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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6
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0answers
49 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 ...
1
vote
0answers
23 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?
-1
votes
1answer
20 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
25 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
24 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
27 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
8 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
22 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
20 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 $f^{k}(\alpha) \not= 0$ ...
9
votes
1answer
73 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
16 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
47 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
60 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
34 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
19 views

polynominal multiply polynominal to get a sequential polynominal

We hava a polynominal $P(x)=\sum_{i=0}^{N-1}\omega (i)x^i,\omega(i)\in\{0,1\}$. Now, we want to get a new polynominal $Q(x)=P(x)P(x^{-1})=\sum_{i=-N+1}^{N-1}\beta (i)x^i$ 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 ...
1
vote
0answers
16 views

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
29 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
16 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
16 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
17 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
28 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
29 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
68 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
108 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
39 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
25 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 $$M=(p\in P_4 : p(2x)=p(x+1)), $$
4
votes
1answer
35 views

Problem involving polynomial and arbitrary continuous function

Let $f\in C^4[0,1]$ and $p$ a polynomial of degree $3$. Suppose: $$f(0)=p(0),\quad f'(0)=p'(0),\quad f(1)=p(1),\quad f'(1)=p'(1)$$ Show that for each $x\in [0,1]$ there exists $\xi\in [0,1]$: ...
0
votes
1answer
80 views

Algebraic element - integral domain

Let $K$ a field and $L$ a subfield of $K$. Let the set $\overline{L}:= \{k \in K: k$ is algebraic over $L$ $\}$ is another subfield of $K$. Show that $\overline{\overline{L}}=\overline{L}$. ...
1
vote
1answer
24 views

Supremum of integral polynomial near origin

Let $P(x,y)$ be a polynomial with integer coefficients that is constant neither in the horizontal nor vertical direction. Prove that $\sup_{-2\leq x,y\leq 2}|P(x,y)|\geq 4$. I suspect we might be able ...
3
votes
1answer
27 views

Rational Points on Fibonacci-like Sequence of Polynomials

Let $\{a_n\}$ be a sequence of polynomials in $\mathbb{Q}[x,y]$ with $a_0=0,a_1=1$, and $$a_n=xa_{n-1}+ya_{n-2}$$ The first few look like $$a_3:y+x^2$$ $$a_4:2xy+x^3=x(2y+x^2)$$ $$a_5:y^2+3x^2y+x^4$$ ...
0
votes
1answer
17 views

Help finding a second homogeneous polynomial of degree 5 that are also harmonic

Essentially I have to find 2 homogeneous polynomial of degree 5 that are also harmonic. Knowing z=(x+iy) is analytic I found my first polynomial to be f(z)=z^5 and that multiples of this would ...
0
votes
0answers
13 views

Using Newton form to find quadratic interpolating (or osculating) polynomial

Let $\ g(t)=\cos(\pi/2 \times t)$ Use Newton form to find the polynomial $\ pt(t)$ which agrees with $\ g(t) $ for each of the sequences $\ [t_0,t_1, t_2] $ The given data is $\ [0,0,2]$ I am having ...
1
vote
0answers
16 views

irreducible polynomials over $\mathbb{F}_p[x_1,x_2]$

Recently I was reading about a irreducibility test for polynomials over the ring $\mathbb{F}_q[x]$. It is layed on the fact that the product of all monic irreducible polynomials whose degree divides ...
1
vote
0answers
13 views

Minimize the rank of a matrix with some entries known

Let $m,n$ be two positive integers, with $m\geq n$. Suppose we have $m$ sets $A_1,\ldots, A_m\subseteq [n]$, with $|A_i|=d_i$. Let $\mathbb F$ be a finite field of size $q$. Let $D$ be the set ...
4
votes
3answers
44 views

Find the two values of $k$ for which $2x^3-9x^2+12x-k$ has a double real root.

Find the two values of $k$ for which $2x^3-9x^2+12x-k$ has a double real root. I've found one method which is to equate $$2x^3-9x^2+12x-k=2(x-r)^2(x-c)$$ Expanding and equating coefficients I ...
11
votes
1answer
1k views

How can I get to Mars with a polynomial?

In order to get to Mars you must win a video game. The video game chooses $10$ points $(a_i,b_i)$ where $a_i$ and $b_i$ are single-digit integers, and places a disk with radius $1/3$ on each of ...
3
votes
1answer
27 views

Alternating polynomial

Let $p(x_1,x_2,\dots, x_n)=\prod \limits_{i<j}(x_j-x_i)$ and $\sigma$ - some permutation of a set $\{1,2,\dots,n\}$. Prove that $$p(x_{\sigma(1)},x_{\sigma(2)},\dots, ...
4
votes
3answers
32 views

Prove that $a(x+y+z) = x(a+b+c)$

If $(a^2+b^2 +c^2)(x^2+y^2 +z^2) = (ax+by+cz)^2$ Then prove that $a(x+y+z) = x(a+b+c)$ I did expansion on both sides and got: $a^2y^2+a^2z^2+b^2x^2+b^2z^2+c^2x^2+c^2y^2=2(abxy+bcyz+cazx) $ but ...