf{B}$ two $m \times n$ real matrices, is there a closed form for the matrix equation \begin{equation} \|\mathbf{X}\|^{2}_{F} - 2\cdot trace(\mathbf{X}^T\mathbf{A}) ...
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LCM of two polynmials when they are represented as point-value.

I`m wondering if we can obtain least common multiple of two polynomial when each polynomial represented as point-value. To be more clear, can we do any computation on these point-values and obtain ...
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Polynomial divisibility question

Let $f_n(a)$ be a polynomial of degree $n-1$ with integer coefficients, such that $f_n(a) > 0$ when $a > 0$, and in fact $f_n(a)$ is monotonically increasing with $a$. If there exists an ...
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Irrational roots of unity?

Is it possible to take irrational roots of unity? For example, say I wanted to solve $f(x)=(x+1)^{\sqrt{2}}=1$. I found that one solution is the obvious $x=0$, and another one can be written nicely as ...
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Degree of the zero polynomial

In “Linear Algebra Done Right” by Axler, while defining the degree of a polynomial, it is stated that the zero polynomial is said to have degree $- \infty$. Why is this so?