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Questions tagged [polynomials]

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 and checking for irreducibility.

0
votes
1answer
28 views

I need help with finding a polynomial and using it to find a perimeter

So I need help with this problem, I’m new to polynomials. Can you also explain how to get the answer? Find the polynomial that models the problem and use it to estimate the quantity: A rectangle ...
3
votes
3answers
30 views

$p^{n-1}x^n - 1$ over $\mathbb{Q}$ for $p$ prime

Consider $f(x) = p^{n-1}x^n - 1 \in \mathbb{Q}[x]$. I want to show that it's irreducible when $p$ is prime. Neither reduction of the coefficients modulo some prime nor Eisenstein seems to work here. ...
1
vote
1answer
33 views

Find all prime ideals of $\mathbb{Z} / n\mathbb{Z} \ [x]$

Let $n=p_1^{r_1}p_2^{r_2}\ldots p_k^{r_k}$ be an integer with $p_i$ being its prime factors. Find all prime ideals of $\mathbb{Z} / n\mathbb{Z} \ [x]$ containing monic polynomials of degree one. I ...
2
votes
1answer
27 views

Showing that a polynomial is zero given that a sum containing its coefficients sum to zero

I've been trying to solve this exam question on an exam in real analysis. Thus, only such methods may be used. The problem is as follows. Assume that $c_0,c_1,\dots,c_n$ are real numbers so that $$\...
2
votes
1answer
25 views

Are two real, two variable polynomials, satisfying the Cauchy-Riemann equations, a complex polynomial?

Let $u, v \in \mathbb{R}[x,y]$ satisfying $u_{x} = v_{y}$ and $u_{y} = -v_{x}$ everywhere in $\mathbb{C}$. Is the function $f(x + iy) = u(x,y) + iv(x,y)$ a polynomial in the variable $z = x + iy$? I ...
3
votes
1answer
22 views

Analogues of the elementary symmetric polynomials for the alternating group

In the case of three variables, the elementary symmetric polynomials are $$ \begin{align} e_1(X_1,X_2,X_3)&:=X_1+X_2+X_3, \\ e_2(X_1,X_2,X_3)&:=X_1 X_2+X_1 X_3+X_2 X_3, \\ e_3(X_1,X_2,X_3)&...
1
vote
1answer
20 views

Number of cycle partition of a set with repeating elements

We have a set $S$ with $E$ elements of which only $N$ are unique. We of course know how many repetitions of each of the $N$ elements are present: element $s_i$ is repeating $t_i$ times. I would like ...
0
votes
3answers
37 views

Finding $(\alpha - \gamma)(\alpha - \delta)$ if they are roots of given quadratic equations

If $\alpha, \beta$ are roots of the equation $x^2 + px - q = 0$. $\gamma , \delta$ are roots of equation $x^2 + px -r$, then find the value of $(\alpha - \gamma )(\alpha - \delta)$. Answer - $q-r$ ...
1
vote
1answer
27 views

Generating Function and Laguerre Polynomials.

The Laguerre polynomial of degree $n$ is $$L_n(x)= \sum_{k=0}^n \frac{(-1)^k \; n\,!}{(k\,!)^2 (n-k)\;!}x^k $$ I am expanding the generating function $$\phi(x,z)= \frac{e^{-xz/(1-z)}}{1-z} $$ to get ...
0
votes
2answers
42 views

Algebra polynomial

if $P(x) = (2x-5)^{2017}+(2x-5)^{2015}+(x-4)^{2017}+(x-4)^{2015}+3x-9 = 0$ find the real roots. I found one real root $3$ by making sum of polynomial equating to o and finding the value of $x$ but not ...
-2
votes
2answers
35 views

If $5x+3$ divides evenly into $10x^3 + x^2 + 32x + k$, find the value of $k$. [on hold]

If $5x+3$ divides evenly into $10x^3 + x^2 + 32x + k$, find the value of $k$. This is a polynomial division question and I'm not sure how to do it. I keep getting the wrong answer. Does anyone have ...
-1
votes
0answers
42 views

complex polynomial

I have a polynomial $P(z) = \lambda z - z^2$. I need to iteratively put $z = \lambda z - z^2$ in it. So, for the first two steps we have: 1) $P(z) = \lambda z - z^2$. 2) $P(z) = \lambda(\lambda z - ...
0
votes
0answers
40 views

To find minimum value of “$b$” in the equation $x^2+ax+b =0$ for some given conditions.

Qs. For a natural no. '$b$', let $N(b)$ denotes the no. of natural numbers '$a$' for which the equation $x^2+ax+b =0$ has integer roots. What is the smallest value of $b$ for which $N(b) =20$? Now, ...
3
votes
0answers
20 views

Given a field extension E/F, $f(x)\in E[x]$, and $f(\alpha )\in F, \forall \alpha \in F$. Does $f(x)\in F[x]$?

Given a field extension E/F, $f(x)\in E[x]$, and $f(\alpha )\in F, \forall \alpha \in F$. Does $f(x)\in F[x]$ ? I have tried the thought that to lower its degree as below. Let $f(x)=\alpha_n x^n+...+...
1
vote
2answers
67 views

How to finds primes $p$ with the property that both $10p^2+9$ and $8p^2-9$ are also primes using Wolfram Alpha?

I want to know the number of primes $p$ that satisfies the condition that $10p^2 + 9$ and $8p^2 - 9$ are both primes, for primes $p$ only. How can I do it using Wolfram Alpha (or any other ...
0
votes
0answers
19 views

BN Elliptic Curves over Field prime p with order n question [on hold]

I have five polynomials over Fp with prime p A(x), B(x), C(x), H(x), Z(x) I need to check that A(x)*B(x)-C(x) = H(x)*Z(x) . ...
2
votes
2answers
15 views

Uniqueness Monic Polynomial Division Arbitrary Ring

Artin states "Let $R$ be a ring, $f$ be a monic polynomial and let $g$ be any polynomial, both with coefficients in $R$. There are uniquely determined polynomials $q$, $r$ in $R[x]$ such that $g = fq ...
0
votes
0answers
28 views

$f^{(j)}(a) = 0$ except for the last derivative. Under which conditions $a$ can be a minimizer of $f$?

Let $f:\mathbb{R}\to\mathbb{R}$ and suppose that $f^{(j)}(a) = 0, j=0,\cdots,n-1$ and $f^{(n)}(a) \neq 0$. Under which conditions the point $x=a$ can be a minimizer of $f$? Based on your answer: $f(...
5
votes
5answers
66 views

Find all matrices that commute with $A$

Given $$A = \begin{bmatrix} 3 & 1 &0 \\ 0 &3 & 1\\ 0 &0 & 3 \end{bmatrix}$$ find matrices $B$ such that $AB=BA$. Trivially $B=A^{-1}$ and $B=kI$ are the solutions Also we ...
3
votes
1answer
63 views

Sequence of polynomials converging to $\frac{1}{z}$ [duplicate]

Is there a sequence of polynomials converging uniformly to $\frac{1}{z}$ in $K:=\{z\in\mathbb{C}\mid 1<|z|<2\}$? My first attempt was to use the theorem of Runge which would apply if $K$ would ...
0
votes
1answer
36 views

Gauss' Lemma prove $\mathbb{Z}[x]$ UFD

I am trying to deduce that $\mathbb{Z}[x]$ is a UFD given the fact that the product of two primitive polynomials $fg$, given $f,g\in{\mathbb{Z}[x]}$, is primitive (I have managed to prove this myself)....
14
votes
3answers
2k views

Polynomial cannot have all roots real?

Let $P \in \mathbb R[x]$ be a degree-$n$ polynomial with real coefficients such that $P(a) \neq 0$, where $a$ is real. If $P'(a) = P ''(a) = 0$ then prove that $P$ cannot have all roots real. Can ...
0
votes
1answer
31 views

Bounds of roots for a parametric quartic equation

I have the following quartic equation $$\omega_4 x^4+\omega_3 x^3+\omega_2 x^2+\omega_1 x+\omega_0=0$$ where $\omega_i$ depend on several real parameters. I'm not interested in searching its roots, ...
0
votes
0answers
12 views

Given $P_1,P_2\in \mathbb R^2$, integer $n>1$, and $f\in \mathbb R[X,Y]$, $f=g_1+g_2$, partial derivatives of order $<n$ of $g_i$ vanishing at $P_i$

Let $P_1,P_2\in \mathbb R^2$ and let $n\ge 1$ be an integer. Given $f(X,Y)\in \mathbb R[X,Y]$, how to show that there exists $g_1(X,Y),g_2(X,Y)\in \mathbb R[X,Y]$ such that $f(X,Y)=g_1(X,Y)+g_2(X,Y)$ ...
0
votes
0answers
36 views

Which is correct for polynomial? The number of degrees or the number of exponents?

I'm sorry for asking a petty question. But, I think here is the best place to ask. If we have a polynomial $a(x)$ of degree $k-1$ $a(x) = \sum_{i=0}^{k-1}a_i x^i$ I know we can say that the degree ...
1
vote
4answers
46 views

Show that in the ring $R = \mathbb{Q} [x, y]$ there are ideals that require at least two generators (the ideal $I =\{f\in R: f (0,0) = 0\}$)

Show that in the ring $R = \mathbb{Q} [x, y]$ there are ideals that require at least two generators (for example, the ideal $I =\{f\in R: f (0,0) = 0\}$) What would be the generators for the example ...
-1
votes
1answer
28 views

Can these data points be used to form a polynomial?

If I have the following data points: $(-7,-1)(-5,1)(-1,-3)(3,1)(6,-2)$ With the correct assumptions, is this enough information to derive a $4^{th}$ degree polynomial function?
1
vote
1answer
26 views

Finding the least sum of digits possible for an outcome of a function in prime numbers.

Let $f(n)=p^4-5p^2+13$ simplified as $f(n)=(p^2-{5\over2})^2+{27\over4}$ where $p$ is an odd prime. Find the least possible sum of digit of $f(n)$. My findings: After putting $p=(3n,3n+1,3n+2)$ in $...
0
votes
3answers
39 views

Factorising cubic polynomial given 1 factor

Given that $x^3-x^2-17x-15 = (x+3)(x^2+bx+c)$ where $b$ and $c$ are constants, find the values of $b$ and $c$. I don't know how to easily solve this question. I could use polynomial long division but ...
0
votes
0answers
25 views

Prove that maximal number of monomials … [duplicate]

I stumbled across this problem and I realize that it's propably easy but somehow I can't imagine the problem properly. Prove that the maximal number of monomials (that are not similar) of polynomial $...
0
votes
1answer
14 views

Find number of polynomials in $P_n(F)$ where $F$ is a finite field with cardinality $m$. [duplicate]

Firstly, I should mention that the above question hits my mind when I am solving a problem in Abstract Algebra which is- Find the number of elements in $\frac{\Bbb{Z}_3[x]}{<x^3+2x+1>}$. I have ...
2
votes
4answers
51 views

Find the values for $a,b,c,d$

Given $$x^3+4x^2y+axy^2+3xy-bx^cy+7xy^2+dxy+y^2=x^3+y^2$$ for any real number $x$ and $y$ , find the value of $a,b,c,d$
1
vote
3answers
47 views

Real value of equation $(x-\frac{1}{x})^\frac{1}{2}+(1-\frac{1}{x})^\frac{1}{2}=x$

Find the real value of x in the equation $(x-\frac{1}{x})^\frac{1}{2}+(1-\frac{1}{x})^\frac{1}{2}=x$ I tried to square the whole term and after expansion not getting the result.
0
votes
3answers
30 views

Finding the number of zeros of a polynomial in the closed disk [duplicate]

Find the number of zeros of $f(z)=z^6-5z^4+3z^2-1$ in $|z|\leq1$. My attempts have not gotten far. I know we can examine the related equation $f(w)=w^3-5w^2+3w-1$ in $|w|\leq1$, letting $w=z^2$. ...
1
vote
2answers
76 views

Let $a$, $b$, $c$, $d$ be the roots of $x^4 + x + 1 = 0$. Find $a^4 + b^4 + c^4 + d^4$. [on hold]

Let $p(x) = x^4 + x + 1 = 0$, and let $a$, $b$, $c$, $d$ be its roots. Find $a^4 + b^4 + c^4 + d^4$. I have no idea how to start solving this problem.
1
vote
1answer
33 views

Polynomial problem with two conditions

I have to find $P(0)$ from the polynomial with minimum degree given that $$(x-1)^3|(P(x)+1)$$ $$(x+1)^3|(P(x)-1)$$ Plugging in $x=\pm 1$ gets something nice, also division by a polynomial of third ...
1
vote
2answers
24 views

Ideal quotients - when does $I:h^2 = I:h$ hold?

A professor gave me the following problem: prove the fact that $I : h^2 = I:h$, where $I \subset k[x_1,\dots,x_n]$ is a zero-dimensional ideal, and $h$ has the property $I + (h) = I + (h^2)$. Now I ...
0
votes
2answers
31 views

Non-constant polynomial factors with leading coefficient 1

How many (nonconstant) polynomial factors with leading coefficient 1, with the other coefficients possibly complex, does $x^{2015} + 18$ have? I don't know much about this problem, all I know is that ...
2
votes
2answers
28 views

Is $(x)$ as a $k[x]$ module free?

Is $(x)$ as a $k[x]$ module free? I think it is free because it seems the basis element is $x$ and it is not annihilated by any element of $k[x]$. Thanks!
1
vote
1answer
30 views

What does it mean for a member of formal power series over a field to be algebraic over polynomial ring of that field?

What does it mean for a member of formal power series over a field to be algebraic over polynomial ring of that field? For example what does it mean for a $f$ in $k[[t_1 ,...,t_n ]]$ which is ...
0
votes
1answer
28 views

Computing $\int_{0}^1 \prod_{x \in \mathcal{X}} (p_xy + (1-p_x)) dy$

How would I compute this integral: $$\int_{0}^1 \prod_{x \in \mathcal{X}} (p_xy + (1-p_x)) dy$$ I have tried using the chain rule and not made any headway on that front. Any advice is welcome.
5
votes
4answers
445 views

Find the roots of $3x^3-4x-8$

It is given that $\alpha$, $\beta$ and $\gamma$ are the roots of the polynomial $3x^3-4x-8$. I have been asked to calculate the value of $\alpha^2 + \beta^2 + \gamma^2$. However I am unsure how to ...
0
votes
0answers
22 views

Is there a theory for piecewise differentiable regression polynomials?

I have an interesting question, I would like to have answered... I have a very noisy signal $f$, that I want to smoothen out. Using a global regression cannot work, as I don't have a model of the ...
1
vote
3answers
44 views

If $ V_n= \alpha^n+\beta^n$ and $\alpha,\beta$ are roots of $x^2+x-1=0$, then $V_n+{V}_{n-3}=2{V}_{n-2}$?

If $ V_n= {\alpha}^n+{\beta}^n$, where ${\alpha}$ and ${\beta}$ are roots of the equation $x^2+x-1=0$. Then prove that $V_n+{V}_{n-3}=2{V}_{n-2}$ (n is whole number). I have tried to manipulate ...
4
votes
0answers
31 views

Can a positive polynomial on sphere be represented as the sum of squares of spherical harmonics

Let $p\in {\mathbb{R}}[x_1,\ldots, x_d]$ be a homogenous polynomial degree $2n$. We know that if $p$ is positive on $[-\pi,\pi]^d$, $p$ is sum of squares polynomial, i.e. $p$ can be witten as sum of ...
2
votes
1answer
44 views

On the proof of Weils hyperelliptic theorem

Let $p$ be a large prime. Consider $F_p$ Theorem: Let $P$ be an element in $F_p[x]$ of degree $k$, assume that $P$ is not a constant multiple of a square. Then the number of solutions $(x,y)$ in $(...
0
votes
0answers
32 views

Function-induced polynomial evaluation trees

My question relates to the Fast polynomial evaluation and composition article by Guillaume Moroz available online at https://hal.archives-ouvertes.fr/hal-00846961v3/document/. It demonstrates notions ...
4
votes
3answers
75 views

Find $x$, given: $x^2 + \frac{9x^2}{(x+3)^2} = 16$

Here's an equation: $$x^2 + \frac{9x^2}{(x+3)^2} = 16$$ First, I subtracted 16 from both sides and factored $x^2$ so I would get a quadratic equation, but with no success. Also, I can see that the ...
2
votes
2answers
29 views

Sum of coefficients of high degree terms in multivariate polynomial expansion

I want to expand the following multivariate polinomial $$\left(\sum_{i=1}^{m} x_i\right)^{n}$$ where $m\geq n$ are both integers. For a fixed integer $k\in\{1,...,m\}$, how to find the sum of ...
0
votes
3answers
52 views

Factorising fourth power polynomial with 5 terms [duplicate]

I've searched all over the internet and cannot seem to factorise this polynomial. $x^4 - 2x^3 + 8x^2 - 14x + 7$ The result should be $(x − 1)(x^3 − x^2 + 7x − 7)$ What are the steps to get to that ...