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.

learn more… | top users | synonyms

3
votes
1answer
56 views

How to remove duplicate roots from a polynomial?

Given a polynomial equation (with real coefficients of any degree with any number of repeating roots), let say $x^5 + 6x^4 - 18x^3 - 10x^2 + 45x - 24 = 0$, ... (A) it can be written as $(x-1)^2 ...
0
votes
1answer
31 views

What is the difference between the largest and smallest possible positive roots?

I am faced with the following question: What is the difference between the largest and the smallest possible positive roots of $4x^5 + 3x^3 -5x^2 + 7x - 12$? Now, my first attempt was to try ...
5
votes
2answers
138 views

Differentiate polynomials in $\mathbb{Z}_2[x]$

It seems suggested that the differential of a polynomial in $\mathbb{Z}_2$ is as I would expect: $$\begin{align} &f = x^6 + x^3 + x + 1 \\ &f' = 6x^5 + 3x^2 +1 \mod 2 \\ &f'= x^2 + 1 \\ ...
1
vote
1answer
25 views

Generate polynomial

Given solutions of a n degree polynomial , how can we find the polynomial. Eg. : I have solutions like : For x = 3, answer = 2 ...
1
vote
0answers
33 views

Proof verification: any n-th order complex polynomial has at most n distinct roots

Here is a proof I came up with in the exam I just took. But I suspect there may be some issues since I think it seems too simple. Proof Let $p_n(x)$ denote a complex polynomial of order $n$ ...
1
vote
1answer
25 views

Algebraic vs. analytic definition of the multiplicity of a polynomial's root

Let $f(x) = a(x - c_1)^{d_1}(x - c_2)^{d_2} \dots (x - c_n)^{d_n}$ be a polynomial over the complex numbers ($n, d_i \in \{1, 2, \dots\}$, $a \in \mathbb{C}\setminus \{0\}$), where the roots $c_1, ...
3
votes
1answer
31 views

Polynomials and Divisibility Rule.

The question is this - If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $h(x)=xf(x^3)+x^2g(x^6)$ is divisible by $x^2+x+1$, then which of the following are true? 1. $f(1)=g(1)$ ...
0
votes
2answers
31 views

Puzzled question of remainder of cubic polynomial

Let $f(x)$ be a cubic polynomial. If $f(x)$ is devided by $x+2$, the remainder is $-10$. If f(x) is divide by (x-1), the remainder is 20. (a) If $f(x)$ is divided by $x^2+x-2$, find the remainder. ...
0
votes
1answer
35 views

Question on direct sum of vector spaces

I have the following linear algebra question on direct sums: I am given the vector spaces: $ V = R_4[x] $ $ W = span\{x^4-x^2,3x^4-x^3+1 \}$ I am asked to find the complement to the direct sum i.e. ...
0
votes
0answers
17 views

Stuck on polynomial equation in optimization problem

I've been trying to solve an optimization problem, but I am completely stock on one step. I had the following Langrangian: $$\nabla\mathcal{L}(x,\lambda)= e\frac{\sum_{t\in I}e^t \Delta P(t)( x^t ...
0
votes
1answer
30 views

little question about a notation of polynomials

In algebra we often consider the ring of polynomials $K[x]$ with coefficients in a field $K$ for example. If you write out a polynomial $p\in K[x]$, sometimes I see different things: ...
0
votes
0answers
53 views

Find the sum of the coefficients in front of the even degrees of x in the normal form of a polynomial

Find the sum of the coefficients in front of the even degrees of x in the normal form of a polynomial $$(x^6 + x + 1)^{2015} + (x^6 + x - 1)^{2015}$$ I am familiar with the binomial theorem , ...
3
votes
1answer
37 views

Bounding $x^2+6x$ between consecutive cubes when solving $y^3=x^2+6x$

I am familiar with the method of bounding a polynomial between consecutive squares to prove it is not a square. For example, this method can prove $y^2=x^2+x+1$ has no solutions since ...
1
vote
1answer
13 views

Linear forms which vanish on commutators

In some exercise, $E$ denotes the vector space $\mathbb{R}[X]$ and $\mathcal{L}(E)$ the algebra of endomorphisms of $E$. I am asked to determine all the linear forms $T \, : \, \mathcal{L}(E) \, ...
1
vote
0answers
54 views

Polynomial division, multivariable, indeterminates

Trying to understand something in the proof of Nullstellensatz, if we have a polynomial $p(x_1,...,x_n,t) \in k[x_1,...,x_n,t]$ with $f(t)$ divides $p(a_1,...,a_n,t)$ for all fixed $(a_1,...,a_n) ...
0
votes
1answer
19 views

Divisibility in a certain ring and divisibility in integers.

Divisibility in the ring $\mathbb{Z}[x,y]$ implies divisibility in $\mathbb{Z}$ ? Let $P(x,y)=Q(x,y)\cdot R(x,y)$ with $P,Q,R$ polynomials with integer coefficients, evaluating in $(x,y)=(a,b)$ with ...
-1
votes
2answers
47 views

Prove that a linear transformation $T$ over linear space of real polynomials of $deg \leq n$ only has one eigenvalue=1 [closed]

Let $V$ be the linear space of polynomials p(x) of degree $\leq n$. If $p\in V$ define $q = T(p)$ to mean $q(t) = p(t+1)$, for all real $t$. Prove $T$ has only the eigenvalue 1. What are the ...
0
votes
1answer
19 views

Finding roots of complex polynomial with conjugates

I am having problem with the following question... I know that I should use De Moivre's formula somewhere... but can't quite get to it $$ (-15w + 34\bar{w})^4 = -1 $$ will be happy to get some help, ...
1
vote
2answers
26 views

Polynomial long division with mod. Trouble with fractions.

For example $4x^4 + x + 1$ divided by $3x + 1$ is $\frac{4x^3}{3} - \frac{4x^2}{9} + \frac{4x}{27} + \frac{23}{81}$ remainder $\frac{58}{81}$. Now I want to do the same division mod $9$, but I can't ...
0
votes
0answers
26 views

Identifying a sequence of numbers from an optimization problem in $L^1$

Question Does there exist general closed form solutions (or some sort of recurrence relation) to the system of equations: $$\begin{align} x_0 &= -1\\ x_{k+1} &= 1\\ \sum_{j = 0}^k (-1)^j ...
1
vote
0answers
14 views

Linear transformation of vector ARMA processes

Can someone help me to solve the following problem. Referring to the one above the bottom equation: I was managed to get the left hand side and first term of the right hand side. But couldn't ...
1
vote
0answers
20 views

Roots of polynomial in terms of “odd radicals”

I'm trying to show that it is not possible to find a formula for the roots of $x²-S_1x+S_2$ only in terms of odd radicals in $C(x,y)$. Here $S_1$ and $S_2$ are the symmetric polynomials in terms of ...
0
votes
3answers
37 views

Ideals Generated by polynomials

So I am currently studying a course in commutative algebra and the main object that we are looking at are ideals generated by polynomials in n variables. But the one thing I don't understand when ...
0
votes
1answer
61 views

example of proper ideal of C[x,y]

I am stuck in this problem for a while, and the main idea will be important for some exercises, so I really want to know how to find an example like this I need an example of an proper ideal, ...
2
votes
2answers
51 views

Roots of a Polynomial Minus It's Constant Term

Suppose we have a sequence of integers $a_1,\dots,a_n$. Is there any way to determine the roots of the polynomial $$P(x) = (x+a_1)\dots(x+a_n) - a_1\dots a_n$$ Clearly $P(0) = 0$, but can anything ...
0
votes
1answer
27 views

Prove that the following polynomial is not dividable over Q

Let $a\neq b$ $|a,b \in\mathbb N$ and let $P(x)=x^3+ax^2+bx+1$ Show that P is irreducible over $\mathbb Q$. Any idea?
0
votes
0answers
21 views

Taylor expansion of a power function

I was wondering about Taylor expansions of functions of the form $x^p$, where p is a real number, about $x = 0$. It seems clear how to do it about any other point, but what happens to the series as I ...
0
votes
1answer
29 views

Integer-valued polynomial question

Let us have an $f(x)$ Integer-valued polynomial, which gains the value $1$ in $4$ different places. Prove, that in that case, it can't gain the value $-1$ on integer places. I tried with $f(x)-1=0$, ...
0
votes
2answers
60 views

Is there a link between the Bunyakovsky conjecture and the Twin Prime conjecture?

Can the proof of one conjecture be considered a proof of the other conjecture? The general method of building an infinite number of prime producing quadratic polynomials was given in the link ...
-1
votes
0answers
24 views

How to evaluate the coefficient of power series?

For some reason(or trick), I need to calculate something like the coefficient of $X^6$ in $f(X)=\frac1{(1-X)(1-X^2)}\times\frac1{(1-X)(1-X^2)(1-X^3)(1-X^4)}$ evaluated as power series. How should I ...
3
votes
1answer
67 views

Irreducibility in $k((t))[y]$

Let $k$ be an algebraically closed field of char $0$ and suppose $f(y) \in k[y]$ (need not be monic). Let $t$ be an indeterminate and consider the fraction field $k((t))$ of power series ring ...
0
votes
0answers
13 views

Legendre polynomial related simple proof question

Given the set of orthogonal polynomials {Qi(x)}i=0 to n , a polynomial Pn(x) of degree ≤ n, can be written as: Pn(x) = a0*Q0(x) + a1*Q1(x) + · · · + an*Qn(x) for some a0, a1, . . . , an. Please help ...
2
votes
2answers
57 views

Prove that $ (a+b\sqrt{2})^n $ is of the form $k+l\sqrt{2}$.(a,b,k,l,n are integers; n>1)

I have previously proved it for n=1. Using induction, assume $(a+b\sqrt{2})^{x-1}$ is true; it is of the form $k+l\sqrt{2}$. for $(a+b\sqrt{2})^x$; how do i proceed from here? Binomial theorem for ...
2
votes
2answers
51 views

Help me to prove this statement about quadratic equations? (from Gelfand's Algebra).

$ x^2+px+q=0 ${p,q are integers; a,b are roots}. Prove $a^n+b^n$(n is any natural number) is an integer. This is the third part of the problem.I have previously proved that $a^2+b^2$ and $a^3+b^3$ ...
1
vote
1answer
41 views

Find all Real polynomials.

"Find all real polynomials for which $f(2) = 3, f(3) = 5$ and $f(5) = 2$." Well my first thought was, since we have three points i can determine a polynomial of second degree such that it satisfies ...
1
vote
1answer
46 views

Find all polynomials $P(x) \in \mathbb{Z}[X]$ such that $P(n) \mid 2^n-1 $ [closed]

Find all polynomials $P(x) \in \mathbb{Z}[X]$ such that $P(n)\mid 2^n-1 $ where $n \in \mathbb{Z}^+$ I don't have any ideals, may be it relates to number theory ?
2
votes
2answers
47 views

Find $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 ,\ n\ge 1$ has $n$ roots $x_1,x_2,\ldots,x_n \le -1$ and such that $a_0^2+a_1a_n=a_n^2+a_0a_{n-1}.$

Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 ,\ n\ge 1$ have $n$ roots $x_1,x_2,\ldots,x_n \le -1$ and $a_0^2+a_1a_n=a_n^2+a_0a_{n-1}$. Find all such $P(x)$ (Poland 1990). I used Viete Theorem ...
2
votes
1answer
34 views

three cubic homogeneous polynomials satisfy a cubic polynomial

Question: How can we show algebraically that three cubic homogeneous polynomials in two variables satisfy a cubic polynomial of three variables? More specifically, let ...
1
vote
1answer
44 views

how to show the derivative of the polynomial is bounded by itself in certain space.

How to prove that for every positive integer $d$, there exists $C(d)>0$, such that: For every polynomial with degree $\leq d$, we have $\max\limits_{x\in [0,1]}|p'(x)|\leq C(d)\max\limits_{x\in ...
1
vote
2answers
41 views

Greatest common divisor of polynomial in Finite Field(256), AES

Have assigment and will use it as example, found solution computationaly, want to understand idea. It is about SubBytes procedure in AES, particulary about finding inverse of polynomial. Suppose we ...
1
vote
1answer
73 views

Non trivial solutions of a polynomial equation

In a question a user asked for a polynomial which solves $$2P(2x^2-1)=(P(x))^2-1.$$ There are two solutions I could provide, namely the two constant ones. However in the comments to my answer it has ...
3
votes
1answer
137 views

Find a polynomial P(X)

Find a polynomial $P(x)$ such that it satisfies $$2P(2x^2-1)=(P(x))^2-1$$ How to find all of them?
1
vote
1answer
42 views

The number of zeros of a polynomial that almost changes signs

Let $p$ be a polynomial, and let $x_0, x_1, \dots, x_n$ be distinct numbers in the interval $[-1, 1]$, listed in increasing order, for which the following holds: $$ (-1)^ip(x_i) \geq 0,\hspace{1cm}i ...
1
vote
1answer
23 views

Is Gershgorin bound of roots sharp?

Gershgorin circle theorem tells that the eigenvalues of a matrix $A$ lie in the union of the associated Gershgorin circles. $A=\begin{pmatrix} 0 & 0 & \dots & 0 & -a_0 \\ 1 & 0 ...
0
votes
1answer
49 views

What does the notation $\mathbb R[x]$ mean?

What does the notation $\mathbb R[x]$ mean? I thought it was just the set $\mathbb R^n$ but then I read somewhere that my lecturer wrote $\mathbb R[x] = ${$\alpha_0 + \alpha_1x + \alpha_2x^2 + ... + ...
1
vote
1answer
23 views

Derivation of the discriminant of a cubic polynomial by algebraic manipulation.

The problem was asked before: Using Vieta's theorem for cubic equations to derive the cubic discriminant . I tried to solve it by purely algebraic manipulation but was faced with an explosion of ...
0
votes
0answers
30 views

Monic polynomial

Recently I've learned that when a given polynomial is a monic polynomial, then this polynomial root has to be a rational root. As far as I know, to figure out if a given polynomial is a monic ...
1
vote
1answer
55 views

What can we learn from prime generating polynomials?

Here's a simple polynomial that generates quite a few primes (not necessarily consecutive). $p(n) = n^2 + 23n + 23$ with $n=0,1,2... $ What can such polynomials tell us about primes? Thanks. ...
0
votes
0answers
18 views

Question about Karp reduction

friends. I have a curiosity about Karp reduction. What we need to do for reduction from problem X to problem Y is that 1) Transformation from Instance of problem X to Instance of problem Y can be ...
1
vote
1answer
26 views

Relation between roots an coefficients in a generic equation: $a_0+a_1\cdot x+\cdots+a_n\cdot x^n$

In a generic equation $$a_0\cdot x+a_1\cdot x^2+ a_3\cdot x^3+\cdots+a_n\cdot x^n$$ there are some relations between roots ($x_1, x_2,\ldots,x_n$) and coefficients ($a_0, a_1,\ldots,a_n$). How can i ...