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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All my notes together on $\Bbb{Z}_p$-theoretic comp. complexity theory.

Def 1. A $\Bbb{Z_p}^k$-machine is a theoretical computer with $k$ data memory slots and $p$ is a prime number. All the operations on the machine are done one at a time in the ring $\Bbb{Z}_p$. No ...
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Not every polynomial in $\Bbb{Z}_p[x]$ can be factored, but can you do next best?

If $f \in R = \Bbb{Z}_p[x]$ is irreducible or doesn't have many factors then it could be hard to compute? Possibly, I'm not saying, but... any way, what if $f = h - g$ where $h, g$ are heavily ...
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Do there exist polynomials not computable in polynomial time?

Motivation: Computing a problem in $k$ memory slots Do there exist polynomials in $R = \Bbb{Z}_p[z_1, \dots, z_k]$ that can't be computed in time polynomial in $k,p$? Thanks... Good luck! Edit. I ...
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Any problem computable in $k$ memory slots can be computed with polynomials.

Let our memory slots be represented by elements of $\Bbb{Z}_p$ for a prime $p$. $k$ memory slots would be $k$ copies of the ring: $R = (\Bbb{Z}_p)^k$. Suppose that for a problem $f : X \to Y$, ...
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Comparing coefficients in finite field

We start with the wrong proof of the following theorem: $p| \binom{p}{k}$ for a prime $p$ and $0<k<p.$ Proof: $(1+x)^p \equiv 1+x \equiv 1+x^p \pmod{p}$ by Fermat's little theorem. Comparing ...
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Proof that the coefficients of a polynomial are real

How does one prove that all the coefficients of this polynomial: $$(x+i)^{10}+(x-i)^{10}$$ are real numbers, without using Newton's Binomial Theorem?
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Showing $\sin{\frac{\pi}{13}} \cdot \sin{\frac{2\pi}{13}} \cdot \sin{\frac{3\pi}{13}} \cdots \sin{\frac{6\pi}{13}} = \frac{\sqrt{13}}{64}$

I would like to show that $$ \sin{\frac{\pi}{13}} \cdot \sin{\frac{2\pi}{13}} \cdot \sin{\frac{3\pi}{13}} \cdots \sin{\frac{6\pi}{13}} = \frac{\sqrt{13}}{64} $$ I've been working on this for a few ...
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Find the parameter $\alpha$ that …

My question is: For which value of the real parameter $\alpha$ the following equation has a root with the multiplicity higher than $1$. $$3x^4+4x^3-6x^2-12x+\alpha=0$$ $Thank $ $you$ $!!!$
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Minimal polynomial over Q

Let $\omega$ be a primitive 7th root of 1 over $\Bbb Q$ .Let $\alpha= \omega+\omega^6$. Find the minimum polynomial of $\alpha$ over $\Bbb Q$. What I have so far is; $\omega^7=1$ ...
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Invertibility of a Polynomial map.

Given following polynomial map $f:\mathbb{R}^2\to V\subset \mathbb{R}^3 $: $$ (z_1,z_2)\mapsto (2z_1-z_2, 2z_1^2-z_2^2, 2z_1^3-z_2^3) $$ Is this map a bijection? If so, how?
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A polynomial identity

Let $x_1<x_2<\dots<x_n$ be $n$ real numbers. I'm trying to prove the following polynomial identity : $$ P(Y):=1+Y+Y^2+\cdots+Y^{n-1}= \sum_{k=1}^n \prod_{\underset{j\neq k}{1\le j \le n}} ...
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Is there a way to compute if(i < j) k := (a + b)c with polynomials over $\Bbb{Z}_p$?

Let $p$ be a prime and let all variables be in $\Bbb{Z}_p$. Then you can write the result of if(i > 0) k = (a + b)c; (C code) as a polynomial $k := ...
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PSD matrix and non-negative polynomial

So I'm trying to prove that if there exists a $5 \times 5$ matrix $Q$ such that $$Q \succeq0,\,\, a_{l-1} = \sum\limits_{i+j=l} Q_{ij} , l=1,\ldots,5$$ then there exists a fourth degree polynomial ...
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How do I distribute this polynomial expansion?

Ok, so for some reason, I cannot seem to get this simple polynomial multiplication correct no matter how many times I do it. I am working in $\mathbb{Z}/13\mathbb{Z}$. $$ (4x+11)(5x+(3x^2+1)) $$ ...
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Roots of polynomial in $F_3[x]$

Let $\alpha$ be a root of $x^2 + x + 2 = 0$ in $F_3[x]$. I am asked to show that $x^3 + x + 1$ has roots $\alpha$, $\alpha^2$ and $\alpha^4$. I started by observing that $\alpha^2 + \alpha + 2 = 0 ...
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Are these computational models equivalent?

Let $f : X \to Y$ be a problem that you want to compute. Say we have an $O(1)$-computable maps, $\phi, \psi$, such that $X \xrightarrow{\phi} (\Bbb{Z}_n)^k \xrightarrow{\psi} Y$. After all, ...
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Number of roots of a polynomial (Proof)

What might be a simple proof to show that the maximum number of roots of a polynomial is equal to the degree of the polynomial? For example a quadratic polynomial can have a maximum of 2 roots. Can ...
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Uniform convergence of Lagrange polynomials

There is a well-known theorem that states that on a closed interval $[a,b]$ any continuous function is the limit of a uniformly convergent sequence of polynomials. Proofs for this theorem usually ...
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Polynomial identity for a sum

If $$f(x) = \sum_{i=0}^{n}A_i x^i \quad \text{ and } \quad g(x) = \sum_{i=0}^{n}B_i x^i$$ are two degree $n$ polynomials, then we can say that the polynomial $$h(x) = \sum_{k=0}^{2n}C_k x^k \quad ...
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How can i optimize this type of equation?

Given an equation, a polynomial for example, how can i optimize it? see the equation below. $$y = -0.266x^6 + 48.19x^5 - 3424.x^4 + 12170x^3 - (2\times 10^6)x^2 + (2\times 10^7)x - (6\times 10^7)$$ ...
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Limit of a rational function to the power of x

Ok so I have been trying for days already to find a solution to this all around the web and in math books but to no success. The problem is to evaluate a limit of a function composed by polynomial ...
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Why all such polynomials have $-1$ as a root?

Why all polynomials of this form have $-1$ as a root? $ x^5+x^4+x^3+x^2+x+1 $ and similar polynomials like $ x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1$
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Find all polynomials p with real coefficients

Find all polynomials $p$ with real coefficients such that $p(x+1)=p(x)+2x+1$. I feel like in this question you let $x+1=x'$.
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Number of integer roots possible of the following polynomial

Let $p(x)$ be polynomial with integer coefficients, such that $p(0)$ and $p(1)$ are both odd. What is the maximum possible number of integer roots this polynomial can have?
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Addition in $\operatorname{GF}(2^4)$

How can I compute $A(x)+B(x) \mod P(x)$ in $\operatorname{GF}(2^4)$ using the irreducible polynomial $P(x)=x^4+x+1$. What is the influence of the choice of the reduction polynomial on the computation? ...
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Coefficients of even powers

Just a simple thought experiment I was running in my head. Say I have a nonnegative even degree polynomial such as $f(x) = ax^4 + bx^3 + cx^2 + dx + e$. Is it true that the coefficients of the even ...
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Divisibility by $z-z_0$ if $z_0\in \mathbb{C}$ [duplicate]

I have a problem I'm working on, and I'm just not getting it. Suppose that $z_0\in\mathbb{C}$ is fixed. Show that if $P(z)=c(z^k-z_0^k)$, then there exists a polynomial $Q(z)$ such that ...
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Non negativity condition for quartic polynomials?

Say I have a quartic polynomial $f(x) = ax^4 + bx^3 + cx^2 + d$. I am told that $f(x)$ is nonnegative iff it can be expressed as a sum of squares as follows. $f(x) = \sum_{i=1}^4 q_i(x)^2$. As an ...
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How find this system $a^2+b^2=3,a^2+c^2+ac=4,b^2+c^2+\sqrt{3}bc=7$

Find the this system real solution $$\begin{cases} a^2+b^2=3\\ a^2+c^2+ac=4\\ b^2+c^2+\sqrt{3}bc=7 \end{cases}$$ I think that one can use Geometry to solve this system. Maybe there exist an ...
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Polynomials division algebra problem

Find sum of coefficients of the quotient obtained in: $$\frac{2x^n+x^{n-1}+x^{n-2}+...+x^2+x+5}{x-\frac{1}{2}}$$ I got "n" as the answer but according to the book is wrong, I don't know what is ...
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Existence of a det. poly-time algo for problem $f: X \to Y$.

$f : X \to Y$ is a deterministic polynomial-time algorithm for problem inputs $x \in X$ and problem outputs $f(x) = y \in Y \iff $there exists a polynomial $P_f \in \Bbb{Z}[x_1]$ such that $C\cdot ...
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Möbius transformation that permutes roots of a cubic polynomial

The roots of the polynomial $x^3-3x-1$ can be permuted by the function $z\mapsto \dfrac{-1}{1+z}$ which is easily checked by a direct calculation. Is there a simple formula for a Möbius ...
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finding the equation of a polynomial given its graph

I have a graph of polynomial and I would like to know how to determine its equation. Please, this isn't homework. What I'd like to do is actually reproduce this graph. Thanks.
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Splitting a multivariable polynomial into homogeneous components

In Wikipedia's proof of the fundamental theorem of symmetric polynomials, it states that the proof focuses on the case where the polynomial is homogeneous, and that "The general case then follows by ...
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Finding the Remainder

Given the polynomials $$P(x) = nx^n+(n-1)x^{n-1}+(n-2)x^{n-2}+\cdots+x+1$$ and $$Q(x)=x(x-1)^2$$ find the remainder of the division $\dfrac{P (x)}{Q (x)}$.
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Finding real, distinct eigenvalues for arbitrary constants

Let $A= \begin{bmatrix} 1 & 1 & 0 \\ -4 & -3 & 1 \\ k & 0 & 0 \end{bmatrix}$. Find all values of $k$ such that $A$ has three real distinct eigenvalues. I have obtained the ...
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Factors of integers of the form $k^2-k+1$

Factorisation of arbitrary integers is of course a computationally hard problem. But what if the integers I'm interested in factorising are all of the form $k^2-k+1$ ? Is there some way to compute ...
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Too many independent cubic polynomials in an ideal $I\subset \mathbb C[x,y,z]$

Let us consider the ideal $I=(x^2-x,y,xz)\subset \mathbb C[x,y,z]$. I want to prove that $I$ contains (exactly) $5$ linearly independent polynomials of degree $3$. In three variables, we have ...
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Linear Independence of Powers of “roots vector” [duplicate]

Let us be working over the field of complex numbers. Suppose $f(x)= a_n x^n + \cdots +a_1 x + a_0$ is a degree $n$ polynomial with $n$ distinct roots $z_1,\ldots,z_n$. Is the following matrix always ...
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Prove that the equation $1+x+\dfrac{x^2}{2!}+\cdots+\dfrac{x^n}{n!}$ cannot have a multiple root.

Prove that the equation $$1+x+\dfrac{x^2}{2!}+\cdots+\dfrac{x^n}{n!}$$cannot have a multiple root. Using induction and the result that $f(x)=0$ have a root $\alpha$ of multiplicity $r\implies ...
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Solving a nonlinear system of equations.

Given that $x,y,z\in\mathbb R$, solve $$\begin{cases}6x^2-12x=y^3\\6y^2-12y=z^3\\6z^2-12z=x^3\end{cases}$$ I've tried adding the equalities but to no avail. I'd add what I've tried, but it'd be ...
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A Polynomial with square values

Can I find the number of values ​​of the variable X for which the value of the polynomial $100X^2+160X+M$ is a perfect square, depending on M
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dimension of subspace - polynominals evaluated on f

I need to prove that the dimension of the subspace of endomorphisms is less or equal m, if m is the degree of a polynomial p of K[t] \ {0} with p(f) = 0 (f is endomorphism). In a second step I ...
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Find all integers $m$ and positive integers $n > 1$ so that $m + \sum_{k=1}^n x^k/k!$ has a rational root

If $m = 1$, then $m + \sum_{k=1}^n x^k/k!$ has no rational root for $n > 1$. And clearly the polynomial has a rational foot for all integers $m$ if $n = 1$. So, besides those cases, for what ...
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Consider the quadratic equation $ax^2-bx+c=0, a,b,c \in N. $ If the given equation has two distinct real root…

Problem : Consider the quadratic equation $ax^2-bx+c=0, a,b,c \in N. $ If the given equation has two distinct real roots belonging to the interval $(1,2) $ then the minimum possible values of a is ...
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Space of complex poynomials

Let $\mathbb{C}_n[z]$ be the space of polynomials (of degree $\le n$) with complex coefficients, let the inner product be $(p,q):=\int_{-1}^1p(t)\overline{q(t)}dt$. There is one and only one $K_{w} ...
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$n$ degree polynomial $p(k)=\frac{k}{k+1}$ for integral $k=0$ to $n$.

Here's the problem statement: Given an $n$ degree polynomial $p(k)$ such that: $$p(k)=\frac{k}{k+1}$$ for all integer $k$ from $0$ to $n$, determine $p(n+1)$. Any ideas on how to solve it?
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Creating Polynomial

By relative prime factor theorem $$R = (Zm,+,.)$$ where R is the ring structure the input is $e_0 = 0$ and $e_1=1$ output is $$S_0 = { k : \gcd(m,k)>1 }$$ $$S_1 = { k : \gcd(m,k) = 1}$$ Now ...
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Show that $(x-a,x-b)=1$

Knowing that $K$ is a field, $a,b \in K$ different from each other,show that $x-a,x-b$ co-primes. We suppose that $\exists f(x) \in K(x)$ such that: $f(x)|x-a$ and $f(x)|x-b$ Then $\deg f(x) \leq ...
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a question about Jacobi polynomials

Imagine if I have a defined function $\omega(\alpha, \beta, \gamma)$, where $0<\alpha<2\pi$, $0<\beta<\pi$, and $0<\gamma<2\pi$. I can then expand this function into series just like ...