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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Similarity of polynomials

I'm looking for a method that takes two simple polynomials (cubic) and gives a value of how visually-similar they are to each other on some specified domain. How could I do this?
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158 views

Justify $\gcd$ of $f(x) = x^3 - 6x^2 + x + 4$ and $g(x) = x^5 - 6x +1$

Let $f(x) = x^3 - 6x^2 + x + 4$ and $g(x) = x^5 - 6x +1$. Using Euclidean algorithm I find $\gcd[f(x), g(x)] = 1$. How could I JUSTIFY that $h(x) = 1$ is the ACTUAL $\gcd$ of $f(x)$ and $g(x)$? ...
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Find the greatest common divisor (gcd) of $f(x) = x^2 + 1$ and $g(x) = x^6 + x^3 + x + 1$

Find the greatest common divisor (gcd) of $f(x) = x^2 + 1$ and $g(x) = x^6 + x^3 + x + 1$. Since $x^6 + x^3 + x + 1 = (x^2 + 1)(x^4 - x^2 + x + 1)$, $\mathrm{gcd}[f(x),g(x)] = x^2 + 1$. My question ...
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4answers
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Galois field and polynomials?

Show there are only two polynomials of degree 3 over $\mathbb{F}_2$ such that it is irreducible and all other degree 3 polynomials can be reduced. So $x^2 = x$ and $x = -x$ I cant think of anything ...
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$f \in \mathbb{C}[x_1,\ldots,x_n]$ and its zeroes [duplicate]

Given a polynomial $f \in \mathbb{C}[x_1,\ldots,x_n]$, then how can I prove that $f(a_1,\ldots,a_n) = 0$ implies $f$ is a summation of factors of $x_i - a_i$ for $i \in \{1,\ldots, n\}?$ This is not ...
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1answer
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Compound quadratic problem

The first issue I have is that I am not sure why this is called a 'compound quadratic problem', but anyway to proceed: Suppose that $x-y=14$ and $$(x+y)(x^2+y^2)(x^4+y^4)=a(x^b-y^b)$$ where $a$ and ...
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1answer
72 views

Vanishing of a multivariable polynomial on a lattice

Let be $p(x_1,...,x_n) \in K[x_1,...,x_n]$ be a polynomial of degree $d$. Suppose there is a $n$-dimensional hyperbox $B = I \times \stackrel{n}{...} \times I = I^n$. Divide $I$ to $d$ segements by ...
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4answers
907 views

Prove that $f(x)=x^3 −x−1$ has at least one real root.

How would I go about proving this? Would I try finding a value for $x$ that will make $f(x) = 0$?
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1answer
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Chebyshev polynomials with non-negative constants

Please let help me solve the following problem that I encountered while engaging in my research. I'm dealing with a class of functions, in which each function has a unique series representation of ...
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polynomial with integer coefficients

Question: Let $\Pi_{j=1}^n (z-z_j)$ be a polynomial with integer coefficients. Is also $\Pi_{j=1}^n (z-z_j^k)$ for $k=1,2,3,\dots$ a polynomial with integer coefficients? In fact, this is a question ...
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1answer
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any way to approach a sequency with a polynomial?

Suppose there is a finite sequence $(a_0, a_1, a_2, \cdots, a_k)$, is there any way to use a polynomial $\alpha_2 x^2 + \alpha_1 x + \alpha_0$ to generate that sequence such that each coefficient ...
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2answers
348 views

Solving system of multivariable 2nd-degree polynomials

How would you go about solving a problem such as: \begin{matrix} { x }^{ 2 }+3xy-9=0 \quad(1)\\ 2{ y }^{ 2 }-4xy+5=0 \quad(2) \end{matrix} where $(x,y)\in\mathbb{C}^{2}$. More generally, how would ...
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1answer
264 views

Fast way to find the smallest root $\mod M$ of a polynomial

Suppose you're given a polynomial of degree $d$ with integer coefficients: $$ P(x) = \sum_{i=0}^{d}{a_i x^i} $$ Is there a fast way to find the smallest root modulo $M$, where $M$ is some composite ...
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1answer
51 views

Monic polynomials with integer coefficients

We have $\Pi_{j=1}^n (z-z_j)$ a polynomial with integer coefficients. Is also $\Pi_{j=1}^n (z-z_j^k)$ for k=1,2,3,... a polynomial with integer coefficients?
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210 views

Polynomials in nature

What polynomials occur in "nature"? I am interested in polynomials of degree three and higher. I am aware of Stefan Boltzmann Law and Chemical Equilibrium Examples.
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Necessary and sufficient conditions for a polynomial $p$ to satisfy $\|x\|\to\infty\implies p(x)\to\infty$?

I'm looking for a necessary and sufficient conditions (I'm not even sure these exist) for a polynomial $p:\mathbb{R}^n\to\mathbb{R}$ to be "radially unbounded", that is $$\|x\|\to\infty\implies ...
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4answers
781 views

For what values of m are the roots of $x^2 +2x+3 = m(2x+1)$ real and positive

I am only able to show that to be real, $m <-1$ or $m\geq2$ Don't know how to finish solution Answer is $2 \leq m < 3$ So far: After expanding and factorising, $x^2 + 2(1-m)x + (3-m) = 0 $ ...
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1answer
309 views

Partial Fractions Theorem Proof

I'm having trouble solving this problem in my text book and I'm just not sure where to start/what the hint is trying to tell me. Let $p(x)$ be an irreducible polynomial of degree $m$, and let ...
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2answers
171 views

Skew Laurent Polynomial Ring.

Let $R$ be a ring and $R[x^{\pm 1}]$ the Laurent Polynomial Ring. $R[x^{\pm 1}]$ is a domain since $R$ is. How to show this? Let $R$ be a ring and $R[x^{\pm 1}]$ the Laurent Polynomial Ring. If ...
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1answer
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Polynomials that DON'T have certain roots

How many degree $\leq$ $d$ mod($p$) polynomials are there such that $P(a_1),...,P(a_k) \neq 0$ for $k < d$ and $0 < a_1 <...< a_k < p$, all integers? I considered subtracting out ...
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Primitives and minimal polynomials

got this assignment from my coding class and don't know if I've made it correct. Can someone tell if my methods for solving the tasks are correct? Let $f(x) = 1 + x ^3 + x ^4$ . It is given that ...
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Conditions for polynomial $f$ such that $f(n) \in \mathbb{N}$ for enough $n \in \mathbb{N}^+$ implies $f$ has rational coefficients

This question is suggested by this one: prove: coefficients of $f(x)$ are rational numbers What are the weakest sufficient conditions and strongest necessary conditions on a set of positive integers ...
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What are some techniques for reducing the dimension of an arbitrary Diophantine polynomial?

A set $S \subset \mathbb{N}^k$ is Diophantine if $$(x_1, \dots, x_k) \in S \iff \exists y_1, \dots, y_d \, p(x_1, \dots, x_k, y_1, \dots, y_d) = 0$$ for some Diophantine (integer coefficients) ...
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Counting Polynomials Question

This is a problem on my midterm review and I'm not sure at all how to approach it. If anyone could provide me with a solution including steps, I would be truly grateful as my exam is tomorrow morning. ...
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206 views

Cyclic linear codes and idempotents

Got this assignment from coding class and would be very thankful for checking if my solutions are correct. a) Find all idempotents modulo $1 + x^{17}$ of degree at most $15$ So first i find $r$ from ...
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2answers
562 views

How to solve this symmetric system of equations?

How many solutions are there to this equation? $$\begin{align*} x^2-y^2&=z\\ y^2-z^2&=x\\ z^2-x^2&=y \end{align*}$$
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Solve the following symmetric equations:

Solve the following equations: \begin{align}\left\{ \begin{array}{c} x_1+x_2+x_3+x_4=6 \\ x_1{}^2+x_2{}^2+x_3{}^2+x_4{}^2=10 \\ x_1{}^3+x_2{}^3+x_3{}^3+x_4{}^3=18 \\ ...
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3answers
197 views

prove: coefficients of $f(x)$ are rational numbers

$f(x)$ is polynomial with complex coefficients. $\forall n\in Z$, $f(n)$is integer, prove: coefficients of $f(x)$ are rational numbers, and give some examples about rational case. Prove: consider ...
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490 views

Finding irreducible polynomials and factorization

Need some explanation and checking if my thinking on the solution is correct for the assignment given below: (In these problems you may use without proof which polynomials of degree 2 and 3 are ...
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1answer
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polynomial question (out of practice)

Find all polynomials $p(x)$ such that $(x+3)p(x) = x p(x+1)$ for all real x. Ok, I am out of practice with this stuff. Here is what I have tried: making $x = -3$ and making $x = -1$ does not help ...
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1answer
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Finding polynomial $f(x)$ from $f(1)$ and $f(f(1))$

Let $f(x)=a_0+a_1x+a_2x^2+\dots+a_nx^n$, where $a_i\ge0$ Given f(1)=p and f(f(1))=q, we have to find $a_0$, $a_1$, $a_2$, $a_3$, $\dots$, $a_n$, where such $f(x)$ exists. Or we have to confirm if ...
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cubic equations which have exactly one real root

Question is to check : For any real number $c$, the polynomial $x^3+x+c$ has exactly one real root . the way in which i have proceeded is : let $a$ be one real root for $x^3+x+c$ i.e., we have ...
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1answer
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Prove that $x^4+x^3+x^2+x+1 \mid x^{4n}+x^{3n}+x^{2n}+x^n+1$

Problem: Prove that $x^4+x^3+x^2+x+1$ divides $x^{4n}+x^{3n}+x^{2n}+x^n+1$ for all positive $n$ that are not multiples of $5$. I'd like to get some pointers about how to solve this. No full ...
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1answer
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Are coefficients and values for x in F[x] in the same set?

I'm trying to understand the construction of $F[x]$ where $F$ is a field. As far as I understand it now, all coefficients and roots for all polynomials $f(x) \in F[x]$ lies in $F$. But what about the ...
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What is the value of $f(0)+f(8)$?

Suppose $f$ is a polynomial of degree $7$ which satisfies $f(1) =2$, $f(2)=5$, $f(3)=10$, $f(4)=17$, $f(5)=26$, $f(6)=37$ and $f(7)=50$. What is the value of $f(0)+f(8)$?
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1answer
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Find x in polynomial given value of inverse

I'm studying for a test and this question has me really stumped: $f(x) = 2x^3+5x+3$. Find x if $f^{-1}(x) = 1$ I don't know how I am supposed to figure out the inverse of this polynomial. I used ...
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Given a cubic $f(x)$ with specified negative real roots $-a,-b,-c$, what happens when we search for solutions to $f(x)=d$?

Noting Roots of a Certain type of Cubic Equation, what if we have the following simpler form for real $d$: $$(x+a)(x+b)(x+c)=d\tag{1}$$ (With $a,b,c\in \mathbb R^+$.) Clearly, depending on $d$, the ...
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Getting K heads out of N biased coins problem (formula generation ).

Problem- Given a set of coins n with each coin i having Pi probability to give heads. Find the probability of getting k heads, when all coins are tossed together. hi i have solved this problem ...
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Is each Chebyshev Polynomial orthogonal with respect to the weight function?

I computed the term $(T_{3})$ in the Chebyshev polynomials on Wolfram Alpha: http://www.wolframalpha.com/input/?i=integrate%28%284x%5E3-3x%29%2F%281-x%5E2%29%5E%281%2F2%29%2Cx%2C-1%2C1%29 After ...
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Help Factoring Quadrinomial

I know factoring questions are a dime a dozen but I can't seem to get this one. $-2x^3+2x^2+32x+40$ Factor to obtain the following equation: $-2(x-5)(x+2)^2$ Do I have to use division (I'd prefer ...
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1answer
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homogenization of irreducible polynomial

This is the last detail in an exercise that I'm working on in hartshorne and I can't seem to figure it out. If $f$ is an irreducible polynomial in $k[x_{0},\cdots,x_{n}]$ (where $x_{i}$ does not ...
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1answer
50 views

Using Legendre polynomial to approximate any polynomial

How can I show that any polynomial can be approximated by using linear combination of Legendre polynomial?
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Determine reducibility of a polynomial

Determine whether the polynomial $ x^3-9 $ is reducible over $\mathbb{Z}_{31}$, and over $\mathbb{Z}_{11}$. Since $ x^3-9 $ is of degree $\leq 3$, it is reducible over a field $F$ if it has a ...
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Proving that $\sum_{k=1}^{\infty} \frac{3408 k^2+1974 k-720}{128 k^6+480 k^5+680 k^4+450 k^3+137 k^2+15 k} = \pi$

I am trying to prove that $$\sum_{k=1}^{\infty} \frac{3408 k^2+1974 k-720}{128 k^6+480 k^5+680 k^4+450 k^3+137 k^2+15 k} = \pi$$ This is what I've tried to simplify the sum: $$\frac{3408 k^2+1974 ...
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2answers
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Minimal polynomial of $\omega:=\zeta_7+\overline{\zeta_7}$

Let $\omega:=\zeta_7+\overline{\zeta_7}$, where $\zeta_7$ is a primitive $7$th root of $1$. I want to find the minimal polynomial of $\omega$ over $\mathbb{Q}$. I've found ...
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3answers
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Prove that $x^4-x-1$ is irreducible over $\mathbb{Q}$

Prove that $f(x)=x^4-x-1$ is irreducible in $\mathbb{Q}[x]$. All methods I know failed. I can only exclude that $f$ admits a factorization with a factor of degree 3, because in this case $f$ would ...
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0answers
31 views

Are there any simple functions which map $\mathbb Z^n\to \mathbb Z\setminus \{k\}$ for given integer $k$?

Obviously, a function could be explicitly constructed as the set of all points in $\mathbb Z^n$ and what they are mapped to such that the given integer $k$ is not in the range. I am hoping to find a ...
2
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4answers
234 views

How to find $\sqrt{1+{4\over x}+{4\over x^2} }$?

If $$abx^2 = (a-b)^2(x+1)$$ then what is $$\sqrt{1+{4\over x}+{4\over x^2} }$$ (A) $a+b \over a-b$ (B)$a-b\over a+b$ (C) $a^2+ab$ (d) None EDIT: What I've done is this: ...
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1answer
60 views

Multivariate polynomial with univariate factor

Suppose $f \in \mathbb{Q}[x_1,x_2,\dotsc,x_i]$. How do I prove the following: There exists $x_1 \in \mathbb{C}$ such that for all $x_2,\dotsc,x_i \in \mathbb{C}$, $$f(x_1,x_2,\dots,x_i) = 0$$ if and ...
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1answer
50 views

Reference of the polynomials satisfying $f(t+u, v)+f(t,u)=f(u+v, t)+f(u,v)$.

Are there some reference of the polynomials satisfying $f(t+u, v)+f(t,u)=f(u+v, t)+f(u,v)$? I would like to know some of there properties. I am asking this question because these polynomials appear in ...