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.

learn more… | top users | synonyms

0
votes
1answer
26 views

Which polynomial has similar properties with Legendre?

I am looking for an kind of polynomial such as Legendre properties that polynomial sequence of orthogonal polynomials such as bellow image. Could you suggest to me one polynomial? Is B-spline correct? ...
6
votes
2answers
280 views

Polynomial rings — Inherited properties from coefficient ring

To avoid mixing up things, I wanted to collect properties a polynomial ring is inheriting from the coefficient ring and what property implies another. Let $R$ be a ring (what else do I need at which ...
1
vote
0answers
33 views

seeking sufficient conditions for polynomials to have no positive roots

I encountered several polynomials as below: $$f(x)=7 + 91 x - 385 x^2 + 1659 x^3 - 1379 x^4 + 553 x^5 - 35 x^6 + x^7$$ $$g(x)=33 + 110 x + 495 x^2 - 252 x^3 + 335 x^4 - 18 x^5 + x^6$$ $$h(x)=71 + ...
3
votes
1answer
54 views

Polynomial-closed properties of rings [on hold]

If $R$ is a ring with certain property, sometimes when we pass to the polynomial ring in one variable, the ring $R[x]$ still has the same property. For instance, it's a theorem that if $R$ is a UFD ...
1
vote
1answer
116 views

Extremely difficult: Polynomials $f,g,h$ such that…

I've been trying to solve this for hours and got nowhere, so I can only assume it's a really difficult problem. Problem: Find polynomials $f,g,h$ with integer coefficients such that: ...
2
votes
1answer
328 views

The volume in cubic meters of water in an aquarium…

The volume in cubic meters of water in an aquarium is given by the polynomial $$V(x) = x^3 - 16x^2 + 79x - 120\;.$$ If the depth in feet (it really does say feet) can be represented by $x-3$, what ...
6
votes
1answer
47 views

Cyclotomic polynomials, properties.

Let $F$ be a field of characteristic prime to $n$, and let $F^a$ be an algebraic closure of $F$. Let $\zeta$ be a primitive $n$th root of unity in $F^a$. I know that the monic polynomial $\Phi_n(X)$ ...
1
vote
0answers
6 views

Question on real polynomial in projective space

Hi all I was given this question and desperately in need of help. I am given a homogeneous polynomial of degree 4 of two variables x and y, with real coefficients with 4 real distinct projective roots ...
2
votes
3answers
67 views

How to solve $(z^1+z^2+z^3+z^4)^3$ using Pascals Triangle?

In an exercise it seems I must use Pascal's triangle to solve this $(z^1+z^2+z^3+z^4)^3$. The result would be $z^3 + 3z^4 + 6z^5 + 10z^ 6 + 12z^ 7 + 12z^ 8 + 10z^ 9 + 6z^ {10} + 3z^ {11} + z^{12}$. ...
1
vote
1answer
16 views

Constructing matrices with eigenvalues equal to roots of a given polynomial

Suppose we are given a polynomial e.g. $$x^4+Ax^3+Bx^2+Cx+D,\tag1$$ and we need to construct a matrix, whose eigenvalues would be equal to roots of this polynomial. One way, rather inelegant, is to ...
2
votes
2answers
687 views

Approximate a polynomial function using a sum of sine waves

I have a polynomial function which I need to approximate by a sum of sine waves with constant amplitude along a given domain. From what I hear, this might be a good time to make use of Fourier ...
-4
votes
0answers
54 views

Please I would like to have the solution of the exercise 3.7 on page 105 of The Arithmetic of elliptic curves second edition (J. H. Silverman) [closed]

Please I would like to have the solution of the exercise 3.7 on page 105 of The Arithmetic of elliptic curve second edition (J. H. Silverman)
5
votes
1answer
64 views

Function equation, find the function evaluated at the certain point.

Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$, $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$ The constant term, $a_0 = f(0) = 1$. Let: ...
5
votes
1answer
164 views

$15a+6b+4c+8d=0$ implies $ax^3+bx^2+cx+d$ has a positive root

Let $a,b,c,d$ be real numbers such that $15a+6b+4c+8d=0$. Show that $f(x)=ax^3+bx^2+cx+d$ has a positive root. (Komal, Problem N. 170.) I want to try to use the intermediate value theorem, showing ...
3
votes
1answer
20 views

finding polynomials to approximate a multivariable function

Let $U := B_1(0) \subseteq \mathbb{R}^2$, with $B_1(0) := \{(x, y) \in \mathbb{R}^2,\space \|(x, y)\| _1 < 1\}$. Now consider the function: $$g: U \to \mathbb{R}^2, (x, y) \mapsto ...
-1
votes
1answer
174 views

Ladder against a wall. [duplicate]

Having a bit of a problem with a question. There is a 4m ladder leaving against a wall. There is a box in between The ladder and wall. The box is a cubic metre. I have found a quartic to find the ...
1
vote
3answers
1k views

How to factorize $x^3 - 7x + 6$?

How do you factorize this polynomial: $${x^3 - 7x + 6}$$ Some online solver doesn't even work saying: using GCF method doesn't work, but sites like Mathway.com gave me the answer, is there a ...
2
votes
1answer
34 views

Polynomial division simple

I am trying to divide polynomials but i am ending up with different outcomes. Let's assume i have : $\dfrac{s^4+3s^3+4s^2+4s+1}{2s^3+2s^2+3s+1}$ Can anyone solve this step by step (long or synthetic ...
-1
votes
1answer
35 views

Any shortcut method to compare the roots of two quadratic equations? [closed]

The given equations are(for example) $81x^2-9x-2=0$ and $56y^2-13y-3=0$. How do i compare the roots of these equation without using the Quadratic formula? Any suggestions please? Thanks.
11
votes
3answers
992 views

Are polynomials infinitely many times differentiable?

Are polynomials infinitely many times differentiable? If so, does it only mean that at some point we reach 0 and then we keep on getting 0? Thank you!
4
votes
3answers
591 views

Why are the coefficients always equal?

Take the equation $ax^{2} + bx + c = 3x^{2} + 4x + 53$. Why is it always true that $a = 3, b = 4$ and $c = 53$? I've seen many examples like this where the coefficients are equated, and was just ...
7
votes
3answers
135 views

How can I prove irreducibility of polynomial over a finite field?

I want to prove what $x^{10} +x^3+1$ is irreducible over a field $\mathbb F_{2}$ and $x^5$ + $x^4 +x^3 + x^2 +x -1$ is reducible over $\mathbb F_{3}$. As far as I know Eisenstein criteria won't ...
0
votes
0answers
11 views

Measure of variation(?) of multidimensional polynomial function

I have a multidimensional function $$\mathbf{f}(x) = [f_0(x), ... , f_N(x)]$$ where $f_n$ are real-valued trigonometric polynomials. I want to measure how much $\mathbf{f}(x)$ varies over some ...
3
votes
2answers
58 views

Shamir's secret sharing interpolation problem

I try to understand this protocol - Shamir's secret sharing - threshold scheme. I got my data and I made interpolation basing on examples published on Wikipedia. You can see them below (sorry, I am ...
2
votes
3answers
135 views

Why can't a non-zero polynomial satisfy some equations?

I'm having a hard time visually picturing/understanding how to explain why a non-zero polynomial function cannot satisfy the equation: $f''(x)$ = $-f(x)$ So is it basically asking to explain why a ...
0
votes
2answers
304 views

Graeffe's root finding method

What are the practical applications of Graeffe's root finding method?I searched a lot but couldn't find.I found that it is used in aerodynamics and electric circuit analysis.But don't know much about ...
1
vote
3answers
29 views

Constructing Polynomial Function from Set of Points and Slopes

I only have a basic knowledge of calculus but I would like to know if it's possible to, given a set of points each with their own slopes, construct the simplest (or any) polynomial function that ...
0
votes
3answers
70 views

Polynomial whose one of its roots is $\cos(\pi/7)$

Let $P(x)$ be a 3rd-degree polynomial with integer coefficients, one of whose roots is $\cos(\pi/7)$. Compute $\frac{P(1)}{P(-1)}$ I saw this question in a contest math problem, and I know that it ...
-1
votes
4answers
69 views

Let $f(x)$ be polynomial of degree four [closed]

Let $f(x)$ polynomial of degree four where: $$f(1)=1,f(2)=4,f(3)=9,f(4)=16, f(7)=409$$ Find $$f(5)=??$$
-4
votes
4answers
77 views

Suppose that $\alpha$ root of the equation [closed]

Suppose that $\alpha$ root of this equation: $$x^4+x^2-1=0$$ Find the value of $$\alpha ^{6}+2\alpha ^{4}$$ "I want the way, not the roots of the equation." I tried, but I couldn't find any thing.
0
votes
2answers
32 views

Three polynomials as unknowns of an equation

If three polynomials $f,g,h\in\mathbb R[x]$ are such that $[f(x)]^2 –x[g(x)]^2+[h(x)]^2=0$, what can we conclude about $f, g, h$?
0
votes
0answers
12 views

Trace of an element in a separable field extension

Let $L=K(\alpha)$ be a finite separable field extension of $K$ of degree $n$ and let $\alpha$ have minimal polynomial $f(X)\in K[X]$ with roots $\alpha=\alpha_1,...,\alpha_n$. Write ...
2
votes
1answer
29 views

Example of $Q((x))$ that doesnt match field of fractions of ring $F[[x]]$

Let $F$ be a commutative ring without zero divisors and $Q$ -its field of fractions. Let $Q(x)$ be also field of fractions of ring $F[x]$. How can field $Q((x))$ not match field of fractions of ring ...
90
votes
1answer
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 ...
0
votes
2answers
36 views

Find the sum of the roots of the exponential equation

The equation $$2^{333x - 2} + 2^{111x + 2} = 2^{222x + 1} + 1$$ has three real roots. Find their sum. I'll simplify it first as: $$\frac{1}{4}2^{333x} + (4)2^{111x} = (2)2^{222x } + 1$$ Let ...
2
votes
3answers
53 views

Find the sum of the roots given no multiple roots.

Find the sum of the roots, real and non-real, of the equation $$ x^{2001} + \left( \frac{1}{2} - x \right)^{2001} = 0 $$ given that there are no multiple roots. I am in a weird situation here. ...
0
votes
2answers
41 views

Proof of associativity of polynomials product (infinite variables)

The product of polynomials in $R[X_i]_{i\in I}$ where $I$ is not necessarily finite is associative ($R$ commutative ring), but I can't find any detailed proof of this fact. Either it is left in ...
1
vote
1answer
391 views

Algorithm to find the maximum/minimum of a polynomial without graphing.

For a quadratic equation of the form $y=ax^2+bx+c$ the max/min occurs at $x=-\frac{b}{2a}$. Is there any hard and fast equation like this for polynomials of degree $\geq 4$?. For such polynomials the ...
1
vote
3answers
40 views

Multiplicity of a root of a polynomial

:) It's true that, if a polynomial has a root (let's say, k, for example) with multiplicity n (n>1, for n integer), then it's true that the derivate polynomial have k as a root with multiplicity ...
1
vote
1answer
73 views

A polynomial that satisfies $x^pf(1-x) + (1-x)^pf(x) = 1$

The context of this question is the construction of the Daubechies wavelet. $f$ is a polynomial of degree $p-1$ which satisfies the equation: $$ x^pf(1-x) + (1-x)^pf(x) = 1 \tag{1} $$ Since $$ ...
2
votes
1answer
249 views

Let $f:R\longrightarrow S$ be a surjective ring homomorphism. If $R$ is PID, then $S$ is PIR.

Let $f:R\longrightarrow S$ be a surjective ring homomorphism. If $R$ is PID, then $S$ is PIR. I think I have proved this: Let $J$ be an ideal of $S$. Then $f^{-1}(J)=(a)$ is a principal ideal of ...
0
votes
1answer
13 views

module isomorphism inbetween two equivalence classes of polynomials

Let $g \in \mathbb{R}[t]$ be a normed irreducible polynomial of degree 2, meaning that $g(t) = (t - \lambda)(t - \overline{\lambda}$) for a $\lambda = a + b i$, with $a, b \in \mathbb{R}$, $b ≠ 0$. I ...
-2
votes
2answers
42 views
6
votes
2answers
81 views

Proving that $x^m+x^{-m}$ is a polynomial in $x+x^{-1}$ of degree $m$.

I need to prove, that $x^m+x^{-m}$ is a polynomial in $x+x^{-1}$ of degree $m$. Prove that $$x^m+x^{-m}=P_m (x+x^{-1} )=a_m (x+x^{-1} )^m+a_{m-1} (x+x^{-1} )^{m-1}+...+a_1 (x+x^{-1} )+a_0$$ on ...
3
votes
1answer
160 views

What is the minimum degree of a polynomial for it to satisfy the following conditions?

This is the first part of a problem in the high-school exit exam of this year, in Italy. The differentiable function $y=f(x)$ has, for $x\in[-3,3]$, the graph $\Gamma$ below: $\Gamma$ exhibits ...
2
votes
0answers
8 views

Special class of Brenke Polynomials

I was wondering if there are any particular papers dealing with a particular class of Brenke Polynomials, defined as $$A(t)B(xt)=\sum_{n\ge 0}P_n(x)t^n$$ where $A=B$ or, where $A(t)=C(B(t),t)$ for a ...
4
votes
3answers
158 views

A lot of confusion in the “Polynomial Remainder Theorem”?

Lately I've been reading about Polynomial Remainder Theorem from various sources, mainly from the wikipedea article, this post and some high school books. Wikipedea says that if we divide a polynomial ...
4
votes
2answers
88 views

Is there any irreducible polynomial over $\mathbb{Q}$ whose Galois group is $S_4$?

As simple as that: Can we find an irreducible polynomial in $\mathbb{Q}$ such that, if $K$ is its splitting field over $\mathbb{Q}$, $\operatorname{Gal}(K|\mathbb{Q})\cong S_4$? I've thought a lot ...
14
votes
5answers
874 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 ...
6
votes
1answer
66 views

When proving that f(z) is a polynomial, is it enough to consider just one point instead of keeping z arbitrary?

I think so - but I'd rather ask the MSE community too. Say I am given the bound |f(z)| < $|z|^3$, and that f is entire. Show f must be a polynomial. I used Cauchy's Integral Formula for ...