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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Even polynomial with odd ccoefficient

Does there exist an even polynomial (i.e $f(-x)=f(x)$ for all $x$) that has at least one odd power of $x$?
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Partial derivatives of polynomial in two variables

Let $k \in \mathbb N$, $a_{ij} \in \mathbb R$ for $i,j \in \mathbb N$, $i+j \le k$. A function $f:\mathbb R^2 \to \mathbb R$ $$f(x,y) = \sum_{i+j \le k} a_{ij}x^iy^j$$ is called polynomial of degree ...
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Find how many solutions has the following equation…

Determine how many real solutions has the following equation: $$x^2(|x|-6)=-15$$ I noticed that $|x|-6$ should be negative because $x^2$ is always a positive value. Thus, $x\in(-6;6)$. I made a ...
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Creating non-linear function to calculate points per distance in reversed order.

I have a small game in which I want to give points according to closeness to location, so the maximum points will be given for the minimal number = 1. It's similar to GeoGuesser game. Here is the data ...
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3answers
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Problem related to remainder

A polynomial in $x$ leaves a remainder $2$ and $3$ when divided by $x-1$ and $x+1$. What is the remainder, when divided by $x^2-1$ ?
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If $(1 + 2i)$ and $(3 - 2i)$ are two roots of $x^5 + ax^4 + bx^3 + cx^2 + dx + 4$, then $a$ =?

Consider the polynomial $x^5 + ax^4 + bx^3 + cx^2 + dx + 4$ where $a, b, c, d$ are real numbers. If $(1 + 2i)$ and $(3 - 2i)$ are two roots of this polynomial then what is the value of a? Well, I ...
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Formula for calculation score based on distance

I try to write function for calculating scoring from distance in my game. I found something similar : link But I need the distance(x) to be between 0-31855000 meters and score between 0 to 1000. ...
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3answers
199 views

A polynomial with integer coefficient

I'm struggling with this question: Suppose $P(x)$ is a polynomial with integer coefficients such that non of the values $P(1),...,P(2010)$ is divisible by $2010$. Prove that $P(n)\neq 0$ for all ...
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1answer
21 views

Comparing real roots of $P(x)$ and $P'(x)$

Let $a$ be a real number and $P(x)$ be a polynomial with real coefficients. 1) Prove that $P'(x)$ doesn't have more non real roots than $P(x).$ 2) $aP(x)+P'(x)$ doesn't have more non real zeroes ...
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1answer
22 views

Every irreducible polynomial in $\mathbb{F}_p[x]$ is separable?

How can I show this? I tried proving the contrapositive statement but didn't get anywhere. I think I may have to do something involving automorphisms of the splitting field of such a polynomial, and ...
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256 views

Simply put, what are the similarities between integers and polynomials?

The Princeton Companion to Mathematics mentions that polynomials (for instance, ones with rational coefficients) share similarities with integers, thus leading to the idea of a general structure of ...
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Eisenstein Criterion

Why is it that the Eisenstein Criterion would work when substituting $x$ with $x + 1$? Why is it OK to do this for polynomials in $\mathbb{Q}[x]$? Thank you
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1answer
16 views

integer solutions to bivariate polynomial of second degree

I am trying to determine if there is a way to quickly determine if an equation of the following type $$0 = axy+x-y-A$$ has integer solutions ($a,A$ are integers). If anyone knows how to do this or ...
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32 views

Proving the area of an equilateral triangle

How do you prove that How do you prove that for any equilateral triangle with side length s, area is $\frac{s^2 √3}{4}$ ? I tried using an equilateral triangle in a square, but I keep coming up with a ...
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14 views

Descartes rule of signs

I'm trying to write an algorithm that gets a polynomial and gives how many roots does it have in the interval [0 $x_0$]. I'm supposed to do it by Descartes law, I know that by Descartes law you Know ...
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1answer
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find non-zero polynomial p(x) to satisfy,very important [on hold]

How can I solve this problem.It's very important to learn until this weekend for exam Find a non-zero polynomial p(x) satisfying, p(-1)=0,p'(-1)=0,p''(-1)=0,p(1)=0,p'(1)=0,p''(1)=0
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If there exists a polynomial of best approximation of degree n, there also exists a polynomial of best approximation of degree n+1.

First I'd like to say that although this question was asked before (here) and is from the same text, the answer used methods that were not introduced in the text. Let $P_n(x)$ be a polynomial of ...
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36 views

Does the geometric sum formula have any useful variants?

Suppose $a$ is a constant sequence in $\mathbb{C}$ with constant value $A \in \mathbb{C}$. Then the geometric sum formula says: for all natural $n$ and all complex $z$, we have $$\sum_{k = ...
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1answer
48 views

Irreducibility of a polynomial in two variables

Anyone knows how to verify that the polynomial $$(ax)^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0-y^n$$ is irreducible, where $n\geq 2$, $a,a_i\in\mathbb{Z}$, $a\neq 0$, and $(ax)^n+a_{n-1}x^{n-1}+\cdots ...
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45 views

Is a constant such as 8 considered an expression?

The question asked was "Which of the following expressions are considered polynomials?" 8 was one of the answers, and though it is clearly a monomial, it was part of the answer and I'm confused as to ...
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1answer
42 views

Concrete FFT polynomial multiplication example

I have read a number of explanations of the steps involved in multiplying two polynomials using fast fourier transform and am not quite getting it in practice. I was wondering if I could get some help ...
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monotonicity of $\sum_0^n x^i$ for odd $n$

I am trying to prove the strict monotonicity of $\sum_{i=0}^n x^i$ for odd $n$. This is not homework; just something I have noticed to appear true, and thus my brain bugs me until I have a proof. I ...
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Minimum degree of polynomial assuming exactly k prime values

Dirichlet's theorem states that there are infinitely many primes of the form $an+b$ for coprime integers $a$ and $b$. This implies that The minimum degree of a polynomial $f \in \mathbb{Z}[X]$ ...
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34 views

Galois field splitting a polynomial

Can someone explain to me how i would go about doing a problem like this? I don't really know where to start. GF refers to a Galois field. I'm struggling to even understand exactly what they want me ...
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Divisor in $\mathbb{C}[X]$ $\implies$ divisor in $\mathbb{R}[X]$?

let $P \in \mathbb{R}[X]$ be a real polynomial divisible by a polynomial $Q \in \mathbb{R}[X]$ in $\mathbb{C}[X]$. How can I easily show that $P$ is also divisible by $Q$ in $\mathbb{R}[X]$? A simple ...
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What do the different parameters in a general sinusoidal model physically correspond to?

Considering the following equation for a generalized sinusoidal, $$ y(t) = e^{(a+bt+ct^2)} * e^{j(d+et+ft^2)} $$ what do the parameters $a,b,c,d,e$ and $f$ represent physically?
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Discriminant of $f(x)=x^3+ax+b$

Suppose we have the polynomial $f(x)=x^3+ax+b$, with roots $\alpha, \beta, \gamma$ in $\mathbb{C}$, and let $\Delta = (\alpha - \beta)(\beta - \gamma)(\gamma - \alpha)$. Is there any quick way of ...
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1answer
37 views

$x^5+4x^4+2x^3+3x^2+-x+5$ is irreducible or reducible.

Question. $f(x)=x^5+4x^4+2x^3+3x^2+-x+5$ is irreducible over $\Bbb Q$? $\;$ I know that use the prime number $p$ to $\bar f_{p}(x)$. let $p=2$. then $\bar f_{2}(x)=x^5+x^2-x+\bar 1$. but $x=1$, ...
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Is factoring polynomials easier than factoring integers?

I was reading the book Algebra: Chapter 0 , by Paolo Aluffi, and came across the following assertion, in page 290, Exercise 5.9: It is in fact much harder to factor integers than integers ...
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Let $\{f_n(x)\}$ be a sequence of polynomials defined as $f_1(x) = (x - 2)^2$, $f_{n+1}(x) = (f_n(x) - 2)^2$; $n \ge 1$. [duplicate]

Let $\{f_n(x)\}$ be a sequence of polynomials defined as $$f_1(x) = (x - 2)^2$$ $f_{n+1}(x) = (f_n(x) - 2)^2$; $n \ge 1$. How to find the constant term and the coefficient of $x$ in $f_n(x)$.
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$f(x^2 + 1) = f(x)g(x)$ $\forall$ x $\in\mathbb{R}$ $\Rightarrow$ no. of roots of $f(x)=$?

If two real polynomials $f(x)$ and $g(x)$ of degrees $m$ $(\ge2)$ and $n$ $(\ge1)$ respectively, satisfy $f(x^2 + 1) = f(x)g(x)$; for every $x \in \mathbb{R}$ , then what can be said about the ...
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Finding the minimum value of a 6th degree polynomial algebraically

Is it possible to answer this question using methods of basic algebra? Find the least value of the expression $a^6 + a^4 - a^3 - a + 1$ for real value of $a$. This question is from the 2013 Philippine ...
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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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666 views

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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32 views

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
124 views

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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27 views

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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92 views

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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36 views

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 ...