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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23
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9answers
1k 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. ...
104
votes
15answers
10k 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 ...
15
votes
5answers
3k views

How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?

How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$? Right now I'm able to prove that it has no roots and that it is separable, but I have not a clue as to ...
5
votes
2answers
5k 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 ...
16
votes
2answers
4k 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!
32
votes
7answers
43k 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 ...
3
votes
4answers
389 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 — ...
7
votes
2answers
2k 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 ...
1
vote
4answers
375 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 ...
46
votes
5answers
1k 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 ...
3
votes
3answers
623 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 ...
1
vote
3answers
2k views

How many irreducible polynomials of degree $n$ exist over $\mathbb{F}_p$?

I know that for every $n\in\mathbb{N}$, $n\ge 1$, there exists $p(x)\in\mathbb{F}_p[x]$ s.t. $\deg p(x)=n$ and $p(x)$ is irreducible over $\mathbb{F}_p$. I am interested in counting how many such ...
46
votes
7answers
5k 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!
23
votes
2answers
1k 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 ...
21
votes
2answers
1k views

Is factoring polynomials as hard as factoring integers?

There seems to be a consensus that factorization of integers is hard (in some precise computational sense.) Is it known whether polynomial factorization is computationally easy or hard? One thing I ...
3
votes
7answers
288 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 ...
4
votes
2answers
1k views

Zero divisor in $R[x]$

Let $R$ be commutative ring with no (nonzero) nilpotent elements. If $f(x) = a_0+a_1x+\cdots+a_nx^n$ in $R[x]$ is a zero divisor, how do I show there's an element $b \ne 0$ in $R$ such that ...
9
votes
2answers
971 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 ...
5
votes
2answers
1k 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 ...
1
vote
4answers
1k views

How to factor the quadratic polynomial $2x^2-5xy-y^2$?

How do I factor this polynomial: $2x^2-5xy-y^2$ ?
36
votes
5answers
2k 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 ...
15
votes
3answers
2k 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 ...
15
votes
5answers
1k views

Polynomial approximation of circle or ellipse

Trying again, with a somewhat simpler sounding question, since my previous one (Generalizations of equi-oscillation criterion) got zero response: Let $F:[0,1] \to R^2$ be a parametric polynomial ...
13
votes
1answer
190 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 ...
12
votes
3answers
925 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. ...
9
votes
5answers
287 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} \\ ...
4
votes
1answer
370 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 ...
8
votes
2answers
813 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
1answer
1k 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 ...
7
votes
0answers
464 views

Enestrom-Kakeya Theorem [duplicate]

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 ...
4
votes
2answers
229 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". ...
4
votes
7answers
862 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 ...
5
votes
2answers
336 views

Eisenstein Criterion with a twist

As opposed to the generic polynomial form for utilizing the Eisenstein Criterion ($a_nx^n+a_{n-1}x^{n-1}+\dots+a_0\in\mathbb{Z}[x]$ is irreducible in $\mathbb{Q}$) how do we prove that if $p$ is a ...
9
votes
4answers
973 views

showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$

I studied the cyclotomic extension using Fraleigh's text. To prove that Galois group of the $n$th cyclotomic extension has order $\phi(n)$( $\phi$ is the Euler's phi function.), the writer assumed, ...
2
votes
3answers
1k 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 ...
4
votes
2answers
679 views

Proof the Legendre polynomial $P_n$ has $n$ distinct real zeros

I need a proof to show that the inequality $m < n$ leads to a contradiction and $P_n$ has $n$ distinct real roots, all of which lie in the open interval $(-1, 1)$.
11
votes
2answers
1k views

Vandermonde Determinant

This is an exercise from Ian Stewart's Galois Theory, $3^{rd}$ edition: If $z_1,z_2,\ldots,z_n$ are distinct complex numbers, show that the determinant $$D=\left|\begin{array}[cccc] ...
7
votes
4answers
276 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 ...
5
votes
1answer
835 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
1k 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?
6
votes
3answers
664 views

Three-variable system of simultaneous equations

$x + y + z = 4$ $x^2 + y^2 + z^2 = 4$ $x^3 + y^3 + z^3 = 4$ Any ideas on how to solve for $(x,y,z)$ satisfying the three simultaneous equations, provided there can be both real and complex ...
5
votes
2answers
424 views

Polynomial $p(a) = 1$, why does it have at most 2 integer roots?

The question that I am trying to answer is : Suppose is $p(x)$ is a polynomial with integer coefficients. Show that if $p(a) = 1$ for some integer a then $p(x)$ has at most two integer roots. I have ...
4
votes
2answers
355 views

Meaning of $\mathbb{R}[x]$

I ran into this expression in a paper I was reading, and I'm confused about part of the meaning. Here $u$ and $v$ are two polynomials. $$u, v \in \mathbb{R}[x]$$ I'm not really familiar with usage ...
13
votes
5answers
2k views

Methods to see if a polynomial is irreducible

Given a polynomial over a field, what are the methods to see it is irreducible? Only two comes to my mind now. First is Eisenstein criterion. Another is that if a polynomial is irreducible mod p then ...
18
votes
1answer
318 views

$[(x-a_1)(x-a_2) \cdots (x-a_n)]^2 +1$ is irreducible over $\mathbb Q$

Suppose that $a_1,a_2, \cdots, a_n$ are $n$ different integers. Then $[(x-a_1)(x-a_2) \cdots (x-a_n)]^2 +1$ is irreducible over $\mathbb Q$. I've no idea why it is true. Thanks very much.
9
votes
1answer
273 views

$\epsilon>0$ there is a polynomial $p$ such that $|f(x)-e^{-x}p|<\epsilon\forall x\in[0,\infty)$

Could any one tell me how to solve this one? Given $f\in C[0,\infty)$ such that $f(x)\to 0$ as $x\to\infty$ we need to show that for any $\epsilon>0$ there is a polynomial $p$ such that ...
23
votes
1answer
524 views

An awful identity

We take place on $\mathbb C(x_1,...,x_r,x'_1,...,x'_p,u_0,...,u_r,u'_0,...,u'_p)$ with $r,p\in \mathbb N$ Show that : $$\displaystyle{\sum_{i=1}^r \left( \frac{\prod_{j=0}^r (u_j-x_i) ...
15
votes
5answers
1k views

Do we really need polynomials (In contrast to polynomial functions)?

In the following I'm going to call a polynomial expression an element of a suitable algebraic structure (for example a ring, since it has an addition and a multiplication) that has the form ...
11
votes
7answers
1k views

Factorize the polynomial $x^3+y^3+z^3-3xyz$

I want to factorize the polynomial $x^3+y^3+z^3-3xyz$. Using Mathematica I find that it equals $(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$. But how can I factorize it by hand?
24
votes
3answers
769 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 ...