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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How many expansion methods exist in math?

For now I know about polynomial expansion and fractional expansion, but what other methods exist that I can use to rewrite and maybe simplify an algebraic expressions ? Is there something strictly ...
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Does $f^{(n)} = 0$ imply that complex $f$ is a polynomial?

Let $f$ be a complex function with the property that $f^{(n)} = 0$. Does this imply that $f$ is a polynomial? If so, why? Upon thinking about this problem myself, I can easily observe that every ...
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Matrix Representation of a Polynomial Linear Operator

Sorry to ask a similar question, but: $$T: P_2(R) \to P_2(R)$$ defined by $$T(f(x))= (x+1)f'(x)$$ to the standard basis $$b=\{1,x,x^2\}$$ I calculated: $$T(1)= (1+1)f'(1)= (2)(0)= 0 \implies 0·1 + ...
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Matrix Transformation of a Polynomial (Linear Algebra)

I'm having trouble understanding matrix representations of polynomials. The question in particular is: T: P2 (R)-> P2(R) defined by T(f(x))= f"(x) +2f'(x) - f(x). I know for this question we have to ...
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If a polynomial $g$ divides $f$ and $f'$, then $g^2$ divides $f$?

Here's a homework problem from Artin's Algebra that I'm having a lot of trouble with Let $f(x) \in F[x]$ (where $F$ is a field of characteristic $0$). If $g$ is an irreducible polynomial that is ...
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Let $f(x)$ and $g(x)$ be two monic polynomials of the same degree such that adding $1$ to the roots of $f(x)$ we get the roots of $g(x)$.

Let $f(x)$ and $g(x)$ be two monic polynomials of the same degree such that adding $1$ to the roots of $f(x)$ we get the roots of $g(x)$. Then does their any relations between the constant term of ...
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Restrictions on the coefficients that allow a polynomial in a field of characteristic 0 to be solvable by radicals and the associated formula.

We know that a general polynomial $p(x) \in \mathcal{F} \left[ x \right]$, $\deg{ p } = n$, (char(${\mathcal{F}}) = 0$) is not solvable by radicals if $n \geq 5$. However, I was wondering what ...
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What is this sequence of polynomials?

NovaDenizen says the polynomial sequence i wanted to know about has these two recurrence relations (1) $p_n(x+1) = \sum_{i=0}^{n} (x+1)^{n-i}p_i(x)$ (2) $p_{n+1}(x) = \sum_{i=1}^{x} ip_n(i)$ == i ...
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How to prove this polynomial expression.

Let the polynomial be in $f$ be a map from $\Bbb{Z}_2^k \to \Bbb{Z}_2$, defined by $f = 1 + \sum_{i=1}^k x_i + \sum_{i\neq j; i,j = 1}^k x_i x_j + \dots + x_1 x_2 \cdots x_k$ Then I want to show ...
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Can we introduce new operations that make quintics solvable?

I have heard from various sources that the typical arithmetic operations (addition, subtraction, multiplication, division, rational exponentiation) are not sufficient to express in general the roots ...
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What are some algorithms that can be used to test if a number is transcendental or not?

Well according to the definition of transcendental numbers I find that its any number that doesn't have any polynomial equation of any degree with integer coefficients summing up to 0. So ...
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1answer
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What is a fast method for evaluate this trigonometric series?

$$\sum_{n=1}^{11}\sin^{14}\left(\theta+\frac{2n\pi}{11}\right)=?$$ By wolfram alpha, we know that ...
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Properties of smallest expressions for polynomials, and potential proof.

See here for an intro. Smallest expressions for polynomials is analogous to smallest grammars for strings. Let $R = \Bbb{Z}_p[x_1, \dots, x_k]$. My goal is to prove that for any $\ \ h(k,p) = \max ...
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Writing in Lag form and finding the characteristic polynomial, MA(2) with constant

I'm wondering how to write an MA(2) model with a constant in lag form such that I can calculate the characteristic polynomial and get the roots (to see if it's stable). The model is given by $Y_t = ...
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Finding roots of a fractional exponential equation.

If we consider a polynomial equation its easy to find the number of roots associated with the expression by applying Descartes Rule. This method, however, doesn't work with non integer exponents. ...
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Minimal polynomial: is $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}=1$?

I was wondering about the minimal polynomial of real number $$u=\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$$ over field $\mathbb{Q}$. As you can see here, I worked out that $u$ is a root of monic ...
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How to solve this third degree polynomial?

Can you explain me how to solve this kind of polynomial? $$x^3 - 3x^2 = 320$$
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Find the roots of the following polynomial equation..

how would you solve this exercise: Find the solutions of the following equation knowing that one of these solutions belongs to $R$: $$x^3+(3i-2)x^2-(1+4i)x+2+i=0$$ I used the condition set in the ...
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3answers
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Euclidean algorithm in the ring of polynomials over a field

I need some help with the following division proofs. I suppose my biggest problem is not being able to visualize the algebra for one GCD equaling another GCD. I'm not sure of how to arrange the ...
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Criterion to decide the invertibility of polynomial maps

Consider a polynomial map $f:\mathbb{R}^{n-1}\to V\subset\mathbb{R}^n$ where $V$ is $n-1$-dimensional variety in $\mathbb{R}^n$. Are there any conditions on $f$ to determine whether it defines ...
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Can I modify a polynomial to return only multiples of a given number?

I'm attempting to create a polynomial equation for a project of mine, with a shape similar to the following: $${3x^5\over500}+{x^4\over25}+x^3+40 x^2+100 x$$ However, one of my goals is to have the ...
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Prove that a certain sequence of polynomials is symmetric

Given $p_0(x,y,z)=1$, $p_{n+1}(x,y,z)=(xy+yz+zx)p_n(x,y,z+1)+z^2(p_n(x,y,z+1)-p_n(x,y,z))$. Prove that all $p_n(x,y,z)$ are symmetric polynomials.
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Are there known patterns among minimal expressions?

Let $R = F[z_1, z_2, \dots]$ be the finite-degree polynomials in a countable number of variables. Let $\mathcal{E}(R)$ be the set of all expressions of polynomials. Note that there could be an ...
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Necessary and sufficient conditions for $\rm P \neq NP$ maybe?

Please review the $\rm P \neq NP$ problem here. I'm working on an algebraic approach to this problem, and all my notes are currently here. Conjecture 1 For all $f \in F[x_1, \dots, x_k]$, a minimal ...
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Show that $(x + 1)^{2n + 1} + x^{n + 2}$ can be divided by x^2 + x + 1 without remainder

I am in my pre-academic year. We recently studied the Remainder sentence (at least that's what I think it translates) which states that any polynomial can be written as P = Q*L + R I am unable to ...
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An exercise regarding polynomials

I guess it is a simple exercise though I'm not very good at polynomials. It asks: Find $m,n,p,q$ natural numbers such that the polynomial $X^m+X^n+X^p+X^q$ is divisible by $x^3+x^2+x+1$. Thank you in ...
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Birational Variety

Given a polynomial map $f:\mathbb{R}^2\to V\subset \mathbb{R}^3 $ defined as follows: $$ (z_1,z_2)\mapsto (2z_1-z_2, 2z_1^2-z_2^2, 2z_1^3-z_2^3) $$ This map defines a Variety ($V$) of dimension $2$ in ...
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Irreducibility of a Polynomial over Q

How do I show that for any odd prime $p$ the polynomial $f(x)=x^p-9$ is irreducible over $\mathbb Q$?
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Number of zeros of $ z^7+4z^4+z^3+1$

How many zeros does $z^7+4z^4+z^3+1$ have in each of the regions |z|<1 and |z|<2? I know I should use Rouche's Theorem but I can't find a $|f(z)| > |p(z)-f(z)|$ and $f(z)$ have equal number ...
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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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1answer
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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 (I found this identity in an undergrad book): $$ P(Y):=1+Y+Y^2+\cdots+Y^{n-1}= \sum_{k=1}^n ...
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Find three numbers given their sum, product and sum of their squares

Given three unknown positive integers. Is it possible to find the three numbers if we are given their Sum->(a+b+c) = X Product-> (abc) = Y Sum of Squares-> (a^2 + b^2 + c^2) = Z
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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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1answer
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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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1answer
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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 ...