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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Two sets of polynomials with distinct roots build the ring of polynomials.

Definitions: $i \in K$ $U_{i}:=\{f\in K[X] |f(i)=0 \}$ $K[X]$ is the ring of polynomials HINTS: K[X] is a vector space Every $U_{i}$ is a vector subspace of $K[X]$ Question: (i) With $s \neq ...
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Does the equality $\partial^\alpha(x^\alpha)(0)=\alpha!$ hold?

Do we have $\partial^\alpha(x^\beta)(0)=\alpha!=\beta!$ if $\alpha=\beta$ and $0$ else? I tried to proof it on induction, can include my attempts if needed, but they seem to have failed anyway...
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polynomial of $4^\text{th}$ degree, prove

There is a polynomial $f$ of integer coefficients such that $\text{deg(f)} \geq 4$. Let's assume that there are four integers $a,b,c,d$ for which $f(a)=f(b)=f(c)=f(d)=5$. Prove that there is no ...
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Prove that $x-1$ is a factor of $x^n-1$

Prove that $x-1$ is a factor of $x^n-1$. My problem: I already proved it by factor theorem† and by simply dividing them. I need another approach to prove it. Is there any other third ...
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transformation of $y=3(4-x)^3-6$

I am looking for the expansion of $y=3(4-x)^3-6$. I got confused about the $(4-x) $ part. Please help, thanks!
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Polynomial rings, division algorithm

Let $m,n$ be non-negative integers and $m>n$. Find polynomials $g(x),r(x)$ from the ring $R[x]$ such that $x^m -1 =q(x)(x^n-1) + r(x)$ , $r(x)=0$ or $\deg(r(x))<n$. In which case $x^n -1|x^m - ...
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I need help to solve this problem [duplicate]

Let $m,n$ be negative integers and $m>n$. Find polinomials $g(x),r(x)$ from the ring $R[x]$ such that $x^m -1 =q(x)(x^n -1) + r(x)$ , $r(x)=0$ or $deg(r(x))<n$. In which case $x^n ...
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Do monomials' degrees always depend on the whole-number exponent of the variable or whether it's a constant (having a degree of zero)?

Is it true that the monomial $4x^4$ has a degree of $4$ because of the exponent? Also, I think $-2x$ has a degree of $1$ because it has an exponent of $1$ when it's also written like this: $-2x^1$. ...
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Polynomial equation $f(x)f(2x^2)=f(2x^3+x)$

Find all polynomials $f(x)$, for which $f(x)f(2x^2)=f(2x^3+x)$. I have no idea how to do it.
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If all the roots of a polynomial P(z) have negative real parts, prove that all the roots of P'(z) also have negative real parts

If all the roots of a polynomial $P(z)$ have negative real parts, prove that all the roots of the derivative $P'(z)$ also have negative real parts. Could anyone provide a proof for this please?
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Chevalley's theorem proof

I'm trying to prove Chevalley's theorem stating that $$ \text{If } f \in \mathbb{Z}[x_1, \dots, x_n] \text{ is a form of degree } r < n \text{,}$$ $$ \text{then there exists a nonzero solution of ...
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Why is 105th cyclotomic polynomial interesting?

According to wikipedia the 105th cyclomatic polynomial is interesting because 105 is the lowest integer that is the product of three distinct odd prime numbers and this polynomial is the first ...
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Polynom equality modulo p

I found these two equations: (a) $$X^4 + 1 \equiv (X + 1)^4 \mod \ 2$$ (b) $$X^4 + 1 \equiv (X^2 - X - 1)(X^2 + X - 1) \mod \ 3$$ I would like to understand the concept of modulo for Polynoms. How ...
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>Find all pairs of positive integers $(m, n)$, so that $1 + x + x^2 +\ldots+x^m \mid 1 + x^n + x^{2n} +\ldots +x^{mn}$

Find all pairs of positive integers $(m, n)$, so that $1 + x + x^2 +\ldots+x^m \mid 1 + x^n + x^{2n} +\ldots +x^{mn}$ I have to find $(m, n)$ such that ...
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Polynomial $(x − a)^2(x − b)^2 + 1$ is not the product of two polynomials with integral coefficients

Let $a, b$ be integers. Then the polynomial $(x − a)^2(x − b)^2 + 1$ is not the product of two polynomials with integral coefficients. Suppose $(x − a)^2(x − b)^2 + 1 = p(x)q(x)$ then ...
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Irreducible polynomials over the reals

Everybody knows that the degree of irreducible polynomials over the reals is either one or two. Is it possible to prove it without using complex numbers? Or without using fundamental theorem of ...
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Roots less than 1 if at least one coefficient is greater than one

I have this doubt. If you have this equation with $\alpha_i \in \mathbb R$ $$P(z)=1-\alpha_{1}z-\alpha_{2}z^{2}- \cdots - \alpha_{p}z^{p}=0$$ I believe that if there exist an $\alpha$ greater or equal ...
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Prove relations between the roots of 3 quadratic equations

Let $x_1, x_2$ be the roots of the equation $x^2 + ax + bc = 0$, and $x_2, x_3$ the roots of the equation $x^2 + bx + ac = 0$ with $ac \neq bc$. Show that $x_1, x_3$ are the roots of the ...
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Polynomial prove exercise

$P(x)=x^n + a_1x^{n-1} +\dots+a_{n-1}x + 1$ with non-negative coefficients has $n$ real roots. Prove that $P(2)\ge 3n$ I don't have an idea how to do that, I'm in 4th grade high school, you don't have ...
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Factorise the following polynomial

file://localhost/var/folders/0p/frxrkc9d4_z99dy684t4_9100000gn/T/LaTeXiT-2.6.1/latexit-drag.pdf How do you factorise the above?
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Factorization of a Polynomials

Does Mathematical induction work?
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How show that $\max_{x\in [-1,1]}|f'(x)| \le n^2\max_{x\in [-1,1]}|f(x)|$? [on hold]

Let $f(x)$ be a polynomial of degree $n$. How show that $\max_{x\in [-1,1]}|f'(x)| \le n^2\max_{x\in [-1,1]}|f(x)|$?
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Factorise the following polynomial [on hold]

$$x^6+3x^4+4x^2+2$$ How do you factor this polynomial if it has no real solutions?
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What is the remainder when $x^7-12x^5+23x-132$ is divided by $2x-1$? (Hint: Long division need not be used.

What is the remainder when $x^7-12x^5+23x-132$ is divided by $2x-1$? (Hint: Long division need not be used.) The Hint is confusing!
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What is the minimum degree for a curve that has two different points.

I'm having some difficulty solving this problem. The information I have is the following: What is the minimum degree for a curve that has two different points.( 2 different ordered pairs let s say ...
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Prove that$a^2+b^2$ is composite from the information provided.

Suppose $\alpha$,a,b are integers and $b\neq-1$. Show that if $\alpha$ satisfies the equation $x^2+ax+b+1=0$,then prove $a^2+b^2$ is composite. I am starting with this study course of polynomials and ...
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Bernstein approximation [on hold]

As we all know, for some univariate monomial $x^{m}$ defined on the [0,1], we can get its Bernstein approximation of order $d$, which is ...
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Is the taylor polynomial of degree $2$ near $(0,0)$ of $𝑓(𝑥, 𝑦) = \frac{1}{ 2 - (𝑥 + 𝑦^2)}$ the following:

$ P(𝑥, 𝑦) = \frac{1}{2} + \frac{𝑥}{4} + \frac{𝑥^2}{4} + \frac{𝑦^2}{2}$ Is this right? I can't tell, as I can't seem to see the remainder going to $0$ when divided by $x^2 + y^2$ as $(x, y) → ...
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Can $f(g(x))$ be a polynomial?

Let $f(x)$ and $g(x)$ be nonpolynomial real-entire functions. Is it possible that $f(g(x))$ is equal to a polynomial ? edit Some comments : I was thinking about iterations. So for instance ...
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How do I show that the polynomial $f(x) = x^2 + x + 3$ $∈$ $Z_7[x]$ is a primitive polynomial?

I understand that a primitive polynomial is a polynomial that generates all elements of an extension field from a base field. However I am not sure how to apply this definition to answer my question. ...
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Will someone explain this polynomial regression equation?

I am in high school and I need to write a program that does polynomial regression to any degree on a set of data for a personal project. I think that this Wikipedia Article has the equation that I ...
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How to solve 29 coupled quadratic equations?

I have a set of 29 coupled quadratic equations, with 29 unknown variables. Can anyone offer any advice on how I could go about solving this? 3 days of staring at a wall has so far given me no ...
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Can the natural proof of this algebraic identity be simplified?

Let $x^4+c_3x^3+c_2x^2+c_1x+c_0$ be a real polynomial with no real root. Then there are two pairs of conjugate complex roots, $a_1\pm b_1 i$ and $a_2\pm b_2 i$, and one has the identity $$ ...
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Bézout's identity on a polynome sequence

I'm stuck on an exercise which is split in 3 questions : 1) Prove that : $$\exists (U_n, V_n) \in \mathbb{R}[X]^2 \text{ s.t. } (1-X)^{n+1}U_n+X^{n+1}V_n=1$$ 2) Let $(R_n, ...
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Total number of distinct solution produced by polynomial

I have a function $F(x,y) = ax + by$ where $x,y$ belongs to range $[1..10^{10}]$ and $a$ and $b$ are constants, all are integers. How many distinct values can be produced by this function, please give ...
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Curious Binomial Coefficient Identity

Consider the following set of identities: ${m+1\choose 1}={m\choose 1}+1$, ${m+1\choose 2}=2\binom m 2 - {m-1\choose 2}+1$, ${m+1\choose 3}=3\binom m3-3{m-1\choose 3}+{m-2\choose 3}+1$, ... This set ...
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Interpolating polynomials question?

For a function $f$ , the Newton’s interpolatory divided-difference formula gives the interpolating polynomial: $P_{3}(x)=1+4x+4x(x−0.25)+\frac{16}{3}x(x−0.25)(x−0.5)$ , on the nodes $x_{0}=0$ , ...
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On the proof of Fejér-Riesz theorem

I'm having a course about Analytic Number Theory, and I'm having trouble understanding the proof of Fejér-Riesz Theorem: http://people.virginia.edu/~jlr5m/Papers/FejerRiesz.pdf First of all, I didn't ...
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Solving polynomial equation system to find three dimensional location

For an embedded systems project, I need to solve a system of equations. However, my algebraic skills are limited, and I am not able to solve it. This question consists of the following parts. The ...
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polynomial approximation - basic chebyshev question

I was asked to find the best linear approximation to $f(x)=x^2$ in $x \in [0,1]$ using chebyshev polynomials, meaning, using the known property that $2^{1-n}T_n(x)$ is the best approximation to $0$ at ...
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Is $(X^3 - 18X + 12, 5) \in \mathbb{Z}[X]$ a prime ideal?

I'm trying to determine wheter $A = (X^3 - 18X + 12, 5)$ and $B = (X^3 - 18X + 12, X-1)$ is a prime ideal in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. I know that $A = \mathbb{Q}[X]$ since I can make ...
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How do I find the coefficient of $x^2$ in this polynomial, given its value at three points?

We are given the following data about a polynomial $P(x)$ of unknown degree: $P(0) = 2$, $P(1) = −1$, $P(2) = 4$. Determine the coefficient of $x^2$ in $P(x)$ if all the third-order differences are ...
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Elementary proof of the irreducibility of $T^4 - a T - 1$ in $\mathbf{Q}[T]$ when $a\in\mathbf{Z}-\{0\}$

This is from the exercises of Bourbaki, Algèbre, Chapitre V, first exercise of the exercises concerning the second paragraph of the fifth chapter. (p. 140.) As Gauss Lemma ("if your gcd is ...
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Proof $27X^3 - 13X^2 + 180 \in \mathbb{Q}[x]$ is irreducible without Gauss's lemma

I'm asked to show that $27X^3 - 13X^2 + 180 \in \mathbb{Q}[x]$ is irreducible in $\mathbb{F}_{13}[X], \mathbb{Z}[X]$ and $\mathbb{Q}[X]$. I've managed to proof the first two. In $\mathbb{F}_{13}[X]$ I ...
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Polynomial division with modulo

The following polynomials are in the field of whole numbers $mod$ $5$, so $x =5x$. $f(x)=2+4x⁴+4x⁵+x⁶$ $g(x)=3+x+4x²+x³$ $f=q*g+r$ Is there a solution where the degree of $r$ is smaller than the ...
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Is reducing factoring of integers to finding a polynomial which takes a perfect square value useful?

Below we only consider numbers $N=pq$, where $p$,$q$ are primes of the form $ (6j+1)$. It's easy to show that $N + 9n^2 = d^2$. $9k^2$ is related to $(p-q)$ and $d^2$ is related to $(p+q)$. the $n$ in ...
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Non-constant prime polynomial does not exist [closed]

How can I prove that there does not exist a polynomial that only returns primes? The proof on Wikipedia confused me.
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Newton's forward-difference formula question?

Use Newton's forward-difference formula to construct interpolating polynomials of degree two, and three for the following data. Approximate the specified value using each of the polynomials. I ...
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Different ways to prove Fundamental Theorem of Algebra

This is just a curosity .I know some proofs of the fact that Every non constant polynomial with complex coefficient has a complex root via using Liouville's theorem in Complex Analysis.Proof goes as ...
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To find nature of roots of $Ax^{4} + Bx^{3} + Cx^{2} + Dx - E $

To find nature of roots of $$f (x) = Ax^{4} + Bx^{3} + Cx^{2} + Dx - E $$ Where $A, B, C, D, E$ are all positive. After applying Descartes' Rule of signs to $f(x)$ there is one sign change , so ...