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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0
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1answer
23 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 ...
0
votes
3answers
40 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?
0
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0answers
11 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: $$\chi\left(\omega\right)=\frac{\sum_{n=0}\left(c_n+ic_n^{\dagger}\right)\omega^{n}}{\sum_{n=0}\left(d_n+id_n^{\dagger}\right)\omega^{n}}$$ ...
0
votes
0answers
17 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 + ...
4
votes
3answers
54 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 ...
0
votes
1answer
28 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
vote
0answers
31 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
vote
0answers
25 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
2answers
28 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 ...
1
vote
0answers
14 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
25 views

Completion of a polynomial ring [on hold]

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
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, ...
7
votes
1answer
97 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 ...
0
votes
0answers
36 views

Reduction modulo $p$ of $x^4 + 3x^3 -21x^2 -62x -40$ with a multiple root

Let us consider $$g(x) = x^4 + 3x^3 -21x^2 -62x -40 \in \mathbb{Z}[x].$$ How does one find the primes $p>0$ such that the reduction of $g(x)$ modulo $p$ has a multiple root (without taking into ...
-1
votes
0answers
28 views

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

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
29 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 ...
1
vote
1answer
29 views

Binomial expansion in the form $(1+x^2)^n$

I'm used to dealing with binomial expansion in the form $(1+x)^n$. I understand that if the number is not $1$ then you have to divide the whole bracket by something which would make it $1$. However ...
0
votes
0answers
11 views

Is it possible to convert a general quintic to Brioschi form in one single transformation?

The standard method of converting a general quintic to Brioschi form $X^5-10CX^3+45C^2X-C^2=0$ proceeds in two steps which required the extraction of a square root. One first converts to the ...
0
votes
1answer
26 views

Let $f(x)$ be a polynomial in $x$ of degree greater than $1$ defined by $g_i(x)$ .Find the average of the roots of $g_{89}(x)$.

Let $f(x)$ be a polynomial in $x$ of degree greater than $1$.Degine $g_i(x)$ by $g_1(x)=f(x)$,and $g_{k+1}(x)=f(g_{k}(x))$.Let $r_k$ be the average of the roots of $g_k$.Determine $r_{89}$ if ...
1
vote
1answer
21 views

Proving Multiplicity in Polynomials with derivatives.

After learning multiplicity in polynomials we were given the task of proving that: if $ f(\alpha) = f'(\alpha) = f''(\alpha) = f'''(\alpha) =$ .... $f^{k-1}(\alpha) = 0$ and $f^{k}(\alpha) \not= 0$ ...
11
votes
1answer
85 views

Is there a polynomial such that $F(p)$ is always divisible by a prime greater than $p$?

Is there an integer-valued polynomial $F$ such that for all prime $p$, $F(p)$ is divisible by a prime greater than $p$? For example, $n^2+1$ doesn't work, since $7^2+1 = 2 \cdot 5^2$. I can see that ...
-1
votes
1answer
18 views

Help with equation and explain to me? [on hold]

$5(-3x - 2) - (x - 3) = -4(4x + 5) + 13$ Solve for $x.$ Can anyone solve this equation and explain it to me? I don't understand this equation for some reason and I always keep getting different ...
0
votes
2answers
48 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 ...
-4
votes
1answer
61 views

What is the inverse function of $y=x^2 + 3x +2$? [on hold]

What is the inverse function of $f(x)=x^2 + 3x +2$? Please show your solution method and demonstrate that $f(f^{-1}(x))=x$
1
vote
2answers
40 views

Find the polynomial $P$ of smallest degree with rational coefficients and leading coefficient $1$ such that $ P(49^{1/3}+7^{1/3})=4 $

Find the polynomial $P$ of smallest degree with rational coefficients and leading coefficient $1$ such that $$ P(49^{1/3}+7^{1/3})=4 $$ (Source:NYSML) My attempt Let $$ ...
0
votes
1answer
10 views

factors in polynomial rings with field coefficients

I'm reading through Dummit and Foote Abstract Algebra, and I was looking for an explanation of the following: Proposition 9: $F$ a field and $p(x)\in F[x]$. Then $p(x)$ has a factor of degree one if ...
0
votes
1answer
35 views

Computing the GCD of two polynomials

I'm trying to find the $\gcd$ of $A:= x^4 - x^3 + x^2 - x + 1$ and $B:= x^2 + 2$. Using the Euclidean algorithm, I've found $$(x^4 - x^3 + x^2 - x + 1)=(x^2 - x - 1)(x^2 + 2) + (x + 3)$$ and ...
0
votes
0answers
27 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 ...
2
votes
1answer
38 views

For how many rational $x$ is $P(x)$ such that $54x^n+P(x)=315$?

Given an integer $n >2$,for how many different rational numbers $x$ does there exist a polynomial $P(x)$ of degree $n-1$ with $P(0)=0$,and with all integer coefficients,such that ...
2
votes
0answers
36 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 $t\downarrow 0$: ...
1
vote
3answers
33 views

Why is $\sup_{x∈[0,1]} {|p'(x)|} ≤ A_d\sup_{x∈[0,1]}{|p(x)|}$ for all polynomials $p$ of degree at most $d$?

How can one prove that for any positive integer $d$, there is a constant $A_d < 0$ such that $$ \sup_{x∈[0,1]} {\lvert\, p'(x)\rvert} ≤ A_d\sup_{x∈[0,1]}{\lvert\, p(x)\rvert}, $$ for all ...
1
vote
0answers
19 views

Why are $F(p) := \sup_{x∈[0,1]}{|p'(x)|}$ and $G(p) := \sup_{x∈[0,1]}{|p(x)|}$ both continuous functions of the polynomial $p$?

Why are $F(p) := \sup_{x∈[0,1]}{|p'(x)|}$ and $G(p) := \sup_{x∈[0,1]}{|p(x)|}$ both continuous functions of the polynomial $p$, which is finite and of degree at most $d$ ? Continuity of a function ...
0
votes
0answers
36 views

$R[x]$ has a subring isomorphic to $R$ [duplicate]

$R$ is a commutative ring. We need to prove that $R[x]$ has a subring isomorphic to $R$. Let $S$ be that subring of $R[x]$ which has polynomials of even degree. Now I consider a mapping from $S$ ...
4
votes
1answer
45 views

How many solution are there to equation $f(x)=f(f(x))$ given the following function?

Shown is the graph of $y=f(x)$,a polynomial function of degree $10$ whose domain is restricted to $[1,5]$.Function $f$ is symmetric about $x=3$.Compute the number of solutions to the equation ...
-1
votes
2answers
19 views

Why do the coefficients of all polynomials of degree at most $d$ as coordinates of a vector in $\mathbb{R}^{d+1}$ lie in ${R}^{d+1}$'s unit sphere?

Consider the coefficients of all polynomials of degree at most $d$ as coordinates of a vector in $\mathbb{R}^{d+1}$. Why would it suffice to suffices to assume that this vector lies in the unit ...
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 $k$? $$k^3 + 3k^2 + 2k$$
1
vote
0answers
20 views

Find GCD of polynomials over GF(101)

Hello all I'm teaching myself cryptography, and I'm struggling with polynomial arithmetic over finite fields. I've some what been able to teach myself how to do the arithmetic over $GF(2)$, but when ...
1
vote
1answer
29 views

Is there a rational function $f$ satisfying $f(x) =f\left( \frac{1}{1-x} \right)$ for all $x$?

I would like to find a pair of relatively prime polynomials $p,q \in k[x]$ (where $k$ is a field) such that $$\frac{p(x)}{q(x)} = \frac{p \left( \frac{1}{1-x} \right)} {q\left( \frac{1}{1-x} \right)} ...
0
votes
1answer
54 views

Is it possible to solve the following equation without using the Rational Root Theorem?

Given $f(x)=x^4+2x^3+2x^2-2x-3$, where $x-1$ is a factor of $f(x)$, how is it possible to solve $f(x)$ without the Rational Root Theorem? Here's my progress: $$f(x)=x^4+2x^3+2x^2-2x-3$$ ...
1
vote
2answers
28 views

A Subspace of the Degree $3$ Polynomials Space for which $P(5)=0$

The problem is the following: Given the space of the polynomials $V=P(x)$ of degree $3$ prove the following: The set $U$ which is defined as the set of the elements for which $P(5)=0$ is a ...
0
votes
1answer
42 views

If $P(x) = (x^4+x^3-3x^2+4x-4)\cdot q(x) + (2x^3-5x^2+7x-3)$ find $P(2)$

If for the polynomial $P(x)$ is true that $$P(x) = (x^4+x^3-3x^2+4x-4)\cdot q(x) + (2x^3-5x^2+7x-3)$$ find $P(2)$ I assumed that the polynomial $(x^4+x^3-3x^2+4x-4)$ has $(x-2)$ as one of its ...
-1
votes
0answers
30 views

Buchberger's Algorithm Example

I've been reading Ideals, Varieties and Algorithms and came across an example of Buchberger's algorithm being computed and I am not able to understand how they came to have the final result. The ...
0
votes
1answer
70 views

Do there exist polynomials $f,g$ in $\mathbb{C}[x]$ such that $(x^2 - 1)f + x = g^2$.

Do there exist two polynomials $f, g \in \mathbb{C}[x]$ such that $(x^2 - 1)f + x = g^2$? I know that this cannot happen in $\mathbb{R}[x]$. However, since $\mathbb{C}$ is algebraically closed, this ...
1
vote
3answers
109 views

Find all pairs of nonzero integers $(a,b)$ such that $(a^2+b)(a+b^2)=(a-b)^3$

Find all pairs of nonzero integers $(a,b)$ such that $(a^2+b)(a+b^2)=(a-b)^3$ My effort Rearranging the equation I have \begin{array} \space (a^2+b)(a+b^2)-(a-b)^3 &=0 \\ ...
0
votes
0answers
21 views

The coordinate ring of $\varepsilon: xy-1=0$ [duplicate]

I want to show that the coordinate ring $\mathbb{R}[x,y]/\mathbb{R}[\varepsilon]$ of $\varepsilon: xy-1=0$ is not isomorphic with the polynomial ring of one variable $\mathbb{R}[x]$. To me this is ...
3
votes
1answer
40 views

Fibonacci sequence in the factorization of certain polynomials having a root at the Golden Ratio

I was playing around with the Golden Ratio $\Phi = \frac{1 + \sqrt 5}{2}$ on Wolfram Alpha and I noticed that if $F_n$ denotes the $n{th}$ Fibonacci number, then the polynomial $P_n(x) = x^n - F_n x - ...
5
votes
2answers
69 views

Let $a$ be a root of the cubic $x^3-21x+35=0$. Prove that $a^2+2a-14$ is a root of the cubic.

Let $a$ be a root of the cubic $x^3-21x+35=0$. Prove that $a^2+2a-14$ is a root of the cubic. My effort Working backwards I let $P(x)$ be a polynomial with roots $a,a^2+2a-14$ and $r$. Thus, ...
0
votes
1answer
26 views

problems to understand a special definition of “free graded commutative algebra” from lecture

I have problems to understand a definition from lecture: Let $R$ be a commutative ring with unit and such that $2$ is invertible in $R$. The free graded commutative algebra in generators $a_1, .., ...
0
votes
1answer
17 views

Complementary subspace of $M=(p(2x)=p(x)) ,p\in P_4$

Can anyone please help me with: Find a some base for complementary subspace of $$M=(p\in P_4 : p(2x)=p(x+1)), $$