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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What is $s_3$ and $s_4$ for $x$

$\sum_{i=0}^n i^k = s_k(n)$, $s_k$ polynomial from degree $k+1$ I have already shown for $s_2(x) = \frac{x(x+1)(2x+1)}6$ How from the sum and $s_2(x)$ can be shown for $s_3(x)$ and $s_4(x)$ ...
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Finding a polynomial satisfying the equation

For $$ f: x^6+3x^4-4 \\ g: x^5-x^4+5x^3-5x^2+6x-6 $$ how do I find a polynomial $a \in \mathbb{Q}[x]_{(\deg f-\deg \gcd(f,g))}$ so that a polynomial $b \in \mathbb{Q}[x]$ exists when ...
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Find monic quartic polynomial f(x) with rational coefficients whose roots include…(Algebra)

Find a monic quartic polynomial f(x) with rational coefficients whose roots include $x=2-3\sqrt{2}$ and $x=1-\sqrt{3}$. How could you find the other roots?
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Let $r_0,r_1,…,r_m$ be the real roots of $a_nx^n+a_{n-1}x^{n-1}+…+a_0$.Is there a closed-form expression for $\sum_{i=1}^mr_i -\sum_{i=1}^m1/r_i$?

Let $r_0, r_1, ... ,r_m$ be the real roots of $a_nx^n+a_{n-1}x^{n-1}+...+a_0$, with $a_0\ne0.$ Is there a closed-form expression for $$ \ \ \ \ \ \sum_{i=1}^mr_i - \sum_{i=1}^m \frac{1}{r_i} \ \ \ ...
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A positive integer $n$ is such that $1-2x+3x^2-4x^3+5x^4-…-2014x^{2013}+nx^{2014}$ has at least one integer solution. Find $n$.

A positive integer $n$ is such that $$1-2x+3x^2-4x^3+5x^4-...-2014x^{2013}+nx^{2014}$$ has at least one integer solution. Find $n$.
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General Polynomial Solution to an Infinite Differential Equation

To begin, I would just like to say that I don't know too much about upper level mathematics. I'm a curious highschooler (also homeschooled). I would greatly appreciate if someone would help me answer ...
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Help with understanding the polynomial long division algorithm

I saw this algorithm at Wikipedia: ...
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1answer
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understanding phrasing of taylor polynomial question

Show that $|\sin x - x + \frac{1}{6}x^3| < 0.08$ for $|x| \le \frac{\pi}2$. How large do you have to take $k$ so that the $k$th order Taylor polynomial of $ \sin x$ about $a=0 $ approximates $\sin ...
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Idempotent generators of the four binary QR codes of length 7

I have a coding theory assignment and I thought it would be a good idea to double check before I hand it in. I'm asked to find the idempotent generators of the four binary QR codes C1, C2, C3, C4, of ...
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Give the generator polynomial of a binary cyclic [9, 2] code.

I'm new to Cyclic Codes and I'm not sure the process to find a generator polynomial of a cyclic code. I know what a cyclic code is, but not sure how to find a generator polynomial. Since the question ...
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2answers
61 views

How to show the equation $x^4 + 2cx^3 + 6x^2 + 60x =-11$ has exactly two real solutions?

How can we show that $x^4 + 2cx^3 + 6x^2 + 60x =-11$ has exactly two real roots? $c$ is any element in the interval $(-2,2)$.
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$\int_{-1}^{1}|f(t)|dt \geq C\left(\int_{0}^{2}|f(t)|^2\right)^{1/2}$ for polynomials

Prove that there exists constant $C>0$ that for all $f \in P_n$ we have: $$\int_{-1}^{1}|f(t)|dt \geq C\left(\int_{0}^{2}|f(t)|^2\right)^{1/2}$$ Where $P_n$ is space of polynomials with ...
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How do I find the zero(s) of a rational function?

I am doing homework and have been given this task: You have the function $$g(x)=\frac{2x^2-8}{x^2+4}$$ and I am asked to find the zeros of the function. My teacher shows the solution as $f(x)=0$ and ...
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Existence of a nontrivial solution to a polynomial equation

Let $p \ne 0$ and consider the equation $$ x_1 (x_1 + p)^2 + \dots + x_n (x_n + p)^2 = 0.$$ Does there exist a solution $x \in \mathbb R^n$ to this equation that is not the trivial solution $x=0$?
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Degree-4 Polynomial

Solve the equation $x^4 - 14x^3 + 50x^2 -14x + 1 = 0$. I am not sure about how to best proceed, and would like a solution that does not involved the generalised quartic formula.
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Solving for the roots of a polynomial

Suppose we have a polynomial of the form: $$-x^3+3x^2+9x-27=0$$ Is there an easy way to find the solutions of $x$? I know that they will be factors of $27$, so I begin by factoring $27$ into ...
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1answer
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Polynomial over finite field

I'm currently reading these notes on the simplicity of $PSL_n(F)$. At page 5 it is used that there exists an element x in fields with 4 or more elements such that both: $x\neq 0$ $x^2-1\neq 0$ I ...
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Given multiple polynomial equations find a basis.

I have read several other threads on Math.SE, including the similarly titled: basis of the polynomial vector space I've also checked out a video lecture on Youtube by njwildberger, but I simply have ...
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A question on vectors represented by multilinear polynomials

Let the set of multilinear polynomials of degree atmost $t$ in $\Bbb Z[x_1,\dots,x_n]$ be $\Bbb Z^{t}[x_1,\dots,x_n]$. Let $S=\{0,1\}^n$. Fix an ordering of $S$. For every $f\in\Bbb ...
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Similarities/differences between multivariate polynomials and integers

There are a few questions on this site that asks for similarities between integers and univariate polynomials. I am wondering if multivariate polynomials have any related analogies with integers.
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Proving Theorem: subspace of polynomials of degree two or less?

How can I prove that the set $S$ of polynomials of degree $2$ or less, whose coefficients sum to zero, is a subspace of all polynomials with degree $2$ or less? I know I need to show that $a+b+c=0$ ...
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Is every zonal homogeneous polynomial a polynomial on the unit sphere?

Let $$P_k(x_1\ldots x_n)=\sum_{\lvert \alpha\rvert=k} c_\alpha x_1^{\alpha_1}\ldots x_n^{\alpha_n}, \qquad (x_1\ldots x_n)\in \mathbb{R}^n$$ be a homogeneous polynomial of degree $k$. Assume that ...
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Polynomials over $\mathbb{F}_2$ without multiplicity 1 factors

Let $f \in \mathbb{F}_2[T]$ such that both $f$ and $f + 1$ have the property that every irreducible factor in the unique factorization domain $\mathbb{F}_2[T]$ appears with multiplicity at least $2$. ...
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3answers
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Finding the roots of a polynomial on a complex plane [duplicate]

I use an online calculator in order to calculate $x^5-1=0$ I get the results x1=1 x2=0.30902+0.95106∗i x3=0.30902−0.95106∗i x4=−0.80902+0.58779∗i x5=−0.80902−0.58779∗i I know that this is the ...
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4answers
69 views

Why $(x-5)^2-4$ can be factorised as $(x-5-2)(x-5+2)$

I would like to understand why $(x-5)^2-4$ can be factorised as $(x-5-2)(x-5+2)$ I am particularly concerned with the term, $-4$.
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when do we have polynomial local minimum = to global minimum

When does a multivariate polynimial has only one stationary point so that local minimum is global minimum?
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If $f$ is a unit in a polynomial ring then $a_0$ is unit and all other coeficients are nilpotent.

I'm trying to prove the converse of the following theorem. I think suggestion available at this website are mistaken or I didn't understand them correctly. Theorem. Let $R$ be a commutative ring with ...
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Finding zeroes of $x^3-5x^2+11x+17$

I'm trying to find all the zeros of $x^3-5x^2+11x+17$. I figured the possible zeros as being +/- 1, +/- 17$. The book says that -1 is supposed to be a factor, but I tried dividing the polynomial by ...
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Factoring a polynomial of fourth degree with false roots: $x^4+4$

I want to write this polynomial in factored form: $$x^4+4$$ The reason I want to do this is to be able to make partial-fraction decomposition on it to make an integrand easier to integrate. What's ...
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Polynomial Root Multiplicity Testing.

I would appreciate some help here. Either a reference or a proof or just a statement that helps me to conduct research of my own. Long ago when I was studying polynomials intently I read about a ...
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1answer
25 views

What is this function?

In python I made this function: def f(x): eqStr = '' for y in range(int(x)): eqStr += 'x**%s + ' % (y) eqStr += '0' return eval(eqStr) ...
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4answers
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Two roots of the polynomial $x^4+x^3-7x^2-x+6$ are $2$ and $-3$. Find two other roots.

I have divided this polynomial first with $(x-2)$ and then divided with $(x+3)$ the quotient. The other quotient I have set equal to $0$ and have found the other two roots. Can you explain to me if ...
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Find the remainder of the division of $P(x)$ with $(x-3)$ if $P(x+2)=2x^3-4x^2+2x+3$

I found that $P(x+2)$ is $2x^3+8x^2+10x+7$, what should I do next? I don't know how t find $P(x)$
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Isomorphism problem for two radical extensions of the same degree

Let $n\geq 2$ and let $a,b\in{\mathbb Q}$. Suppose that both the polynomials $A=X^n-a$ and $B=X^n-b$ are irreducible over $\mathbb Q$. We want to know whether ( * ) there is a root $\alpha$ of $A$ ...
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If a differential operator $C$ factors as $AB$, then every solution of $C(y)=0$ has the form $y=y_1+y_2$ with $A(y_1)=0$ and $B(y_2)=0$

Given two constant-coefficient operators $A$ and $B$ whose characteristic polynomials have no zeros in common. Let $C = A B$. First part of question is Prove that every solution of the ...
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Factorize x^3+3

Task: Factorize $x^3+3$ in $\mathbb{R}$ and $GF(7)$ I think the solution is $x^3+3$ in both cases, so the polynomial already is irreductible. Is my assuption is right, how do i show that?
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1answer
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Disprove that this subset of P3 is not a subspace by using a counterexample

The set of all polynomials with degree 3 plus the zero polynomial. A hint would be appreciated to get me going :)
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What does “two polynomials have no zeros in common” mean?

The question is Given two constant-coefficient operators $A$ and $B$ whose characteristic polynomials have no zeros in common. Let $C=AB$... What does that mean by "no zeros in common"?
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Converting expressions to polynomial form

My question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook. Page 57. Exercise 12. Show that the following are polynomials by converting them to the ...
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Finding set of integer pairs for which two integer polynomials intersect

I am wondering if there is a theorem in number theory that addresses the following issue: Suppose we have two polynomials, f and g, with integer coefficients. Is there a general way to find elements ...
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Polynomial curve fit

Well I have a 2 (or 3) data points - and some extra limits - and a polynom needs to be fitted through those points (exactly). The polynom needs to be of the smallest order, and not a least square, it ...
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3answers
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Prove that one of x,y,z is smaller than 3 and one is bigger than 5 if…

If $x+y+z=12$ and $x^2+y^2+z^2=54$ then prove that one has to be smaller or equal to 3 and one has to be bigger or equal than 5. So I got that $xy+yz+zx=45$ and with that I had a function with x,y,z ...
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1answer
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Is the Frobenius automorphism a polynomial?

Let $p$ be a prime and $\Bbb{Z}/\Bbb{Z}_p$ the field of integers mod $p$, and since $\Bbb{Z}/\Bbb{Z}_p$ is a field we have the ring of polynomials in $X$ with coefficients in $\Bbb{Z}/\Bbb{Z}_p$ ...
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Proof Explanation: Vector Space of Polynomials with Average Value 0 around a circle

The question is from Putnam 2009 B4. Problem: Say that a polynomial with real coefficients in two variable, $x,y$, is balanced if the average value of the polynomial on each circle centered at the ...
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On polynomials over finite fields?

Pick prime $q\in\Bbb Z$ such that $q>B^{3t}>(mB)^{2t}$. Suppose I have a multilinear polynomial $g(x)\in \Bbb Z[x_1,\dots,x_n]$ of degree $t$ with $m^t$ non-zero coefficients that bound by ...
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How to determine the smallest interpolation degree required?

Given a set of $n$ points $(x_k, y_k)\ (k\in\{1,...,n\})$, of course a polynomial of degree $n$ can fit all points. However, in some cases the coefficient of the higher degrees actually vanish and one ...
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1answer
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$\frac{-1}{2}$ Zero of odd powers sum polynomials?

Consider the polynomial $S_k(x) \in \mathbb{Q}[x]$ such that $S_k(n)=\sum_{i=1}^{n}i^k, \forall n \in \mathbb{N}$. Now if i recall correctly the definition it should be that ...
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Number of roots of $x^3+2x-1$ in $\mathbb Q$

How many roots does $x^3+2x-1$ have in $\mathbb Q$? I know that it has one real and two complex conjugate roots because the determinant is $-59$.
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What is the name for a polynomial with all coefficients equal to 1?

I am looking for a good google search word for polynomials that have all coefficients equal to 1. An example of a such polynomial is: $$1+x^{23}+x^{57}+x^{101}$$ One such polynomial could also be ...
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1answer
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$P_n(x):=1+ \sum_{m=1}^n\dfrac{x^m}{m!}$ has no real root for even $n$ and exactly one real root for odd $n$

Is it true that $P_n(x):=1+ \sum_{m=1}^n\dfrac{x^m}{m!}$ has no real root for even $n$ and exactly one real root for odd $n$ ? I can only prove that the polynomial cannot have any multiple roots . Am ...