This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

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15
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2answers
720 views

Characteristic polynomials exhaust all monic polynomials?

Let $A$ be an $n\times n$ matrix, then $\mathrm{char}_A(x):=\det(xI-A)$ is a monic polynomial of degree $n$. It is called the characteristic polynomial of $A$. My question is the converse: Let ...
15
votes
5answers
2k 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 ...
15
votes
1answer
326 views

Why is $x^3-5x$ injective on the rationals?

I've found the statement on the internet that the polynomial $x^3-5x$ is injective on the rational numbers, but without any comments on how to prove it. I think it means it must be easy, but I don't ...
15
votes
4answers
6k 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. ...
15
votes
2answers
3k views

Why does the discriminant of a cubic polynomial being less than 0 indicate complex roots?

The discriminant $\Delta = 18abcd - 4b^3d + b^2 c^2 - 4ac^3 - 27a^2d^2$ of the cubic polynomial $ax^3 + bx^2 + cx+ d$ indicates not only if there are repeated roots when $\Delta$ vanishes, but also ...
15
votes
1answer
619 views

Is factoring polynomials easier than factoring integers? [duplicate]

I was reading the book Algebra: Chapter 0 , by Paolo Aluffi, and came across the following assertion, in page 290, Exercise 5.9: It is in fact much harder to factor integers than integers ...
15
votes
3answers
565 views

How prove $P(x)=\sum_{k=0}^{n}(2k+1)x^k$ is irreducible over $\mathbb{Q}$

Show that the polynomial $$P(x)=\sum_{k=0}^{n}(2k+1)x^k,\forall n\in N^{+}$$ is irreducible over $Q$. My try: Since $P(x)$ has integer coefficients and the gcd of these coefficients is $1$, by ...
15
votes
1answer
335 views

Understanding proof by infinite descent, Fermat's Last Theorem.

See here. The question is as follows. How do we see that there do not exist nonconstant, relatively prime, polynomials $a(t)$, $b(t)$, and $c(t) \in \mathbb{C}[t]$ such that$$a(t)^3 + b(t)^3 = ...
15
votes
3answers
3k views

Shortcut/trick for integrating a factored polynomial?

If I'm integrating a factored polynomial, say $$\int{x(x+1)(x-2)(x+3)dx},$$ does some shortcut exist that keeps me from having to expand the polynomial? Currently, I'd just do all the multiplication ...
15
votes
2answers
456 views

Necessary and sufficient conditions for a polynomial $p$ to satisfy $\|x\|\to\infty\implies p(x)\to\infty$?

I'm looking for a necessary and sufficient conditions (I'm not even sure these exist) for a polynomial $p:\mathbb{R}^n\to\mathbb{R}$ to be "radially unbounded", that is $$\|x\|\to\infty\implies ...
14
votes
2answers
958 views

Prove that a polynomial has at least one nonreal complex root

Prove that the polynomial below has at least one nonreal complex root $$x^5+\frac{x^4}2+ \frac{x^3}3+\frac{x^2}4+\frac x{24}+\frac 1{120}$$ I have tried to prove that there exist $k\in \Bbb R$, such ...
14
votes
3answers
508 views

Why is the difference of distinct roots of irreducible $f(x)\in\mathbb{Q}[x]$ never rational?

The way I understand it, is that if $f(x)$ is an irreducible polynomial in $\mathbb{Q}[x]$ of degree at least 2, then a difference of distinct roots $a_i-a_j$ is never rational for any of the ...
14
votes
3answers
900 views

Solve $3n^3 + 3n^2 + 4n = n^n$ in positive integers

So my cousin is in the math team (7th grade) and he was asking me for help on one of his problems but I don't know how to solve For what positive integer n does $3n^3 + 3n^2 + 4n = n^n$ anyone know ...
14
votes
2answers
816 views

Can we introduce new operations that make quintics solvable?

I have heard from various sources that the typical arithmetic operations (addition, subtraction, multiplication, division, rational exponentiation) are not sufficient to express in general the roots ...
14
votes
2answers
864 views

Is Vieta the only way out?

Let $a,b,c$ are the three roots of the equation $x^3-x-1=0$. Then find the equation whose roots are $\frac{1+a}{1-a}$,$\frac{1+b}{1-b}$,$\frac{1+c}{1-c}$. The only solution I could think of is by ...
14
votes
9answers
464 views

Prove that $f=x^4-4x^2+16\in\mathbb{Q}$ is irreducible

Prove that $f=x^4-4x^2+16\in\mathbb{Q}[x]$ is irreducible. I am trying to prove it with Eisenstein's criterion but without success: for p=2, it divides -4 and the constant coefficient 16, don't ...
14
votes
4answers
3k 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 ...
14
votes
4answers
2k views

Prove that $f(x)$ has no integer roots

Suppose $f(x) \in \mathbb{Z}[x]$ is such that $f(0)$ and $f(1)$ are odd. How do I show that $f(x)$ has no integer roots?
14
votes
10answers
35k views

How to solve an nth degree polynomial equation

The typical approach of solving a quadratic equation is to solve for the roots $$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$$ Here, the degree of x is given to be 2 However, I was wondering on how to solve ...
14
votes
4answers
591 views

Polynomial: $p(x) = p(x+3)$.

Determine polynomial $p(x)$ s.t. $p(x) = p(x+3)$. Just by looking at the above equation, it immediately appears that p has got to be some kind of constant function. I thought it might also be a ...
14
votes
3answers
967 views

Why is the zero polynomial not assigned a degree?

Yesterday, I read in my textbook, We assign degree to every polynomial and even a non-zero constant is assigned a degree $0$ but $0$ itself is not assigned a degree. Why is that? Why we don't ...
14
votes
4answers
757 views

Reducibility over $\mathbb{Z}_2$?

I have seen that $x^{2}+x+1$ and $x^{4}+x+1$ are irreducible over $\mathbb{Z}_2$ and I thought a polynomial of the form $x^{2^m}+x+1$ for $m\ge3$ would be irreducible too. However using WolframAlpha, ...
14
votes
1answer
269 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 ...
14
votes
5answers
911 views

How to solve a cyclic quintic in radicals?

Galois theory tells us that $\frac{z^{11}-1}{z-1} = z^{10} + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1$ can be solved in radicals because its group is solvable. Actually performing the ...
14
votes
1answer
4k views

Using Vieta's theorem for cubic equations to derive the cubic discriminant

Background: Vieta's Theorem for cubic equations says that if a cubic equation $x^3 + px^2 + qx + r = 0$ has three different roots $x_1, x_2, x_3$, then $$\begin{eqnarray*} -p &=& x_1 + x_2 ...
14
votes
1answer
480 views

Existence of Irreducible polynomials over $\mathbb{Z}$ of any given degree

Question is to prove : Irreducibility of $(x-1)(x-2)\cdots (x-n)- 1$ over $\mathbb{Z}$ for all $n\geq 1$ Irreducibility of $(x-1)(x-2)\cdots (x-n)+ 1$ over $\mathbb{Z}$ for all $n\geq 1$ and ...
14
votes
2answers
1k views

Inverse function of $y=\frac{\ln(x+1)}{\ln x}$

I've been wondering for a while if it's possible to find the inverse function of $y=\frac{\ln(x+1)}{\ln x}$ over the reals. This is the same as finding the positive real root of $x^y-x-1$. I realize ...
14
votes
1answer
130 views

Prove that ${x^7-1 \over x-1}=y^5-1$ has no integer solutions

I want to show that $${x^7-1 \over x-1}=y^5-1$$ cannot have any integer solutions. The only observation I have made so far is that the left hand side is the $7$th cyclotomic polynomial $$\Phi_7(x)= ...
14
votes
1answer
170 views

If $p$ is a positive multivariate polynomial, does $1/p$ have polynomial growth?

I wanted to ask a separate question to focus on an elementary issue from my question Does the inverse of a polynomial matrix have polynomial growth?. Let $p : \mathbb{R}^n \to \mathbb{R}$ be a ...
14
votes
1answer
182 views

Natural density of solvable quintics

A recent question asked about the topological density of solvable monic quintics with rational coefficients in the space of all monic quintics with rational coefficients. Robert Israel gave a nice ...
14
votes
1answer
700 views

Multiplicative norm on $\mathbb{R}[X]$.

How to prove that : there is no function $N\colon \mathbb{R}[X] \rightarrow \mathbb{R}$, such that : $N$ is a norm of $\mathbb{R}$-vector space and $N(PQ)=N(P)N(Q)$ for all $P,Q \in \mathbb{R}[X]$. ...
14
votes
2answers
187 views

Is a linear combination of minors irreducible?

Let $X=(X_{ij})_{1\le i,j\le n}$ be a matrix of indeterminates over $\mathbb C$. For choices $I,J\subseteq\{1,\ldots,n\}$ with $|I|=|J|=k$ denote by $X_{I\times J}$ the matrix $(X_{ij})_{i\in I,j\in ...
14
votes
1answer
257 views

If $\gamma\in \Bbb A$ then there exists a $\pm$quadratic coefficiented polynomial for which $\gamma$ is a root.

$\mathbf{1.\space Proposition}$ Let $\gamma$ be a solution to the equation: $$ \sum_{i=0}^n \rm a_i\rm x^i=0, \rm a_i\in\Bbb Z, a_n=1. $$ Then, there exists a polynomial $p\in \Bbb Z[x]$ ...
14
votes
1answer
441 views

Polynomials all of whose roots are rational

I have two questions about the class of integer-coefficient polynomials all of whose roots are rational. Q1. Is there some way to recognize such a polynomial from its coefficients $a_0, a_1, \ldots, ...
14
votes
0answers
692 views

The radical solution of a solvable 17th degree equation

(The question is at the bottom of the post.) Here's a "natural" solvable 17-th deg eqn with small coefficients: $$\begin{align*} x^{17}-6 x^{16}&-24 x^{15}-42 x^{14}-31 x^{13}-23 x^{12}-7 ...
14
votes
1answer
176 views

Existence of rational sequence such that a polynomial is split over $\Bbb{Q}$

Does there exist a sequence $(a_n)_{n\in \Bbb{N}}$ of rationals such that for all $n\in \Bbb{N}$, $a_n\neq 0$ and the polynomial $a_0+a_1X+\cdots+a_nX^n$ is split over $\Bbb{Q}$? I was asked this ...
13
votes
14answers
2k views

How to prove that $k^3+3k^2+2k$ is always divisible by $3$? [on hold]

How can I prove that the following polynomial expression is divisible by 3 for all integers $k$? $$k^3 + 3k^2 + 2k$$
13
votes
5answers
1k views

$x^2 +y^2 + z^2$ is irreducible in $\mathbb C [x,y,z]$

Is $x^2 +y^2 + z^2$ irreducible in $\mathbb C [x,y,z]$? As $(x^2+y^2+z^2)= (x+y+z)^2- 2(xy+yz+zx)$, $$(x^2+y^2+z^2)=\left(x+y+z+\sqrt{2(xy+yz+zx)}\right)\left(x+y+z-\sqrt{2(xy+yz+zx)}\right).$$ ...
13
votes
4answers
513 views

When does the product of two polynomials = $x^{k}$?

Suppose $f$ and $g$ are are two polynomials with complex coefficents (i.e $f,g \in \mathbb{C}[x]$). Let $m$ be the order of $f$ and let $n$ be the order of $g$. Are there some general conditions ...
13
votes
5answers
365 views

Irreducibility of $f(x)=x^4+3x^3-9x^2+7x+27$

Question at hand is: Is $x^4+3x^3-9x^2+7x+27$ irreducible in $\Bbb Q$ and/or $\Bbb Z$. This is for an exam, reasoning is trivial, but no calculators in hand. Clearly, if there is a rational ...
13
votes
4answers
822 views

Do polynomials in two variables always factor in linear terms?

Consider a polynomial of one variable over $\Bbb C$: $$p(x)=a_0+a_1x+\cdots+a_nx^n,\quad a_i\in\Bbb C.$$ We know from the Fundamental Theorem of Algebra that there exists $c,\alpha_i\in\Bbb C$ such ...
13
votes
4answers
315 views

Show $\mathbb{Q}[x,y]/\langle x,y \rangle$ is Not Projective as a $\mathbb{Q}[x,y]$-Module.

Disclaimer: Though I have been re-reading my notes, and have scanned the relevant texts, my commutative algebra is quite rusty, so I may be overlooking something basic. I want to show $\mathbb{Q} ...
13
votes
8answers
378 views

The proofs of the fundamental Theorem of Algebra [closed]

There are many proofs of the fundamental theorem of algebra. Which are the most beautiful proofs?
13
votes
4answers
1k views

Approximating continuous functions with polynomials

Given $\epsilon \gt 0$ and $f \in C^{0}[0,1]$, Weierstrass says that I can find at least one (how many? probably a lot?) polynomial $P$ which approximates f uniformly: $$\sup_{x \in [0,1]} |f(x) - ...
13
votes
4answers
553 views

Intriguing polynomials coming from a combinatorial physics problem

For real $0<q<1$, integer $n >0 $ and integer $k\ge 0$, define $$[k, n]_q \equiv -\sum_{m=1}^{n} q^{m(k+1)} (q^{-n}; q)_m = -\sum_{m=1}^{n} q^{m(k+1)} \prod_{l=0}^{m-1} (1-q^{l-n})$$ ...
13
votes
3answers
16k views

Proving that a polynomial is irreducible over a field

What's the general strategy to show that a particular polynomial is irreducible over a field? For example, how can I show $x^4 - 10x^2 -19$ is irreducible over $\mathbb Q$?
13
votes
5answers
933 views

Irreducibility of Polynomials in $k[x,y]$

I'm working through some Hartshorne problems and have noticed that in order to do certain problems properly one must prove a given polynomial $f\in k[x,y]$ is irreducible. For example, in problem ...
13
votes
3answers
338 views

How to see that the polynomial $4x^2 - 3x^7$ is a permutation of the elements of $\mathbb{Z}/{11}\mathbb{Z}$

This is from Rotman's Group Theory book, although I don't have the specific reference right now, as the book is with a friend. He asks to show that $\alpha (x) = 4x^2 - 3x^7$ is a permutation of the ...
13
votes
2answers
523 views

Number of real positive roots of a polynomial?

Consider the polynomial $$f(x)=x((1+x^n)^n+a^n)-a(1+x^n)^n,$$ where $n\geq 2$ is a positive integer and $a$ is a positive real number. I'm interesting in deducing the number of positive real roots ...
13
votes
2answers
389 views

Does this polynomial have all its roots both distinct and real?

Recently, I wondered about the following problem: let $n\geq 5$ and let $$ P_n(x)=(x-1)(x-2)\ldots (x-n)-1 $$ Is it true that $P_n(x)$ has $n$ distinct real roots for any $n\geq 5$ ? I checked it ...