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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710 views

Sum of derivatives of a polynomial

Let $p(x)$ be a polynomial of degree $n$ satisfying $p(x)\geq 0$ for all $x$. That is, for all $x$, $p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \geq 0$, $a_n\neq 0$. Show that ...
12
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2answers
119 views

Fully factored integer polynomials with constant differences

Given a degree $d$, it is possible to construct a pair $(F,\delta),$ where $F$ is a polynomial in $\mathbb{Z}[X]$ and $\delta$ a non-zero integer, such that $F(X)$ and $F(X)+\delta$ both split into ...
12
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1answer
193 views

If polynomial with rational number is injective on rationals then it is injective on reals?

Let $p:\Bbb{R}\to\Bbb{R}$ is polynomial with rational coefficients. If restriction of $p$ to $\Bbb{Q}$ is injective, then $p$ is injective? I conjectured that $p$ is monotonic, but I don't know how ...
12
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2answers
219 views

Zero of a complex polynomial satisfying one of three assertions.

Let $n$ be a positive integer greater than $1$. Prove that if $x$ is a zero of $ X^n+1+(-1)^n(X+1)^n$ then $|x|=1$ or $|x+1|=1$ or $|x+1|=|x|$. My initial thought was to study the cases $n=2,3,4$ ...
12
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1answer
448 views

Use of FFT in the multiplication of multinomials

I'm aware that one can use a Fast Fourier Transform (FFT) to take the cost of multiplication of two polynomials of degree N from O$(N^2)$ to O$(N \ln N)$ (which is an amazing reduction when dealing ...
12
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1answer
230 views

Semialgebraic conditions that convey properties of Galois group

Let $f \in \mathbb{Z}[x]$ be a polynomial of degree $n$ with integer coefficients and let $G_f$ be the Galois group of $f$ over $\mathbb{Q}$. I am trying to collect results that convey some ...
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1answer
562 views

Simply formulated but very hard problem about certain polynomial

Let $L:=[k_1,\dots,k_n]$ be a list of natural numbers (i.e. $\{1,2,\dots \} $) , repetitions are allowed. How to prove that the sum of the moduli of the coefficients of the polynomial $$ ...
12
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1answer
144 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 ...
11
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5answers
717 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 $(x^2+y^2+z^2)= (x+y+z)^2- 2(xy+yz+zx)$, $$(x^2+y^2+z^2)=\left(x+y+z+\sqrt{2(xy+yz+zx)}\right)\left(x+y+z-\sqrt{2(xy+yz+zx)}\right).$$ ...
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7answers
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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?
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3answers
399 views

$f(f(x))$ has no fixed points if $f(x)$ has no fixed points

Assume that $f(x)=x$ has no real roots where $$f(x) = ax^2+bx+c$$ Prove that $f(f(x))=x$ has no real roots as well. What I've done is, calculating $f(f(x))$: ...
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4answers
675 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) - ...
11
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3answers
264 views

Polynomial always non-negative

Is there an elegant way to demonstrate that (for example) $x^{2016}-1008x^2+1007\ge 0$ $\forall x\in \mathbb{R}$ ? I tried to write it as sum of squares, but I didn't succeed.
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4answers
213 views

Why does differentiating a polynomial reduce its degree by $1$?

This may seem a bit silly but I am wondering: can it intuitively be shown that the derivative of a polynomial is precisely 1 degree lower than itself? I understand the basics of calculus enough to ...
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2answers
258 views

A polynomial determined by two values

From a St. Petersburg school olympiad, 11th grade. Prove or disprove: a non constant polynomial $P$ with non-negative integer coefficients is uniquely determined by its values $P(2)$ and $P(P(2))$.
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467 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 ...
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3answers
337 views

Determinant of Abstract Matrix

Given an $n \times n$ matrix $A$, where $x$ is any real number: $A = \left[ \begin{array}{ c c c c c c c c } 1 & 1 & 1 & 1 & 1 & 1 & \cdots & 1 \\ 1 & x ...
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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] ...
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2answers
398 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 ...
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2answers
507 views

Number of roots of $x^a-1=0$ with $a \in \mathbb{C}$

It is well known that $x^2-1=0$ has two roots in $\mathbb{C}$, namely $\pm 1$. In general $x^n-1=0$ has exactly $n$ roots in $\mathbb{C}$. But what happens when $n$ is non integer (rational or real or ...
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1answer
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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 ...
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206 views

Is a bivariate function that is a polynomial function with respect to each variable necessarily a bivariate polynomial?

Let $ \mathbb{F} $ be an uncountable field. Suppose that $ f: \mathbb{F}^{2} \rightarrow \mathbb{F} $ satisfies the following two properties: For each $ x \in \mathbb{F} $, the function $ ...
11
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1answer
127 views

Will this sequence of polynomials converge to a Hermite polynomial pointwise?

While trying to solve this question my testing lead to an observation that I found interesting in its own right. Consider the linear transformation $L:P\to P$ from the space of polynomial functions ...
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4answers
664 views

Are translations of a polynomial linearly independent?

I've been wondering about the following question: Suppose that $P$ is a polynomial of degree $n$ with complex coefficients. Assume that $a_0, a_1, \dots, a_n \in \mathbb{C}$ are distinct. Are the ...
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2answers
402 views

A property of roots of the truncated series for $\sin(x)$

Let $p_n(x) = \sum\limits_{k=0}^n \frac{(-1)^kx^{2k+1}}{(2k+1)!}$ In other words, $p_n$ is the polynomial made of the first $n$ terms of the Taylor expansion of $\sin(x)$ around $x = 0$. ...
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4answers
144 views

$F$ is a polynomial, $\deg F = 3$, and $(x^2 - 1)(x^2 - 2) | F(F(x)) - x$. Prove that $F$ exists

$F$ is a polynomial, $\deg(F) = 3$, and $(x^2 - 1)(x^2 - 2) | F(F(x)) - x$. Prove that: a) $F$ exists b) There are at least 10 such polynomials What I've tried to do: $(x^2 - 1)(x^2 - 2) \mid ...
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2answers
304 views

Monotonic version of Weierstrass approximation theorem

Let $f\in\mathcal{C}^1([0,1])$ be an increasing function over $[0,1]$. Prove or disprove the existence of a sequence of real polynomials $\{p_n(x)\}_{n\in\mathbb{N}}$ with the properties: $p_n(x)$ ...
11
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1answer
626 views

Zero divisors in polynomial rings

The following is an exercise in Hungerford (Ch. III, ex. 5.6). Let $R$ be a commutative ring with identity. If $f=a_nx^n+\dots+a_0$ is a zero divisor in $R[x]$, then there exists a nonzero $b$ ...
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1answer
578 views

What's the name of a parabola mapped onto a sphere?

It seems that an 'arc' is a line-segment mapped onto the surface of a sphere (although I don't know if that name still holds if the segment wraps around the sphere more than once, i.e., if the angle ...
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1answer
495 views

Hermite's solution of the general quintic in terms of theta functions

Can someone point me at or produce a translation or modern exposition of Hermite's solution of the general quintic in terms of theta functions? (the "before" and "after" steps are on the mathworld ...
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0answers
205 views

Polynomials with rational zeros

Find all polynomials $F(x)={a_n}{x^n}+\cdots+{a_1}x+a_0$ satisfying $a_n \neq0$; $(a_0, a_1, a_2, \ldots ,a_n)$ is a permutation of $(0, 1, 2 ... n)$; all zeros of $F(x)$ are rational.
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0answers
239 views

Is $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ an irreducible polynomial over $\mathbb{Q}[X]$?

Let $a, b \in \mathbb{Q}$, with $a\neq b$ and $ab\neq 0$, and $n$ a positive integer. Is the polynomial $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ irreducible over $\mathbb{Q}[X]$? I know that ...
11
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0answers
571 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 $n \geq 1$ ? In ...
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0answers
439 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 ...
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3answers
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Is this a polynomial?

$$x^4 + x^3 + x + 1$$ Notice how I skipped $x^2$. Do "polynomials" need to have a sequence of exponents that start from $1$ and go up by $1$ and only $1$ each time? Thanks
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4answers
250 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} ...
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4answers
468 views

Axiomatic approach to polynomials?

I only know the "constructive" definition of $\mathbb K [x]$, via the space of finite sequences in $\mathbb K$. It essentially tells a polynomial is its coefficients. Is there a way to define ...
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2answers
605 views

Cyclotomic polynomials explicitly solvable??

I don't know why I'm having trouble with this, but I can't quite see whether the cyclotomic polynomials are considered solvable. Obvioulsy we can write the solution of the nth cyclotomic polynomial as ...
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5answers
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imaginary numbers - how can I understand them - especially as they occur in 'roots' of polynomials?

In another question here, about roots of equations being imaginary, the accepted answer said something interesting about "imaginary" (as a technical word in math) not meaning "not real". I ...
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4answers
342 views

sum of reciprocals of derivative of polynomial at its roots

If $P(x)$ is a polynomial of degree $n > 1$ with only simple roots $a_1,\ldots,a_n$, is it true that $\frac 1{P'(a_1)} + \cdots + \frac 1{P'(a_n)} = 0$, and, if so, what is the proof? I ...
10
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2answers
191 views

If $f\in\mathbb{Q}[X]$ and $f(\mathbb{Q})=\mathbb{Q}$ then $\deg f=1$

Could you give me any hint how to prove that if $f\in\mathbb{Q}[X]$ and $f(\mathbb{Q})=\mathbb{Q}$ then degree of $f$ is equal to $1$? I have tried to prove it by using Bézout's theorem but I guess ...
10
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4answers
769 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 ...
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2answers
172 views

Given a cubic and quadratic share a root, prove $(ac-b^{2})(bd-c^{2})\geq 0$

Here is an interesting problem. Perhaps someone would be so kind as to give me a shove in the right direction?. If $ax^{3}+3bx^{2}+3cx+d$ and $ax^{2}+2bx+c$ share a common root, then prove that ...
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6answers
250 views

Finding the roots of $x^3 - 93x - 308$ extremely quickly?

I was at a quiz bowl competition and one of the questions was to find the roots of this polynomial. In one or two seconds after reading the question, somebody on the other team buzzed and got the ...
10
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5answers
361 views

Finding real roots of $ P(x)=x^8 - x^7 +x^2 -x +15$

Let $ P(x)=x^8 - x^7 +x^2 -x +15 $, Descartes' Rule of Signs tells us that the polynomial has 4 positive real roots , but if we group the terms as $$ P(x)= x(x-1)(x^6+1) +15 $$ we find that $ P(x) $ ...
10
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2answers
241 views

Can all polynomials of a given degree be reducible?

Let $n > 1$ be a fixed integer. Does there exist a field $F$ with the following properties? $F$ is not algebraically closed. Every polynomial $f(x) \in F[X]$ of degree $n$ is reducible. I ...
10
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4answers
296 views

How to prove $f\equiv 0$ without Weierstrass theorem?

Let $\,f:[0,1] \to \mathbb{R}$ continuous. Show that: If $$\int_0 ^1 x^k f(x)\, dx=0,$$ for all $k\in\mathbb N$, then $f\equiv 0$. I know we can use Weierstrass theorem but I'd like to ...
10
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1answer
243 views

Find three polynomials whose squares sum up to $x^4 + y^4 + x^2 + y^2$

Prove that $$p(x,y) = x^4 + y^4 + x^2 + y^2$$ can be written as a sum of squares of three polynomials over $x,y$ for real numbers.
10
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3answers
409 views

Show a Cubic Polynomial over $\Bbb C$ can be factored as a product of linear terms

The title says it all: I want to show that an arbitrary cubic polynomial can be factored as a product of linear terms without appealing to the fundamental theorem of algebra and (preferably) without ...
10
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1answer
304 views

The zeros of a multivariable polynomial

The other day I came across the following statement: A polynomial $f(x,y)$ of degree at most $3$ that vanishes at $8$ of the $9$ points $(x,y)$ with $x, y \in \{-1,0,1\}$ must also vanish at the ...