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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1
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1answer
45 views

Checking whether a given polynomial is reducible or irreducible.

We're given the polynomial $x^{2}-2$ , and we need to prove that it's irreducible in $\mathbb Q$ but reducible in $\mathbb R$. Writing the polynomial as $(x^{2}-2) = 1.(x^{2}-2)$ , $(x^{2}-2)$ ...
0
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0answers
21 views

What is the probability of interpolating the Tutte polynomial of a planar graph from the values at the two hyperbolas?

Crossposted from MO The Tutte polynomial is a bivariate polynomial with positive integer coefficient which is a graph invariant and can be defined recursively. Evaluating it is $\#P$-complete even ...
1
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1answer
35 views

Can a hermitian, rational polynomial have non-zero odd and real coefficients in the numerator/denominator?

Assume that we have a rational polynomial of the form: $$\chi\left(\omega\right)=\frac{\sum_{n=0}\left(c_n+ic_n^{\dagger}\right)\omega^{n}}{\sum_{n=0}\left(d_n+id_n^{\dagger}\right)\omega^{n}}$$ ...
30
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0answers
997 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 ...
0
votes
1answer
53 views

Maximum of polynomial [on hold]

I was studying statics and came across this problem: Find the value $\beta$ such that $P$ has a maximum value in $R^2 - 1000^2 = P^2 + 2000P\cos(75^{\circ}+\beta)$. When $R$ is constant, the ...
3
votes
1answer
53 views

Closed form for the sole positive root of the polynomial ${x^\alpha } + {x^{\alpha - 1}} + \cdots + {x^3} + {x^2} -p$, $p > 0$

For a study I'm making about the minimum and maximum radial values of bounded orbits in a central force system with general power law forces, I came across this special polynomial equation: ...
-6
votes
1answer
50 views

Polynomial Factoring problem [on hold]

Find the value of $p$ and $q$ such that $15x^3 + 26x^2 - 11x - 6$ is a factor of $15x^4 + px^3 - 37x^2 + qx + 6$.
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0answers
19 views

Determining if a Polynomial is a subspace and its Basis

Hi, the question is Which of the subsets of P2 given in Exercises 1 through 5 are subspaces of P2 Find a basis for those that are subspaces. So I know that P'(1) = 1b + 2c And I know that P(2) = a ...
0
votes
2answers
36 views

How to prove if $5/2 < x < (5/4)(1+\sqrt2)$, then $25/(x(2x-5)\ge 8$

if $\frac52 < x < \frac54(1+\sqrt2)$, then $\frac{25}{x(2x-5)} \ge 8$ First I unpacked the conclusion to: $$ 16w^2-40w-25 \le 0 $$ I attempted to solve by manipulating the interval (squaring, ...
0
votes
1answer
44 views

Finding a sixth degree polynomial that goes through 8 points

For a summative math research assignment, I will have to find a sixth degree polynomial that would ideally go through the following points: (0, 20.5625) (10, 27.5625) (30, 14.5625) (50, 14.6875) (60, ...
3
votes
0answers
70 views
+50

Polynomial in the components of the curvature tensor

Consider a closed Riemannian manifold $(M,g)$ of dimension n and let $K(t,x,y)$ be its heat kernel. Then it is known that the heat kernel has an asymptotic expansion as $t\downarrow 0$: ...
0
votes
0answers
32 views

property of complex polynomials

I can't solve the following problem: Let $p(z) = z^n + a_{n-1}z^{n-1} + ... + a_0$ be a complex polynomial of degree $n \ge 1$. Assume that there exist $j \in \{0, 1, ... n-1\}$ such that $a_j \neq ...
1
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0answers
30 views

What is the maximum value of coefficient $f_v$ with the constraint that the matrix is positive semi-definite?

I am trying to solve this equation my self with my knowledge about characteristic polynomials, etc but I have placed it here earlier because I'm not a mathematician and maybe you give me ideas to ...
0
votes
0answers
9 views

Entry Expansion of Power Matrix

Suppose $A:=\{a_{i,j}\}, 1\le i,j, \le n$ is a $n\times n$ matrix with real positive entries. Now replace the constant $a_{1,1}$ with a real variable $x$. Denote by $A_x$ the resulting variable-Matrix ...
7
votes
1answer
70 views

Prove/Disprove : Every polynomial with prime degree and coefficients in $[-1,1]$ has galois-group $S_p$

Conjecture : Let $p$ be a prime number , $f\in \mathbb Z[X]$ an irreducible polynomial with degree $p$ and coefficients in the range $[-1,1]$. Then the galois group of $f$ over $\mathbb Q$ ...
5
votes
2answers
53 views

Roots of $x^p + x + [\alpha]_p \in \mathbb{Z}_p[x]$

Let $$g(x) = x^p + x + [\alpha]_p \in \mathbb{Z}_p[x],$$ where $p$ is prime. For which $\alpha, p \in \mathbb{Z}$ does $g(x)$ have at least one root? And for which $\alpha, p \in \mathbb{Z}$ ...
1
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3answers
60 views

For which $n \in \mathbb{N}$ does $x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$ have at least $7$ distinct solutions?

I have to find one $n \in \mathbb{N}$ such that $$x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$$ has at least $7$ distinct solutions in $\mathbb{Z}_n$ (or, equivalently, $f(x) = x^8 + ...
2
votes
2answers
63 views

Show $p(x)$ is a primitive polynomial

First the definition: Polynomial $q(x) \in \mathbb{Z}_p[x]$ of degree $n$ is called primitive, iff: $q(x) \mid x^{p^n-1}-1$ $\forall k : 1 \leq k \leq p^{n}-1$ : $q(x) \nmid x^k - 1$ ...
1
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2answers
41 views

Linear maps of polynomials, the bases of the space and their corresponding matrix.

Suppose $T \in \mathrm{Hom}(\mathscr{P}_3(\mathbb{R}),\mathscr{P}_4(\mathbb{R}))$ is defined by: $$Tp(x)=(x^2p(x))',$$ for all $x \in \mathbb{R}$ and $S \in\mathrm{Hom} ...
0
votes
0answers
40 views

How do I derive the cubic formula? (without substitutions)

I've heard of a number of ways that people have derived a cubic formula (I've even heard of a number of different ways to show the formula itself too). What I want to know is how to derive it without ...
0
votes
1answer
46 views

Can the galois group be the symmetric group, if the discriminant is a perfect square?

Let $f\in \mathbb Z[X]$ be an irreducible polynomial. Suppose, the discriminant of $f$ is a perfect square. Can the galois group of $f$ over $\mathbb Q$ be $S_d$, where $d$ denotes the degree of ...
0
votes
1answer
41 views

Alternate proof to number of monomials in a given degree - “more” rigorous, formal [duplicate]

Let $s$ be the number of variables and $n$ be the degree of the monomials we want to count in $R[X_1,\dots,X_s]$. Then show, that the count is $$\delta(n,s):=\binom{s-1+n}{s-1}.$$ The question ...
0
votes
0answers
13 views

Identifying indeterminable terms in polynomial fit

I am using SVD to fit a polynomial surface to a set of points, where the number of points may be less than, equal to, or more than the number of polynomial terms. For simplicity, let's assume points ...
0
votes
0answers
16 views

Is this part of a known sequence?

while trying to express as an infinite sum the function $t^x/\Gamma(x)$ I came across some coefficients of the form $a_0=1$ $a_1=-\psi^{(0)}(1)$ $a_2=[\psi^{(0)}(1)]^2-\psi^{(1)}(1)$ ...
0
votes
0answers
13 views

Expected values of Hermite polynomials under Gaussian distribution

On Wikipedia there's a nice result stating that $$E[He_n(X)]=\mu^n,$$ where $He_n$ is the (probabilists') Hermite polynomial of order $n$ and $X$ is a $N(\mu, 1)$ random variable. I'm interested in ...
1
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2answers
24 views

How to show if the following subset $W$ is a subspace of a vector space $V$?

$1.$ $V=P_n(\mathbb{R}), $and $ W=\{p(x)\in P_n(\mathbb{R})\mid p(1)+p(2)+p(3)=0 \}$ $2.$ $V=M_{n\times n}(\mathbb{R}), $and $ W=\{A\in M_{n\times n}(\mathbb{R}) \mid A \text{ is not symmetric}\}$ ...
0
votes
1answer
36 views

Basis for 4th degree polynomials such that integral of $p(x)$ from $-1$ to $1$ equals $0$

Let $U= \{ p \in \mathscr P_4\mathbb{R} \ | \int_{-1}^1 p(x)dx=0\}$. a.) Find a basis for $U$. b.) Find a subspace $W$ of $\mathscr{P_4}(\mathbb{R})$ such that $\mathscr{P_4}(\mathbb{R})= U \oplus ...
1
vote
3answers
47 views

Simplifying an equation from $a - b(c)$ to $a(b) - b(c)$.

A section in a book shows jumping from $$7-3(9-7) = 1$$ to $$4(7)-3(9) = 1$$ I can't see how this happens. What steps have been taken? Where did the four come from?
4
votes
3answers
55 views

A non continuous linear map $A:\Bbb{R}[X]\rightarrow \Bbb{R}$ such that $A(P)=P(1).$

I have a linear map $A:\Bbb{R}[X]\mapsto \Bbb{R}$ such that $A(P)=P(1)$, for the $p-$norm : $\Vert P\Vert=\bigl(\sum_{i=1}^n\vert a_i\vert^p\bigr)^{1/p}$ where $p\in[1,+\infty].$ For the cas ...
7
votes
1answer
107 views

Roots of iterations of polynomials

Let $f \in \Bbb Q[X]$ a polynomial, and let denote by $f^n$ the composition $\underbrace{f \circ \cdots \circ f}_{n \text{ times }}$. Let $R(f^n) \subset \Bbb C$ the roots of $f^n$. I'm interested in ...
5
votes
1answer
2k views

Finding the value of $f(6)$ when $f(x)$ of degree $5$ with leading coefficient

Problem : Suppose $f(x)$ is a polynomial of degree $5$, and with leading coefficient $2009$. If further that $f(1) =1; f(2)=3, f(3)=5, f(4)=7, f(5)=9$. What is the value of $f(6)?$ My work : Let ...
-1
votes
4answers
22 views

Finding a coefficient of the variable in a function

$f(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$ $f(1) = 2$ $f(2) = 3$ $f(3) = 4$ $f(4) = 5$ $f(5) = 6$ $b = \underline{\qquad}$ How should I answer this kind of question? Edit: Actually I had ...
0
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0answers
26 views

Irrational roots conjugate theorem

This theorem seems pretty clear cut at first, but i have read a lot of queries about it. I have found out that if a cubic has only $1$ irrational root, then it cannot be expressed in the form $a + ...
2
votes
1answer
840 views

Sum of Lagrange basis polynomials

Let $L_i(x)$ be Lagrange basis polynomials for $n+1$ points $(x_0,y_0),\ldots, (x_n,y_n)$. How do you prove that $\sum_{i=0}^n (x-x_i)^pL_i(x)=0$ for $p\leq n$?
26
votes
2answers
393 views

Prove $|P(0)|\leq 2n+1$

Let $P(x)$ be a polynomial with degree $\leq n$ and $|P(x)|\leq\frac{1}{\sqrt{x}}$ for $x\in(0,1]$. Prove that $|P(0)|\leq 2n+1$. The idea should be that if $|P(0)|$ is too large, then the polynomial ...
2
votes
1answer
45 views

Kernel of a module homomorphism

Let $F$ be a field and let $R=F[x,y,z]$. Let $M=\langle x,y,z\rangle$ be the ideal of $R$ generated by $x,y,z$. Define an $R$-module homomorphism $$\phi:R^3\to M,\quad (f_1,f_2,f_3)\mapsto ...
0
votes
2answers
30 views

How to solve a quadratic inequality that acts like a quadratic equality?

This will be largely a trivial question. But how do I solve an inequality like this: $3x^4 - 4x^2 + 1>0$ ? Of course, I can treat it like a quadratic inequality by saying $t=x^2$ So I can solve ...
0
votes
1answer
29 views

$f$ is divisible by a square of non-constant polynomial iff $f,f'$ are not relatively prime

Let $R$ be a commutative ring and $f=a_0+ \cdots +a_nt^n \in R[t]$. Define $f':=a_1+2a_2t+ \cdots + na_{n-1}t^{n-1}$. Show that $f$ is divisible by a square of non-constant polynomial if and only ...
-1
votes
0answers
31 views

Completion of a polynomial ring [closed]

Let $R$ be a commutative ring with ideal $I$. Let $J$ be the ideal of $R[x]$ generated by $I$ and $x$. What is the $J$-adic completion of $R[x]$? Is it $S[[x]]$, where $S$ is the $I$-adic ...
1
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0answers
35 views

$\sqrt{1-x^2}|P(x)|\le 1$ for all $x\in [-1,1]$

Let $P(x)$ be a real polynomial with degree $n$ such that $\sqrt{1-x^2}|P(x)|\le 1$ for all $x\in [-1,1]$. Prove that $|P(x)|\le n+1$ for all $x\in [-1,1]$. This question was posted some years ...
1
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0answers
17 views

A determinant that arises when proving the Alternating Sign Matrix Conjecture

Prove that $$\det\bigg(\frac{1-s^{i+j-1}}{1-t^{i+j-1}}\bigg)^n_{i,j=1}=t^{n^3/3-n^2/2+n/6}\prod_{1\leq i<j\leq n}(1-t^{j-i})^2\prod_{i,j=1}^n\frac{1-st^{j-i}}{1-t^{i+j-1}}$$ In his book, D. ...
0
votes
0answers
34 views

polynomial multiply polynomial to get a sequential polynomial

We hava a polynomial $P(x)=\sum_{i=0}^{N-1}\omega (i)x^i,\omega(i)\in\{0,1\}$. Now, we want to get a new polynomial $Q(x)=P(x)P(x^{-1})=\sum_{i=-N+1}^{N-1}\beta (i)x^i$ and all the coefficient ...
1
vote
2answers
39 views

Given $A$, $A^{-1}$ can be expressed with: $A^{-1}=bA+dI$

Given the matrix $A=\begin{pmatrix} -1 &3 &3 \\ 3& -1 & 3\\ 3& 3 & -1 \end{pmatrix}$ then $A$ is invertible and $A^{-1}$ can be expressed with: $A^{-1}=bA+dI, ...
3
votes
6answers
983 views

What are the common solutions of $x^2+y=31$ and $y^2+x=41$?

A friend asked me if I have a certain algorithm to solve $x^2+y = 31$ and $y^2+x=41$ simultanously. We found the solutions but we didn't find a way to solve both equations. Any ideas?
-1
votes
1answer
40 views

how to do isomorphic ideal from root system of degree 5 or more [closed]

update1 i notice solving 1+x+x^2+x^3+x^4+x^5 have 5 solutions two conjugate real number and each of them having conjugate complex number part i change a*b + c to a1*a2*b1*b2 + c1*c2** still can ...
2
votes
1answer
2k views

How to solve polynomial-exponential equation

I'm trying to solve equations like the following one: $$5 + 3x - 4x^3 = e^{x^2}$$ I've tried using the Lambert W function, but I didn't get any success. I must admit I'm relatively new to Lambert W ...
-1
votes
0answers
29 views

Pick out a polynomial such that ideal $J=q(x)R$ , where $q(x)$ is polynomial and $R$ is ring [closed]

In the ring of polynomials $R =\mathbb Z_5[x]$ with coefficients from the field $\mathbb Z_5$, consider the smallest ideal $J$ containing the polynomials, $p_1(x) = x^3 + 4x^2 + 4x + 1$ $p_2(x) = ...
2
votes
1answer
26 views

Degree of Rational Function

This might sound like a very trivial question but I found different answers on the web. Assume one has a rational function $$\frac{f(x)}{g(x)} ,$$ where $f(x)$ and $g(x)$ are polynomials. What is ...
0
votes
2answers
34 views

equation to create unique value

I have a list of n objects say [ apple, orange, carrot, cherry, banana ] Now I am trying to come up with an equation which will generate an unique number for ...
0
votes
2answers
49 views

Is a factorable polynomial invertible?

The reason there exists no quintic formula that finds the roots of a quintic polynomial is simply because some quintic polynomials are irreducible. But reducible quintic polynomials may be invertible ...