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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A question from the mod p irreducibility test's proof

Let $p$ be a prime an suppose that $f(x) \in \mathbb Z[x]$ with $\deg f(x) \geq 1$. Let $f_1(x)$ be the polynomial in $\mathbb Z_p[x]$ obtained from $f(x)$ by reducing all the coefficients of $f(x)$ ...
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What basis and coordinate system is used in this quadratic Bézier triangle equation? $[x,y,z] = A*s^2 + B*t^2 + C*u^2 + D*2st + E*2tu + F*2su$

I have the following equation for a quadratic Bézier triangle, but I'm having a lot of trouble understanding how to describe it: $[x,y,z] = A*s^2 + B*t^2 + C*u^2 + D*2st + E*2tu + F*2su$ ...
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Multivariate Polynomials Sage

Sorry if I'm in the wrong Stackexchange (but sage is a math program...) I'm computing something on multivariate polynomials: I have a primary variable $x$ and several other variables $a, b, c, ...
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If $a^2=b^2+c^2$ and $0<n<2$ prove $a^n<b^n+c^n$

If $a^2=b^2+c^2$ and $a,b,c$ are positive real numbers, prove (a) if $n>2$ then $a^n>b^n+c^n$, (b) if $0<n<2$ then $a^n<b^n+c^n$. Part (a) was easy to prove: $a^2=b^2+c^2$ and ...
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Linear Equation as matrix

Using a series of 3x3 matrices multiplied together, it is possible to create a matrix which will rotate, translate, scale and invert a size 2 vector. Using a 4x4, it is possible to do this to a size ...
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Find a basis and state its dimension of a $C$-vector space polynomial.

The $C$ vector space $V$ of polynomials $P(t) \in C[t]$ of degree at most $n$ and such that $P(a) = P'(a) = 0$ for $a \in C$ fixed. Indication : prove that $P(t) \in V \Leftrightarrow (t − a)^2$ ...
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Changing the order of the elements of the divided difference Polynomial Interpolation

Apparently this is rather trivial but I don't understand why what I've highlighted in green is correct.
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110 views

roots of a polynomial inside a circle

I am asked to show that for $n$ larger or equal to $2,$ the roots of $1 + z + z^{n}$ lie inside the circle $\|z\| = 1 + \frac{1}{n-1}$ Attempt1: Induction for the case $n = 2,$ the roots of $1 + z + ...
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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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How do you solve the coefficients of functions of two variables as part of 2nd order polynomial?

I'm having a major issue in trying to get my head around creating a formula for the graph ΔT=f(V,W) I already have two graphs for that, but they are 2D graphs. Both represent ΔT as Y but one graph ...
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Primitive-recursive functions and polynomial equations

I am looking for examples of primitive-recursive functions $f:\mathbb{N}\rightarrow\mathbb{N}$ that can not be written as a pair of polynomials, i.e. $$f(n) = m \Leftrightarrow P(n,m) = Q(n,m)$$ ...
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How find the polynomial whose roots are $\frac{{a^2 }} {{2a^2 + bc}},\frac{{b^2 }} {{2b^2 + ca}},\frac{{c^2 }} {{2c^2 + ab}}$?

a,b,c are the roots of the polynomial $x^3 - (a + b + c)x^2 + (ab + bc + ca)x - abc$. How find the polynomial whose roots are $\frac{{a^2 }} {{2a^2 + bc}},\frac{{b^2 }} {{2b^2 + ca}},\frac{{c^2 }} ...
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Is $2x^2+4$ reducible over $\mathbb C$?

I am not sure if I making some very fundamental mistake. But Gallian says that $2x^2+4$ is reducible over $\mathbb C$. If $D$ be an integral domain. A polynomial $f(x)$ from $D[x]$ is said to be ...
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About the subspace of polynomial vector space

Why the set of functions in $C\left [ 1,-1 \right ]$ such that $f\left ( -1 \right )= f\left ( 1 \right )$ is the subspace of $C\left [ 1,-1 \right ]$?
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Modulo Quadratic Polynomials

Can you, given a large number N, find a, b, c such that ax^2 + bx + c = 0 has at least N roots? All of this is in any mod you choose.
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Is there a reason for some polynomial quotients to have a remainder equals to zero?

I was helping some highschool students with factorization exercises. They had alternatives to choose the correct factor. Then one of them said to me: We use a calculator and evaluate some prime ...
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Algebra problem about spam and polynomials

How do you know when a set of polynomials span r2 For example, P1 = x+ 5x^2+x^3 and P2 is the same. What constitutes a set to be in the span or a specific space?
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Problems from Polynomial Rings. My attempt shown.

$1.$ Let $f(x) =a_mx^m+a_{m-1}x^{m-1}+ \cdots +a_0$ and $g(x) = b_nx^n+b_{n-1}x^{n-1}+ \cdots +b_0$ belong to $\mathbb Q[x]$ and suppose that $f \circ g \in \mathbb Z[x].$ Prove that $a_ib_j$ is an ...
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Finding a polynomial's constant from its points

Let's say I was given a set of $d+1$ distinct points known to be from a polynomial $P$ of degree $d$. So: $$P = a_dx^d + a_{d-1}x^{d-1} + ... a_1x + c$$ And I have pairs $(x_i, y_i)$ such that: ...
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Roots Of Monic Cubic

I'm currently preparing for the USA Mathematical Talent Search competition. I've been brushing up my proof-writing skills for several weeks now, but one area that I have not been formally taught about ...
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Prove that no polynomial function represents a given sequence

I recently encountered the following question: How to find the $n$th term of the sequence $2,3,6,7,14,15,30,\dots$? I replied to that post, and gave the following answer: $S_n = ...
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Polynomial division problem

Let $f(x) = x^{10}+5x^9-8x^8+7x^7-x^6-12x^5+4x^4-8x^3+12x^2-5x-5. $ Without using long division (which would be horribly nasty!), find the remainder when $f(x)$ is divided by $x^2-1$. I'm not sure ...
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Maps preserving roots of a polynomial function over finite fields

Let $P(x_{1},\ldots,x_{n}):\mathbb{F}_{2}^{n}\rightarrow \mathbb{F}_{2}$ be a polynomial function with degree $d$ and with variables $x_{1},\ldots,x_{n} \in \mathbb{F}_{2}$. Let $S(P)=\{ ...
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23 views

Not understanding steps in Algebraic simplification

The simplification in question is that the expression goes from $(4-x)(6-x)(3-x)-8(3-x)=0$, to $(3-x)(8-x)(2-x)=0$ I don't understand how one goes from the first expression to the second. I ...
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42 views

Proof of subspace, finding a basis in polynomials

Let W = {(f(x)∈ P2[R] : f '(x) + xf(0) = 0} i) Prove that W is a subspace of P2[R]. ii) Find a basis for W. Here's what I have so far: i) I have to verify that ...
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If $F$ be a field, then $F[x]$ is a principal ideal domain. Does $F$ have to be necessarily a field?

If $F$ be a field, then $F[x]$ is a principal ideal domain. Does $F$ have to be necessarily a field? My Thoughts: Suppose instead of $F$, we take the set of polynomials $R[x]$ over a commutative ring ...
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Polynomial joint pdf $f(x,y)$ such that of $f(x) \neq f(y)$

How can I build a polynomial joint pdf $f(x,y)$ for $x \in [x_1, x_2]$ and $y \in [y_1, y_2]$ such that of $f(x) \neq f(y)$ or equivalently, $x$ and $y$ are depended on each other?
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Mistake in a question in Fulton's algebraic curves book?

I'm trying to solve this question in Fulton's book Algebraic Curves: I don't think this is true. Counter-example: $k=\mathbb R$, $n=1$ and $F=X_1^2+1$. Thanks
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Sum of Non Real Roots of Bi Quadratic

Consider $$f(x)=8x^4-16x^3+16x^2-8x+k=0$$ where $k \in \mathbb{R}$,then find sum of non Real roots of f(x). My approach: we have $$f'(x)=32x^3-48x^2+32x-8=0$$ Also ...
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Existence of Irreducible polynomials over Z of any given degree which do not satisfy the Eisenstein's Criterion

I just came across the following interesting question which has been once discussed: Existence of Irreducible polynomials over $\mathbb{Z}$ of any given degree I was wondering if we could find such ...
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find a polynomial whose roots are inverse of squares of roots of $x^3+px+q$

Question is : Given a polynomial $f(x)=x^3+px+q\in \mathbb{Q}[x]$ find a polynomial whose roots are inverse of sqares of roots of $f(x)$ Supposing $a,b,c$ as roots of $f(x)$ we have : $a+b+c=0$ ...
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Division of Polynomial using Euclid's lemma

Find the quotient and remainder when $p(x)=x^3+4x^2-6x+2$ is divided by $g(x)=x-3$ without actually dividing or by long division method or by synthetic division methods. I used Euclid's lemma ...
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242 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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Irreducibility of some multivariate polynomials

Consider the polynomials $xw-yz\in A[x,y,z,w]$ and $x^n+y^n+z^n\in A[x,y,z]$, where $A$ is a commutative ring. I am curious to know what conditions on $A$ (factorial ring, algebraically closed field, ...
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Euclid's proof of the infinitude of primes to prove this question

I'm trying to prove that if $k$ is a field, then there are an infinite number of irreducible monic polynomials in $k[X]$. My attempt of solution is use almost the same strategy of the Euclid's proof ...
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Adding monomials of different degree.

Can you prove that x^m + x^n can never equal x^k, where k is some rational number, and m is not equal to n. I know we've all been doing it since middle school, but is there a mathematical way of ...
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Newton-Cotes Quadrature formula

Im trying to find more information about numerical integration methods. When is a Newton-Cotes Quadrature formula on n nodes exact?
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Projective roots of a homogeneous polynomial

Suppose that $f(X,Y)\in\mathbb C[X,Y]$ is a homogeneous polynomial of degree $n$, then we can consider it as a function on $\mathbb P^1_\mathbb C$. It has at most $n+1$ projective roots (points of ...
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Form of a function I can fit like a polynomial that has an asymptote on x-axis and is always positive

So I have a small list of pairs of the form $(\mathbb{R}, \mathbb{R}_{\geq 0})$ and I want to fit a function to this data. Additionally I know that as $x$ grows large in either direction $y$ tends to ...
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Convert segment of parabola to quadratic bezier curve

How do I convert a segment of parabola to a cubic Bezier curve? The parabola segment is given as a polynomial with two x values for the edges. My target is to convert a quadratic piecewise ...
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weighted inner product of polynomials, can weight function be complex?

I am just learning about inner-products on polynomial space, where the coefficients of the polynomials may be complex: $P_m(\mathbf{F})$ The inner-product given by: $\langle p,q \rangle = \int_0^1 ...
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Polynomial of degree inequality

Let $ P(x,y)\ge 0 $ for all $ x,y $ be a polynomial of degree n such that $ P(x,y)=0 $ only for $ x=y=0 $. Does there exist a constant $ C>0 $ such that following inequality $ ...
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Semialgebraic conditions that convey properties of Galois group

Let $f \in \mathbb{Z}[x]$ be a polynomial of degree $n$ with integer coefficients and let $G_f$ be the Galois group of $f$ over $\mathbb{Q}$. I am trying to collect results that convey some ...
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Find the basis for the subspace of the set of polynomials of degree less than five?

Let U = {p $\in P_4(F): p(2) = p(5) = p(6)$. Find a basis for U. I know how to do this problem if I were given p(2) = p(5). Set the two equal to each other and solve for one of the coefficients. I ...
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Lagrange Interpolation Theorem?

The polynomials $p(x) = 5x^3 - 27x^2 + 45x - 21$ and $q(x) = x^4 - 5x^3 + 8x^2 - 5x + 3$ both interpolate the points $(1,2) , (2,1) , (3,6), (4,47)$. Even though these polynomials are of different ...
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prove $(a+b+c)^n=a^n+b^n+c^n$ if $(a+b+c)^3=a^3+b^3+c^3$

if $(a+b+c)^3=a^3+b^3+c^3$ and n is odd number,prove that: $$(a+b+c)^n=a^n+b^n+c^n$$ hint of the question was: factor this expression $f(a,b,c)=(a+b+c)^3-(a^3+b^3+c^3)$ after factorization ...
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Quartic Polynomial Manipulation

I have a quartic polynomial in $x$ (too long to write here) $f(x,c_1, c_2, c_2)$ where $c_1, c_2, c_3$ are constants which I have complete freedom over how to fix their values, as long as they are ...
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163 views

How can this equality be established by elementary algebraic means?

Let $x \geq 1$. Then is it true that $2x^3 - 3x^2 + 2 \geq 1$? If so, how can I show this using only elementary ideas such as factorisation? Of course, I can demonstrate this using the methods of ...
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Factor Cyclic Polynomial

Factor $(a+b)(b+c)(c+a)+abc$. I know this is a cyclic polynomial, but I don't know how to solve problems like this. What should I do?
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Determine $P_2 = f(0.7)$ when Neville's method is used to approximate $f(0.5)$

Let $f(x) = \ln(x + 1)$. Neville's method is used to approximate $f(0.5)$, giving the following table. $$x_0 = 0 - P_0 = 0$$ $$x_1 = 0.4 - P_1 = 2/8 - P_{0,1} = 3/5$$ $$x_2 = 0.7 - P_{2=?}- ...