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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3
votes
1answer
41 views

Solution to a simple system of quadratic equations

I am hoping to find a closed-form solution to the following system of $n$ quadratic equations: $$ x_j^2 = \sum_{i=1}^n B_{ij}x_i $$ for $j\in\{1,\dots,n\}$, where $B_{ij}\geq 0$. There is a trivial ...
1
vote
1answer
39 views
2
votes
0answers
34 views

A function that is locally a quotient of polynomials but not globally [duplicate]

Let $X =\{ x_1x_4=x_2x_3\;, (x_2,x_4) \neq (0,0)\} \subset \mathbb{C^4}$, i.e. not both of $x_2,x_4$ are zero. Define a function $\phi$ on $X$ by $\phi(x)=\left\{\begin{matrix} \frac{x_1}{x_2} ...
1
vote
1answer
34 views

Polynomial having all integral coefficients $P_n(a)=b$ and $P_n(b)=c$ and $P_n(c)=a$

Let $a,b,$ and $c$ denote three distinct integers, and let $P_n$ a polynomial having all integral coefficients. Show that it is impossible that $P_n(a)=b$ and $P_n(b)=c$ and $P_n(c)=a$. I started ...
3
votes
1answer
677 views

Irreducible polynomials over the reals

Everybody knows that the degree of irreducible polynomials over the reals is either one or two. Is it possible to prove it without using complex numbers? Or without using fundamental theorem of ...
0
votes
0answers
54 views

What does the irreducible polynomial over $\mathbb{Q}$ and $\mathbb{Q}(i)$ mean?

So here'as the problem Find the monic irreducible polynomial $g(x) \in \mathbb{Q}(x)$ for $i+ \sqrt{3}$ over $\mathbb{Q}(i)$ Erm...huh? Okay, so a minimal polynomial $m$ is what I need, and ...
1
vote
3answers
40 views

Is it true that a polynomial is reducible over a field only if the polynomial has a zero in the field?

I am doing some practice problems for abstract algebra and have come across this idea in a couple places, but it seems fundamentally wrong. For example, in order to prove that $f(x) = x^2 + x + 1$ is ...
1
vote
2answers
23 views

Are the “weights” inside a neural network actually “terms” for a polynomial?

This just hit me today. I am not too experienced with math or neural networks, but I am trying to find out about them in my own way so I can some day understand them well. So I was thinking about how ...
11
votes
6answers
7k views

Factorize the polynomial $x^3+y^3+z^3-3xyz$

I want to factorize the polynomial $x^3+y^3+z^3-3xyz$. Using Mathematica I find that it equals $(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$. But how can I factorize it by hand?
0
votes
3answers
86 views

If $x^{15}-x^{13}+x^{11}-x^9+x^7-x^5+x^3-x=7$, prove that $x^{16}>15$.

"If $x^{15}-x^{13}+x^{11}-x^9+x^7-x^5+x^3-x=7$, prove that $x^{16}>15$." The above problem came on a local question paper. I tried to solve it by factorizing and sum of G.P. , But I was unable to ...
1
vote
2answers
22 views

The polynomial ring $K[t_1,\dots,t_n]$ of $n$ variables over the field $K$ has no zero divisors

Show that the polynomial ring $K[t_1,\dots,t_n]$ of $n$ variables over a field $K$ has no zero divisors (except the zero polynomial). When revising some Linear Algebra topics, I got stuck with this ...
2
votes
1answer
348 views

Testing for irreducibility over $R=\mathbb Q[x]/(1+x^2)$

Let $R=\mathbb Q[x]/I$ where $I$ is the ideal generated by $1+x^2$. Then is $y^2 +1$ is irreducible over $R$ ? $y^2+y+1$ is irreducible over $R$ ? $y^2-y+1$ is irreducible over $R$ ? $y^3+y^2+y+1$ ...
1
vote
0answers
41 views

Image of element not square of any element, maximal ideal, field is quadratic extension?

This is a followup to my question here. Say we have $\mathbb{F}_q$ a finite field, $\text{char.} \neq 2$, have $f \in \mathbb{F}_q[x]$, $f \notin \mathbb{F}_q$ be a squarefree element, and let us ...
1
vote
1answer
46 views

Degree of the field extension

I need to determine the degree of the field extension $\mathbb{Q}(\sqrt{(2+\sqrt{2})(3+\sqrt{3}}))/\mathbb{Q}$. I've determined that the minimal polynomial of $\sqrt{(2+\sqrt{2})(3+\sqrt{3}})$ is ...
3
votes
2answers
36 views

The content of a polynomial vs the ideal of its values

Let $f(x) = \sum_i a_i x^i$ be a degree $d$ polynomial over some ring $A$. Define the content of $f$ to be the ideal: $$c(f) = (a_0,\dots,a_d).$$ One can ask for the relation of the above ideal to the ...
-3
votes
2answers
33 views

Basis of polynomial [closed]

I tried to do this but no result. Can anyone please explain me and make me understand this) Let $a \in \Bbb R - \{0\}$, and consider the family of polynomials $$B_a=\{x^2,\ (x - a)^2,\ x^2(x - a),\ ...
0
votes
1answer
48 views

The existence of an irreducible factor of degree at least $k$ of a polynomial [duplicate]

Let $f = a_0 + a_1x + ... + a_nx^n$ be a polynomial in $\mathbb{Z}[X]$ of degree $n$. Suppose that for some $k (0 < k < n)$ and some prime $p: p∤a_n; p∤a_k; p|a_i$ for all $0≤i≤k-1$; and ...
1
vote
1answer
27 views

Solving a multivariate polynomial system involving the power sums

I would like to know if there is a way to solve or simplify the system of equations given by: $$ x_1^1+x_2^1+\cdots x_n^1 = c_1\\ x_1^2+x_2^2+\cdots x_n^2 = c_2\\ \vdots\\ x_1^n+x_2^n+\cdots x_n^n = ...
0
votes
0answers
20 views

prove that specific polynom is a sum of squares of polynoms [duplicate]

Given polynomial P(x) with real coefficients and condition: $P(x) \ge 0 $ for every x, prove that P(x) can be represented as sum of squares of polynomials with real coefficients. I understand, that ...
0
votes
1answer
19 views

Residue at infinity of $zp'(z)/p(z)$

Suppose a polynomial $p\in\mathbb{C}[x]$ of degree $m$ with complex roots $b_1,\ldots,b_m$. Then $$ f(z):=z\frac{p'(z)}{p(z)}=\frac{z}{z-b_1}+\ldots+\frac{z}{z-b_m}.$$ I want to compute ...
-3
votes
0answers
18 views

How to find the coordinate matrix?

I want the coordinate matrix of the conjugated (T*)linear transformation in the R2 real polynomial spaces with scalar product and T linear transformation. The matrix must be in standard basis of ...
0
votes
2answers
48 views

Show that the set of polynomials with 1 as a root form a linear subspace

Let $\mathbb{C}(x)$ be the vector space $\mathbb{C}$ of polynomials $p\left(x\right)$ in one variable $x$ with coefficients in $\mathbb{C}$. Is the set $p(x) \in \mathbb{C}\left(x\right)$ such that ...
4
votes
1answer
57 views

Does $c(f) = \gcd(\{ f(n) | n \in \mathbb{Z} \})$?

Consider $\sum_{i = 0}^n a_i x^i \in \mathbb{Z}[x]$. Recall that the content of a polynomial is the gcd of its coefficients. I'm wondering whether the content is equal to $\gcd ( \{ \sum_{i = 0}^n a_i ...
0
votes
1answer
36 views

Algebraic Long-Division, where the divifing $x$ has a co-efficient

I know how to do algebraic long division when dividing by $(x+a)$ however when a co-efficient is added to the $x$ what do you do? e.g $\frac{x^{3}-4x^{2}+12}{3x-4}$ I recall you having to times by ...
0
votes
0answers
11 views

Product of heights of factors smaller than length of a polynomial with integer coefficients

I have the following question. Given a (univariate) polynomial with integer coefficients, I want to prove, if true, that the product of heights of its (irreducible) factors is smaller or equal to its ...
4
votes
1answer
1k views

Relative Maxima/Minima of polynomial functions

I am taking the Pre Calculus 12 course online. I came across this concept that the online material teaches in 3 different ways, and each one contradicts the other. I find this extremely frustrating. ...
2
votes
1answer
36 views

When you divide the polynomial $A(x)$ by $(x-1)(x+2)$, what remainder will you end up with?

When you divide the polynomial $A(x)$ by $x-1$, you get a remainder of $10$. When you divide $A(x)$ by $x+2$ you get remainder $0$. When you divide $A(x)$ by $(x-1)(x+2)$ what remainder will you end ...
0
votes
2answers
34 views

Zeroes of the polynomial $f(x)$ over the field $F$ of order 256.

Let $F$ be a field with 256 elements and $f \in F[x]$be a polynomial with all roots in $F$. Then (1) $f \neq x^{15} -1$. (2) $f \neq x^{63} - 1$ (3) $f \neq x^2 + x + 1$ (4) if $f$ ...
0
votes
0answers
59 views

Don't understand what the roots of $t^6-8$ are

I am ultimately asked to find the splitting field of the said polynomial over $\mathbb{Q}$. So I must find the roots first and honestly $t^6-8=(t^2-2)(t^4+2t^2+4)$ So at least two of my solutions ...
2
votes
0answers
20 views

Mathematical development with Polynom modulo n

I have to implement a method seen in an article, and I'm stuck with some mathematical development. The article is on iEEE Xplore, so I'll try to be as specific as I can. It's about pairing-based ...
5
votes
2answers
6k views

Finding the discriminant and roots of a polynomial

How is the discriminant of a polynomial determined? I know that for a quadratic function, the roots (where $f(x)=0$) are found by $$x=\frac{-b\pm\sqrt{\Delta}}{2a}$$ and here $\Delta$ is the ...
0
votes
0answers
32 views

Number of ways to represent a number as a sum of K numbers in subset S

Let the set $S = \{ 1 , 2 , 4 , 5 , 10\}$ Now I want to find the number of ways to represent $x$ as sum of $k$ numbers of the set $S$ (a number can be included any number of times). If $x = 10$ and ...
2
votes
1answer
40 views

$f(x)$ is a quadratic polynomial with leading coefficient $1$, $|f(x)| \leq 8 \: \forall \: x \in [1,9]$ find $f(x)$

$f(x)$ is a polynomial of the form ($b,c$ are real numbers) $$f(x) = x^2+bx+c$$ such that $|f(x)| \leq 8 \: \forall \: x \in [1,9]$. Find all $f(x)$ satisfying the given condition. I found ...
1
vote
3answers
81 views

Congruence $16^{(x^ 2+x+1)} \equiv 4 \mod 11$

Given the congruence $16^{x^2+x+1}≡ 4 \mod 11$ I'm not necessarily sure how to approach this problem if someone can help me head in the right direction since 16 is not a primitive root of mod 11 I ...
2
votes
1answer
21 views

real affine varieties are hypersurfaces

In $\mathbb{R}^n$, let X be a Zariski-closed set. then $X=\mathbb{V}(f)$ for some polynomial $f$. Elementary formulation: let $X \subset \mathbb{R}^n$ be the set of common zeroes of some ...
1
vote
1answer
520 views

Newton backward interpolation in Mathematica

I have the following task: Create a function (in Wolfram Mathematica), called $\mathrm{NewtonBackward}$[n_,x0_,h_,f_] which interpolates backwards the function $f(x)$ with nodes {x_i = x_0 + ...
4
votes
3answers
110 views

Find all polynomials $P(x)$ such that $P(x^2)=P(x)^2$

Find all polynomials $P:\mathbb{C}\rightarrow\mathbb{C}$ such that $$P(x^2)=P(x)^2 .$$ Here is what I tried: First, it is easy to see the constant solutions, namely $P\equiv 0,P\equiv 1$. Let ...
0
votes
1answer
8 views

Given this discrete non-linear set of values how can I get an equation for it?

I want to generate a equation in the form f(x) = {...} for this discrete data below. As X doubles Y halves but its a bit more complicated. Using an online Polynomial Interpolation calculator I got: ...
1
vote
1answer
75 views

Problem involving polynomial function and prime numbers

Let $f$ be a polynomial function, with integer coefficients, strictly increasing on $\Bbb N$ such that $f(0)=1$. Show that it doesn't exist any arithmetic progression of natural numbers with ratio ...
2
votes
2answers
79 views

Number of polynomials which are divisible by $x+1$

Let $a,b,c,d$ be four integers (not necessarily distinct) in the set ${1,2,3,4,5}$ . The number of polynomials $f(x)=x^4+ax^3+bx^2+cx+d$ which are divisible by $x+1$ are: $(A)$ Between 55 and 65 ...
1
vote
3answers
68 views

zeroes to polynomials in residue rings of Z

I'm supposed to find zeroes of $x^{12} -16$ in $\mathbb Z_{17}$, seems simple enough but I just can't seem to make any progress. I realize of course that we have $X^{12} = -1$ in $\mathbb Z_{17}$, ...
7
votes
1answer
45 views

Unramified primes of splitting field

I would like to show the following: Theorem: Let $K$ be a number field and and $L$ be the splitting field of a polynomial $f$ over $K$. If $f$ is separable modulo a prime $\lambda$ of $K$, then $L$ ...
3
votes
1answer
67 views

How to write this cubic equation

For an imaginary number $i=\sqrt{-1}$ ,the cubic equation $24x^3+21x^2-72x-7=0$ can be represented in the form ...
0
votes
0answers
19 views

$n$th order Polynomial for $(n+1)$ points

I was reading about Polynomial Fitting and found this sentence: How can one reach this conclusion and prove it?
2
votes
1answer
56 views

Show that $\int_{0}^{1}P(x)f(x)=0$ [closed]

Let $f:[0,1] \rightarrow \mathbb{R}$ a continuous function, such that $\exists (u,v)\in [0,1]$; $ f(u)>0, f(v)<0$. Show that there exists $P$ a polynomial such that $P>0$ over $[0,1]$ and ...
1
vote
4answers
85 views

Find $\alpha^3 + \beta^3$ which are roots of a quadratic equation.

I have a question. Given a quadratic polynomial, $ax^2 +bx+c$, and having roots $\alpha$ and $\beta$. Find $\alpha^3+\beta^3$. Also find $\frac1\alpha^3+\frac1\beta^3$ I don't know how to proceed. ...
1
vote
1answer
92 views

Example of a nonzero polynomial with more roots than its degree [closed]

Obviously normal polynomials won't work. Can you give an example that does?
0
votes
0answers
17 views

Coefficient of bivariate polynomial as a determinant of matrix

Given $$ \begin{bmatrix} a\\ b\\ c\\ d\\ \end{bmatrix}=\begin{bmatrix} a_0t^3+a_1st^2+a_2s^2t+a_3s^3\\ a_4t^2+a_5st+a_6s^2\\ a_7t+a_8s\\ a_9\\ \end{bmatrix} $$ the following equation holds: $$ ...
1
vote
3answers
31 views

Evaluate $\int_1^N \frac{-3N+6t-3}{t^3(N-t+1)^4}dt$ when $N=3$ or $N=5$

Let the Cauchy product $$(\zeta(3))^2=\sum_{n=1}^\infty c_n,$$ where $$c_n=\sum_{k=1}^n\frac{1}{k^3(n-k+1)^3},$$ and $\zeta(3)$ is the Apèry constant. Taking $f(x)=\frac{1}{x^3(N-x+1)^3}$ in Abel's ...
0
votes
2answers
21 views

Why has $p(\lambda)$ exactly one positive zero?

Let $p(\lambda ) = {a_m}{\lambda ^m} + {a_{m - 1}}{\lambda ^{m - 1}} + \cdots + {a_1}\lambda - {a_0}$ and ${a_0} > 0,{a_1},{a_2}, \ldots ,{a_m} \ge 0$, and at least one of the coefficients ...