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

1
vote
1answer
20 views

Write down basis for the set of all polynomials $f(x)$ of degree at most 5 such that $f(2) = 0$.

Write down basis for the set of all polynomials $f(x)$ of degree at most 5 such that $f(2) = 0$. I know there are lots of answers you could write, but would this be correct: $\{(x-2)^5, (x-2)^4, ...
0
votes
0answers
14 views

Polynomial Division for crc

I did this question by just using the xor long division of the binary, but my teacher said he doesn't want it done that way, but want me to use polynomial Division. I have no clue how to do this, and ...
1
vote
1answer
31 views

Find all polynomials such that $P(A)\subset U$ for a countable subset of the unit circle $U$

I recently answered a question, in which I proved that If a polynomial fixes the unit circle then $P$ is a monomial (a classical result),i,e: $$\forall P\in \Bbb C[X]\ \ \ \ (\forall z\in \Bbb C \ \ ...
2
votes
1answer
164 views

Help me understand Gröbner basis result please

I'm practicing a bit with Gröbner bases but I'm not understanding the following result I obtain from Mathematica: ...
2
votes
1answer
30 views

If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$.

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ such that $AB=BA$ and $\det > B\neq 0$. a) If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$. b) Is the ...
0
votes
0answers
9 views

Writing a series of polynomial equations of certain degree from a sequence of binary bits using Magma

How do I write a series of polynomial equations of a specified degree from a sequence of binary bits using Magma. So far, I have the following code for converting a decimal sequence to binary. ...
1
vote
2answers
56 views

Polynomials mod prime $p$

The problem is $5m^2+m+4 \equiv 0\pmod 7$. I am supposed to first convert it to a quadratic whose first coefficient is $1$. But the polynomial cannot be factored, so I am unsure as to how to do ...
-1
votes
0answers
15 views

Let $P(x),Q(x) \in \mathbb{Z}[x]$ such that, exist $a,b \in \mathbb{Z}^+$ and $a<b$: $P(a)=Q(a)$ and $P(b)=Q(b)$

Let $P(x),Q(x) \in \mathbb{Z}[x]$ such that, exist $a,b \in \mathbb{Z}^+$ and $a<b$: $P(a)=Q(a)$ and $P(b)=Q(b)$ Prove that $P \equiv Q$
0
votes
1answer
23 views

For any $n \in \mathbb{Z^+}$ Not extis $P(x) \in \mathbb{R}[x]$ with coefficients in $B$ and all roots of $P(x)$ in $A$

Problem: Let $A=\{a_1,a_2,..,a_m\}$ and $B=\{b_1,b_2,...,b_p\}$ where $a_1,a_2,...,a_m,b_1,b_2,...,b_p \in \mathbb{R}$ Prove that , the following statements is bad : for any $n \in ...
0
votes
1answer
56 views

Help required! Polynomials

Let $D(p) = p^{20} - p^{18} - p^{16} - \dots - p^2 - 2$ Prove that the sum of fourth powers of all the real roots of $D(p) = 8.$ Please help.
1
vote
1answer
17 views

Limit at $\infty$ of a polynomial multiplied by a negative exponential

I am trying to show $\int_0^{\infty} x^2 e^{-2 x} dx = 1/4 $ Integration by parts gets the indefinite integral $$\int x^2 e^{-2 x} dx = \frac{-1}{4} e^{-2 x} (2 x^2+2 x+1)+constant$$ In order to ...
2
votes
1answer
42 views

Determine the units of the ring $A= \mathbb Z[X]/(X^3)$ and the structure of the group $A^*$ [duplicate]

Determine the units of the ring $A= \mathbb Z[X]/(X^3)$ and the structure of the group $A^*$ I've only managed to show that the free coefficient of any unit in $A$ is a unit in $\mathbb Z$.
2
votes
4answers
60 views

Solve the following integral: $ \int \frac{x^2}{x^2+x-2} dx $

Solve the integral: $ \int \frac{x^2}{x^2+x-2} dx $ I was hoping that writing it in the form $ \int 1 - \frac{x-2}{x^2+x-2} dx $ would help but I'm still not getting anywhere. In the example it was ...
0
votes
0answers
16 views

Are there Karnaugh maps over other algebras?

Karnaugh maps are a useful way to minimize or factorize polynomial expressions in Boolean algebra by considering the smallest combinations of logical "subcomponents" of an expression, whose sum is ...
1
vote
2answers
45 views

$x^3+ (5m+1)x+ 5n+1$ is irreducible over $\Bbb Z$

How to prove that the polynomial: $x^3+ (5m+1)x+ 5n+1$ is irreducible over the set of integers for any integers $m$ and $n$? I was trying to put $x= y+p$ for some integer $p$ so that I could apply ...
0
votes
2answers
59 views

How should you go about simplying cubic polynomial: $y(x) = x^3+12x^2+21x+10$

Claim: $$y(x) = x^3+12x^2+21x+10$$ Can be factored into $$(x+1)^2(x+10)$$ But what is the quickest way to see this?
0
votes
1answer
31 views

What are the steps to function design?

So I'm trying to write a program, and I want to use math functions to help it. In this example, I'm trying to change the color of a line based on the position of each pixel on the line. Anyway, I ...
0
votes
2answers
43 views

Same roots, same polynomial? How to prove characteristic polynomial of $AB = BA$?

I'm giving a (simple) proof that the characteristic polynomial of $AB$ = characteristic polynomial of $BA$ (without using the fact that $AB$ and $BA$ are similar). $det(AB) = det(A)det(B) = ...
1
vote
1answer
39 views

How to prove whether $x^{2}+y^{2}+1$ is irreducible over $\mathbb{C}$ or not?

Let's consider a 2-variable polynomial $f(x, y)= x^{2}+y^{2}+1$. It can be established that it's irreducible over $\mathbb{R}$. For example, if it's irreducible over $\mathbb{R}$ as a polynomial of ...
0
votes
0answers
30 views

Lagrange interpolation given a list of points

I have to calculate a value in which I use Lagrange interpolate to calculate numerator and denominator individually. On dividing the interpolated numerator and denominator I don't get the required ...
0
votes
0answers
7 views

A question related Kharitonov's Theorem for Hurwitz stable interval polynomials

Definition: An interval polynomial is the family of all polynomials $$p(s)= a_0 + a_1 s^1 + a_2 s^2 + ... + a_n s^n\tag{1}$$ where each coefficient $a_i \in \mathbb{R}$ can take any value in the ...
4
votes
4answers
116 views

$tr(A)=tr(A^{2})= \ldots = tr(A^{n})=0$ implies $A$ nilpotency

Let's consider a $n \times n$ matrix and the sequence of traces $tr(A)=tr(A^{2})= \ldots = tr(A^{n})=0$. How to prove that $A$ is a nilpotent matrix (a matrix so that $A^{k} \times u = 0$ for all $u ...
0
votes
1answer
33 views

Factoring and solving a cubic polynomial

When can we not use synthetic division to solve for a cubic polynomial? For example we can use synthetic division to solve $-t^3 -4t^2 +20t +48$. When I can't use synthetic division what are my other ...
0
votes
0answers
30 views

Polynomial optimization and AM-GM inequality

I want to maximize the function $f(\mathbf{x},\mathbf{y}) = \sum \limits_{k=1}^{K}p_k(\mathbf{x})q_k(\mathbf{y})$, where $0 < p_k(\mathbf{x}) \leq \delta_k$ and $0 < q_k(\mathbf{y}) \leq ...
3
votes
1answer
54 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
137 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
32 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
24 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, ...
2
votes
0answers
28 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
33 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
35 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
45 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
36 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
60 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?