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Questions tagged [polynomials]

For both basic and advanced questions on polynomials in any number of variables, including, but not limited to solving for roots, factoring, and checking for irreducibility.

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298 votes
22 answers
49k 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 ...
idonno's user avatar
  • 3,929
195 votes
5 answers
8k views

A Topology such that the continuous functions are exactly the polynomials

I was wondering which fields $K$ can be equipped with a topology such that a function $f:K \to K$ is continuous if and only if it is a polynomial function $f(x)=a_nx^n+\cdots+a_0$. Obviously, the ...
Dominik's user avatar
  • 14.4k
172 votes
15 answers
357k views

Is there a general formula for solving Quartic (Degree $4$) equations?

There is a general formula for solving quadratic equations, namely the Quadratic Formula, or the Sridharacharya Formula: $$x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a } $$ For cubic equations of the ...
John Gietzen's user avatar
  • 3,501
170 votes
1 answer
5k views

Rational roots of polynomials

Can one construct a sequence $(a_k)_{k\geqslant 0}$ of rational numbers such that, for every positive integer $n$ the polynomial $a_nX^n+a_{n-1}X^{n-1}+\cdots +a_0$ has exactly $n$ distinct rational ...
user84673's user avatar
  • 2,027
163 votes
19 answers
21k views

What actually is a polynomial?

I can perform operations on polynomials. I can add, multiply, and find their roots. Despite this, I cannot define a polynomial. I wasn't in the advanced mathematics class in 8th grade, then in 9th ...
Travis's user avatar
  • 3,396
151 votes
4 answers
9k views

Does $R[x] \cong S[x]$ imply $R \cong S$?

This is a very simple question but I believe it's nontrivial. I would like to know if the following is true: If $R$ and $S$ are rings and $R[x]$ and $S[x]$ are isomorphic as rings, then $R$ and $...
Richard G 's user avatar
  • 3,925
135 votes
6 answers
10k 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 $\mathbb C$,...
MJD's user avatar
  • 65.4k
132 votes
0 answers
3k views

If polynomials are almost surjective over a field, is the field algebraically closed?

Let $K$ be a field. Say that polynomials are almost surjective over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. ...
Eric Wofsey's user avatar
129 votes
1 answer
4k views

Is There An Injective Cubic Polynomial $\mathbb Z^2 \rightarrow \mathbb Z$?

Earlier, I was curious about whether a polynomial mapping $\mathbb Z^2\rightarrow\mathbb Z$ could be injective, and if so, what the minimum degree of such a polynomial could be. I've managed to ...
Milo Brandt's user avatar
128 votes
9 answers
23k 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 ...
Harry Altman's user avatar
  • 4,662
107 votes
13 answers
12k views

Why would I want to multiply two polynomials?

I'm hoping that this isn't such a basic question that it gets completely laughed off the site, but why would I want to multiply two polynomials together? I flipped through some algebra books and ...
user3818's user avatar
  • 1,079
107 votes
1 answer
3k views

All polynomials with no natural roots and integer coefficients such that $\phi(n)|\phi(P(n))$

Let $P(x)$ be a polynomial with integer coefficients such that the equation $P(x)=0$ has no positive integer solutions. Find all polynomials $P(x)$ such that for all positive integers $n$ we have $\...
Amir Parvardi's user avatar
104 votes
8 answers
31k views

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

Let $p$ be a prime. 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 ...
MathTeacher's user avatar
  • 1,559
101 votes
4 answers
4k views

A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...
Dan Ismailescu's user avatar
96 votes
7 answers
4k views

Polynomials such that roots=coefficients

Here is my question : Are there monic polynomials with degree $\geq 5$ such that they have the same real all non zero roots and coefficients ? Mathematically, prove or disprove the existence of $n \...
Gabriel Romon's user avatar
95 votes
5 answers
4k views

Reversing an integer's digits is multiplicative for small digits

So my 7 year old son pointed out to me something neat about the number 12: if you multiply it by itself, the result is the same as if you took 12 backwards multiplied by itself, then flipped the ...
BCA's user avatar
  • 793
83 votes
3 answers
12k 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 ...
spin's user avatar
  • 12k
82 votes
1 answer
14k views

How to solve fifth-degree equations by elliptic functions?

From F. Klein's books, It seems that one can find the roots of a quintic equation $$z^5+az^4+bz^3+cz^2+dz+e=0$$ (where $a,b,c,d,e\in\Bbb C$) by elliptic functions. How to get that? Update: How to ...
ziang chen's user avatar
  • 7,771
73 votes
17 answers
173k views

What is a real world application of polynomial factoring?

The wife and I are sitting here on a Saturday night doing some algebra homework. We're factoring polynomials and had the same thought at the same time: when will we use this? I feel a bit silly ...
Dan's user avatar
  • 871
71 votes
8 answers
10k 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 ...
J. M. ain't a mathematician's user avatar
70 votes
3 answers
31k views

Characterizing units in polynomial rings

I am trying to prove a result, for which I have got one part, but I am not able to get the converse part. Theorem. Let $R$ be a commutative ring with $1$. Then $f(X)=a_{0}+a_{1}X+a_{2}X^{2} + \cdots +...
user avatar
67 votes
2 answers
32k views

Number of monic irreducible polynomials of prime 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 $p$ do exist over $F$? Thanks!
IBS's user avatar
  • 4,155
66 votes
9 answers
61k 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!
papercuts's user avatar
  • 1,873
65 votes
7 answers
10k views

Compute polynomial $p(x)$ if $x^5=1,\, x\neq 1$ [reducing mod $\textit{simpler}$ multiples]

The following question was asked on a high school test, where the students were given a few minutes per question, at most: Given that, $$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$ and, $$Q(x)=x^4+...
joeblack's user avatar
  • 1,013
65 votes
7 answers
6k views

Is there any geometric intuition for the factorials in Taylor expansions?

Given a smooth real function $f$, we can approximate it as a sum of polynomials as $$f(x+h)=f(x)+h f'(x) + \frac{h^2}{2!} f''(x)+ \dotsb = \sum_{k=0}^n \frac{h^k}{k!} f^{(k)}(x) + h^n R_n(h),$$ where $...
glS's user avatar
  • 6,893
61 votes
10 answers
160k views

How to solve an $n$-th degree polynomial equation

The typical approach of solving $$ f_2(x):=ax^2+bx+c=0 $$ is to solve for the roots $$x_{1/2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.$$ Here, the degree of $f$ is given to be $2$. However, I was wondering ...
Ayush Khemka's user avatar
60 votes
10 answers
10k views

Why does the discriminant in the Quadratic Formula reveal the number of real solutions?

Why does the discriminant in the quadratic formula reveal the number of real solutions to a quadratic equation? That is, we have one real solution if $$b^2 -4ac = 0,$$ we have two real solutions if $...
user avatar
60 votes
1 answer
2k views

Is $1+x+\frac{x^2}2+\dots+\frac{x^n}{n!}$ irreducible?

The polynomial $f(x)=1+x+\frac{x^2}2+\dots+\frac{x^n}{n!}$ often appears in algebra textbooks as an illustration for using derivative to test for multiple roots. Recently, I stumbled upon Example 2.1....
Martin Sleziak's user avatar
59 votes
2 answers
11k views

$x^p-c$ has no root in a field $F$ if and only if $x^p-c$ is irreducible?

Hungerford's book of algebra has exercise $6$ chapter $3$ section $6$ [Probably impossible with the tools at hand.]: Let $p \in \mathbb{Z}$ be a prime; let $F$ be a field and let $c \in F$. Then $x^...
user79709's user avatar
  • 593
54 votes
7 answers
29k 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 ...
user avatar
53 votes
12 answers
27k views

Is There A Polynomial That Has Infinitely Many Roots?

Is there a polynomial function $P(x)$ with real coefficients that has an infinite number of roots? What about if $P(x)$ is the null polynomial, $P(x)=0$ for all x?
Карпатський's user avatar
52 votes
5 answers
5k views

Solution to the equation of a polynomial raised to the power of a polynomial.

The problem at hand is, find the solutions of $x$ in the following equation: $$ (x^2−7x+11)^{x^2−7x+6}=1 $$ My friend who gave me this questions, told me that you can find $6$ solutions without ...
user avatar
52 votes
3 answers
18k views

Irreducible polynomial which is reducible 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 $x^4+1=(x+1)^4\bmod 2$. Also $\bmod 3$ we have that $0,1,2$ are not ...
palio's user avatar
  • 11.1k
52 votes
5 answers
3k views

AM-GM-HM Triplets

I want to understand what values can be simultaneously attained as the arithmetic (AM), geometric (GM), and harmonic (HM) means of finite sequences of positive real numbers. Precisely, for what points ...
Srivatsan's user avatar
  • 26.3k
52 votes
1 answer
2k views

On the problem of polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$

The question titled "Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$" which was posed on MathOverflow attracted quite a lot of attention (and may be the question with most wrong ...
Dal's user avatar
  • 8,234
51 votes
2 answers
3k views

Decomposing polynomials with integer coefficients

Can every quadratic with integer coefficients be written as a sum of two polynomials with integer roots? (Any constant $k \in \mathbb{Z}$, including $0$, is also allowed as a term for simplicity's ...
Vandermonde's user avatar
  • 2,674
50 votes
5 answers
11k 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 ...
Ami's user avatar
  • 938
50 votes
8 answers
4k views

Appearance of Formal Derivative in Algebra

When studying polynomials, I know it is useful to introduce the concept of a formal derivative. For example, over a field, a polynomial has no repeated roots iff it and its formal derivative are ...
Dylan Yott's user avatar
  • 7,019
50 votes
5 answers
2k views

Paradox: Roots of a polynomial require less information to express than coefficients?

A somewhat information theoretical paradox occurred to me, and I was wondering if anyone could resolve it. Let $p(x) = x^n + c_{n-1} x^{n-1} + \cdots + c_0 = (x - r_0) \cdots (x - r_{n-1})$ be a ...
chausies's user avatar
  • 2,200
50 votes
1 answer
2k views

Is $ f_n=\frac{(x+1)^n-(x^n+1)}{x}$ irreducible over $\mathbf{Z}$ for arbitrary $n$?

In this document on page $3$ I found an interesting polynomial: $$f_n=\frac{(x+1)^n-(x^n+1)}{x}.$$ Question is whether this polynomial is irreducible over $\mathbf{Q}$ for arbitrary $n \geq 1$. ...
Pedja's user avatar
  • 12.9k
49 votes
3 answers
4k views

Does multiplying polynomials ever decrease the number of terms?

Let $p$ and $q$ be polynomials (maybe in several variables, over a field), and suppose they have $m$ and $n$ non-zero terms respectively. We can assume $m\leq n$. Can it ever happen that the product $...
Chris Brooks's user avatar
  • 7,424
47 votes
1 answer
4k views

Does convergence of polynomials imply that of its coefficients?

Let $\{p_{n}\}$ be a sequence of polynomials and $f$ a continuous function on $[0,1]$ such that $\int\limits_{0}^{1}|p_{n}(x)-f(x)|dx\to 0$. Let $c_{n,k}$ be the coefficient of $x^{k}$ in $p_{n}(x)$. ...
Kavi Rama Murthy's user avatar
47 votes
5 answers
2k views

If $f(x)=x^2-x-1$ and $f^n(x)=f(f(\cdots f(x)\cdots))$, find all $x$ for which $f^{3n}(x)$ converges.

Let $f:\mathbb{R}\to\mathbb{R}$ be the polynomial defined by $$f(x)=x^2-x-1$$ and let $$g_0(x)=f(x),\quad g_1(x)=f(f(x)),\quad\ldots\quad g_n(x)=f(f(f(\cdots f(x)\cdots)))$$ The positive root of $f(x)$...
Simon Parker's user avatar
  • 4,303
46 votes
1 answer
2k views

Every polynomial with real coefficients is the sum of cubes of three polynomials

How to prove that every polynomial with real coefficients is the sum of three polynomials raised to the 3rd degree? Formally the statement is: $\forall f\in\mathbb{R}[x]\quad \exists g,h,p\in\...
Glinka's user avatar
  • 3,192
46 votes
1 answer
3k views

What are ways to compute polynomials that converge from above and below to a continuous and bounded function on $[0,1]$?

Main Question Suppose $f:[0,1]\to [0,1]$ is continuous and belongs to a large class of functions (for example, the $k$-th derivative, $k\ge 0$, is continuous, Lipschitz continuous, concave, strictly ...
Peter O.'s user avatar
  • 931
45 votes
8 answers
8k 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. ...
Qiang Li's user avatar
  • 4,107
44 votes
6 answers
8k views

Find $xy+yz+zx$ given systems of three homogenous quadratic equations for $x, y, z$

This is a question from Math Olympiad. If $\{x,y,z\}\subset\Bbb{R}^+$ and if $$x^2 + xy + y^2 = 3 \\ y^2 + yz + z^2 = 1 \\ x^2 + xz + z^2 = 4$$ find the value of $xy+yz+zx$. I basically do not ...
MIT998's user avatar
  • 477
44 votes
2 answers
1k views

Only 12 polynomials exist with given properties

Prove that there are only 12 polynomials that have all real roots, and whose coefficients are all $-1$ or $1$. (zero coefficients are not allowed, and constant polynomials do not count.) Two of ...
VividD's user avatar
  • 16k
43 votes
5 answers
17k views

Zero divisor in $R[x]$

Let $R$ be commutative ring with no (nonzero) nilpotents. If $f(x) = a_0+a_1x+\cdots+a_nx^n$ is a zero divisor in $R[x]$, how do I show there's an element $b \ne 0$ in $R$ such that $ba_0=ba_1=\cdots=...
Mohan's user avatar
  • 14.9k
43 votes
2 answers
2k views

Which polynomials are characteristic polynomials of a symmetric matrix?

Let $f(x)$ be a polynomial of degree $n$ with coefficients in $\mathbb{Q}$. There are well-known ways to construct a $n \times n$ matrix $A$ with entries in $\mathbb{Q}$ whose characteristic ...
Paul Siegel's user avatar
  • 9,127

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