Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, dividing, factoring and solving for roots.
19
votes
8answers
940 views
why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$
why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$?
I know this is well-known. But how to prove it rigorously? Even mathematical induction does not seem so straight-forward.
...
64
votes
10answers
5k views
Why can ALL quadratic equations be solved by the quadratic formula?
In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use ...
5
votes
2answers
3k views
Reed Solomon Polynomial Generator
I am developing a sample program to generate a 2D Barcode. And i am using reed solomon error correction code. By Going through this article i am developing the program. But i couldn't understand how ...
3
votes
3answers
284 views
Factoring $ac$ to factor $ax^2+bx+c$
I was watching a first-year high-school-algebra student struggle with factoring quadratics last night. Given a quadratic $ax^2+bx+c$ (I'll give you the exact example in a moment), her method — ...
1
vote
3answers
267 views
Solving a recurrence of polynomials
I am wondering how to solve a recurrence of this type
$$p_1(x) = x$$
$$p_2(x) = 1-x^2$$
and
$$p_{n+2}(x) = -xp_{n+1}(x)+p_{n}(x).$$
I am wondering, how could one solve such a recurrence. One way ...
24
votes
7answers
15k views
Is there a general formula for solving 4th degree equations?
There is a general formula for solving quadratic equations, namely the Quadratic Formula.
For third degree equations of the form $ax^3+bx^2+cx+d=0$, there is a set of thee equations: one for each ...
8
votes
2answers
2k views
Number of monic irreducible polynomials of degree $p$ over finite fields
Suppose $F$ is a field s.t $\left|F\right|=q$. Take $p$ to be some prime. How many monic irreducible polynomials of degree $p$ can exist over $F$?
Thanks!
5
votes
2answers
871 views
reducible polynomial modulo every prime
how to show that $x^4+1$ is irreducible in $\mathbb Z[x]$ but it is reducible modulo every prime $p$.
For example i know that $\mod 2 $, $x^4+1=(x+1)^4$ . Also $\mod 3$,we have that $0,1,2$ are not ...
40
votes
5answers
823 views
Why are the solutions of polynomial equations so unconstrained over the quaternions?
An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb ...
8
votes
2answers
539 views
Methods to check if an ideal of a polynomial ring is prime or at least radical
I am looking for methods to check whether a given ideal in $K[x_0,\dots,x_n]$ is prime. I mean something you can effectively use in some concrete non-trivial example.
To be more explicit, I am working ...
10
votes
3answers
741 views
Why are polynomials defined to be “formal”?
Despite the fact that $\forall n, n^3 + 2n \equiv 0 \pmod 3$, I understand that $n^3 + 2n$ (considered as a polynomial with coefficients in $\mathbb Z/3\mathbb Z$) is not equal to the zero polynomial.
...
3
votes
2answers
169 views
For what $(n,k)$ there exists a polynomial $p(x) \in F_2[x]$ s.t. $\deg(p)=k$ and $p$ divides $x^n-1$?
For what $(n,k)$ there exists a polynomial $p(x) \in F_2[x]$ s.t. $\deg(p)=k$ and $p$ divides $x^n-1$?
Motivation: if exists $p(x)$, then ideal generated by $p(x)$ is "cyclic error correcting code". ...
3
votes
3answers
229 views
Irreducibility of $X^{p-1} + \ldots + X+1$
Can someone give me a hint how to the irreducibility of $X^{p-1} + \ldots + X+1$, where $p$ is a prime, in $\mathbb{Z}[X]$ ? Our professor gave us already one, namely to substitute $X$ with $x+1$, but ...
39
votes
4answers
2k views
Prove every odd integer is the difference of two squares
I know that I should use the definition of an odd integer ($2k+1$), but that's about it.
Thanks in advance!
12
votes
3answers
1k views
Why is it so hard to find the roots of polynomial equations?
The question that follows was inspired by this question:
When trying to solve for the roots of a polynomial equation, the quadratic formula is much more simple than the cubic formula and the cubic ...
2
votes
3answers
402 views
Create polynomial coefficients from its roots
Given some roots : $r_1,r_2,\ldots,r_n$, how can we reconstruct polynomial coefficients?
I know the Horner scheme and that we can just go backwards receiving those coefficients.
But I'm curious if ...
8
votes
2answers
569 views
Finding roots of polynomials with rational coefficients
I'm looking for a general approach (or approaches) for finding the roots of polynomials with rational coefficients of higher degrees than $4$. The problem is that I need to find the exact roots and ...
5
votes
2answers
652 views
symmetric polynomials and the Newton identities
I want to write
$P(x,y,z)=yx^{3}+zx^{3}+xy^{3}+zy^{3}+xz^{3}+yz^{3}$
in terms of elementary symmetric polynomials, but I'm getting stuck at the first step. I know I should follow the proof of the ...
4
votes
1answer
628 views
Why not write the solutions of a cubic this way?
For the solution of the cubic equation $x^3 + px + q = 0$ Cardano wrote it as:
$$\sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}+\sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^2}{4} + ...
2
votes
2answers
540 views
Irreducible polynomial of $\mathrm{GF}(2^{16})$
I'm implementing some code for the Galois field $\mathrm{GF}(2^{16})$. Which irreducible polynomial do you recommend that I use?
3
votes
5answers
337 views
How do I come up with a function to count a pyramid of apples?
My algebra book has a quick practical example at the beginning of the chapter on polynomials and their functions. Unfortunately it just says "this is why polynomial functions are important" and moves ...
13
votes
2answers
437 views
Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$
Let $f(x) \in \mathbb{Z}[x]$. If we reduce the coefficents of $f(x)$ modulo $p$, where $p$ is prime, we get a polynomial $f^*(x) \in \mathbb{F}_p[x]$. Then if $f^*(x)$ is irreducible and has the same ...
23
votes
3answers
626 views
Galois groups of polynomials and explicit equations for the roots
Lets say I have calculated the galois group of some polynomial and I also have the subgroup structure. What's an effective procedure to turn the group into equations for the actual roots of the ...
4
votes
1answer
222 views
Holomorphic function of a matrix
A statement is made below. The questions are:
(a) Is the statement true?
(b) If it is, does it appear in the literature?
Here is the statement.
For any matrix $A$ in $M_n(\mathbb C)$, write ...
9
votes
2answers
581 views
Is this a known algebraic identity?
In the course of analyzing a certain Markov chain, I once had to prove the following algebraic identity.
Is there a slick or known proof?
For $n$-tuples $(x_1,x_2,\dots, x_n)$ of positive real ...
7
votes
4answers
242 views
Irreducibility issue [duplicate]
This is a homework question.
Given $f(x)=x^{p-1}+x^{p-2}+\cdots+x+1$, where $p$ is any prime.
Prove that $f(x)$ is irreducible over $\mathbb{Z}[x]$?
Any idea, hint, etc? Hint given by my book was ...
7
votes
2answers
525 views
Is there a General Formula for the Transition Matrix from Products of Elementary Symmetric Polynomials to Monomial Symmetric Functions?
Given the elementary symmetric polynomials $e_k(X_1,X_2,...,X_N)$ generated via
$$
\prod_{k=1}^{N} (t+X_k) = e_0t^N + e_1t^{N-1} + \cdots + e_N.
$$
How can one get the monomial symmetric functions ...
6
votes
1answer
224 views
Iterated polynomial problem
Polynomial $P$ satisfies $P(n)>n$ for all positive integers $n$. Every positive integer $m$ is a factor of some number of the form $P(1),P(P(1)),P(P(P(1))),\ldots $. Prove that $P(x)=x+1$.
4
votes
1answer
728 views
Why $x^{p^n}-x+1$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$
I have a question, I think it concerns with field theory.
Why the polynomial $$x^{p^n}-x+1$$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$?
Thanks in advance. It bothers me for ...
4
votes
4answers
430 views
Factoring $a^{10}+a^5+1$
I'm very interested to know how I can factorise $a^{10} +a^5 +1$ in two factors with integer coefficients. I've tried a lot but I don't have any idea how do that.
3
votes
0answers
382 views
Enestrom-Kakeya Theorem
The Enestrom-Kakeya theorem states that all roots of the polynomial:
$$p(z):=\sum_{k=0}^n a_kz^k$$
lie outside the open unit disk if the sequence $(a_k)$ is positive and decreasing.
A proof can be ...
3
votes
4answers
431 views
Polynomial of degree $-\infty$?
I'm reading E.J Barbeau Polynomials. I'm in a page where he asks a polynomial of degree $-\infty$. Then I thought about $77x^{-\infty}+1$, but when I went for the answers, the answer to this question ...
3
votes
5answers
284 views
How to “Re-write completing the square”: $x^2+x+1$
The exercise asks to "Re-write completing the square": $$x^2+x+1$$
The answer is: $$(x+\frac{1}{2})^2+\frac{3}{4}$$
I don't even understand what it means with "Re-write completing the square"..
...
1
vote
1answer
90 views
Property of an operator in a finite-dimensional vector space $V$ over $R$
Let $L: V\to V$ be an operator in a finite-dimensional vector space $V$ over $R$.
For any $n \geq 0$, let $K_n = \ker (L^n)$, $I_n = \mathrm{Im}(L^n)$.
(a) Prove that there exists $N$ such that ...
5
votes
2answers
141 views
Closure of a number field with respect to roots of a cubic
Consider the following property of subfields ${\mathbb K}$ of ${\mathbb C}$ :
$$
\text{Any polynomial of degree 3 with coefficients in} \ {\mathbb K} \ \text{has a root in } {\mathbb K} \ \ \ \ ...
3
votes
2answers
517 views
Scalar Product for Vector Space of Monomial Symmetric Functions
Suppose a multinomial $P(X_1, X_2,\ldots, X_n)$, that is given as a sum of monomials $m_\lambda$ with coefficients $c_k$:
$$
P(\vec{X})=P(X_1, X_2,\ldots, X_n) = \sum_k c_k m_{\lambda_k} .
$$
Since ...
3
votes
2answers
218 views
If $2 x^4 + x^3 - 11 x^2 + x + 2 = 0$, then find the value of $x + \frac{1}{x}$?
If $2 x^4 + x^3 - 11 x^2 + x + 2 = 0$, then find the value of $x + \frac{1}{x}$ ?
I would be very grateful if somebody show me how to factor this polynomial by hand, as of now I have used to ...
2
votes
2answers
153 views
Test for an Integer Solution
This came up an a training piece for the Putnam Competition and also in Ireland and Rosen.
The question posed was basically:
Let $p(x)$ be a polynomial with integer coefficients satisfying that ...
1
vote
6answers
870 views
How to find the root of $x^4 +1$
I'm trying to find the roots of $x^4+1$. I've already found in this site solutions for polynomials like this $x^n+a$, where $a$ is a negative term. I don't remember how to solve an equation when $a$ ...
19
votes
5answers
2k views
Using Gröbner bases for solving polynomial equations
In my attempts to understand just how computer algebra systems "do things", I tried to dig around a bit on Gröbner bases, which are described almost everywhere as "a generalization of the Euclidean ...
29
votes
5answers
1k views
Continuity of the roots of a polynomial in terms of its coefficients
It's commonly stated that the roots of a polynomial are a continuous function of the coefficients. How is this statement formalized? I would assume it's by restricting to polynomials of a fixed ...
10
votes
1answer
124 views
Polynomial $P(x,y)$ with $\inf_{\mathbb{R}^2} P=0$, but without any point where $P=0$
Recently I've came across such problem: give a polynomial $P(x,y)$, with $\inf_{\mathbb{R}^2} P=0$, but there is no point on the plane where $P=0$. I couldn't solve it after a day, and seriously doubt ...
8
votes
1answer
177 views
When do Harmonic polynomials constitute the kernel of a differential operator?
Let $f$ be a real polynomial of two variables. Let $\partial_f=f\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)$. Let $H$ denote the space of harmonic polynomials, i.e., ...
4
votes
1answer
242 views
rational angles with sines expressible with radicals
An angle x is rational when measured in degrees. sin(x) is can be written using radicals. What are the conditions on x? If nested square roots are allowed?
What I know so far:
If sin(x) can be ...
14
votes
4answers
2k views
Finding the minimal polynomial of $\sqrt 2 + \sqrt[3] 2$ over $\mathbb Q$.
I have to find the minimal polynomial of $\sqrt 2 + \sqrt[3] 2$ over $\mathbb Q$. The suggested way of doing it is to prove that $\mathbb Q[\sqrt 2 + \sqrt[3] 2]=\mathbb Q[\sqrt 2,\sqrt[3] 2]$ first.
...
8
votes
5answers
221 views
Constructing a degree 4 rational polynomial satisfying $f(\sqrt{2}+\sqrt{3}) = 0$
Goal: Find $f \in \mathbb{Q}[x]$ such that $f(\sqrt{2}+\sqrt{3}) = 0$.
A direct approach is to look at the following
$$
\begin{align}
(\sqrt{2}+\sqrt{3})^2 &= 5+2\sqrt{6} \\
...
7
votes
1answer
94 views
Vandermonde identity in a ring
Let $R$ be a commutative $\mathbb{Q}$-algebra. For $r \in R$ and $n \in \mathbb{N}$ we can define the binomial coefficient $\binom{r}{n}$ as usual by $\binom{r}{0}=1$ and ...
5
votes
2answers
332 views
Are quartic minimal polynomials over $\mathbb{Q}$ always reducible over $\mathbb{F}_p$?
This situation arose while studying biquadratic extensions.
Let $\mathbb{Q}(\alpha)$ is some biquadratic extension, with $m(x)$ the minimal polynomial of $\alpha$. Suppose that ...
1
vote
1answer
227 views
Multiple choice question - number of real roots of $x^6 − 5x^4 + 16x^2 − 72x + 9$
The equation $x^6 − 5x^4 + 16x^2 − 72x + 9 = 0$ has
(A) exactly two distinct real roots
(B) exactly three distinct real roots
(C) exactly four distinct real roots
(D) six distinct real roots
9
votes
2answers
261 views
