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.

learn more… | top users | synonyms

0
votes
1answer
18 views

Undefined derivative of a third degree polynomial inverse?

I have the following equation: $$r_t=kr_s^{3}+r_s$$ I need to get $r_s'$, i.e. derivative of $r_s$ w.r.t. $k$, provided that value of $r_t'$ is already known. Taking derivatives of both sides w.r.t. ...
3
votes
1answer
118 views

For which polynomials $f$ is the subset {$f(x):x∈ℤ$} of $ℤ$ closed under multiplication?

You surely know about the Brahmagupta–Fibonacci identity, $$(a_1^2 + b_1^2)(a_2^2 + b_2^2) = (a_1a_2 \pm b_1b_2)^2 + (a_1b_2 \mp a_2b_1)^2$$ which tells us that the product of two numbers, each of ...
2
votes
1answer
29 views

A question on roots of a polynomial of degree $n$

Under what conditions on cofficients of a polynomial $p(x)$, the roots of $p(x)$ are real and positive?
2
votes
0answers
20 views

Why polynomial $\psi^\top(t) A^{-1} \psi(t)$ attains maximum on $[-1, 1]$ at $\pm 1$, where $\psi_k(t) = t^k$?

Problem. Let $\psi(t) = (1, t, t^2, \ldots, t^{p-1})$ - a polynomial basis. Suppose there is a matrix $$ A = \int_{-1}^1 \psi(t) \psi^\top(t) dt, \ \text{i.e. } \ A_{ij} = [2 \, | \, i+j] \cdot ...
1
vote
1answer
34 views

Noether's normalization lemma in practice (example)

I would like to know how to use the Noether's normalization lemma in practice. Noether's normalization lemma Let $k$ an infinite field, and $k[a_1,\dots ,a_n]$ be a finite $k$-algebra. There ...
6
votes
0answers
90 views

Product of two random polynomials

Let $\alpha,\beta$ be two polynomials of the form $$\alpha(X)=\sum_{i=0}^{n}\alpha_iX^i,\quad \quad \beta(X)=\sum_{j=0}^n\beta_jX^j$$ where each coefficient is $1$ with a probability of $p$ and $0$ ...
2
votes
1answer
45 views

Integer coefficients in a polynomial function

The problem: Find the polynomial $r(x)$ with the smallest degree and integer coefficients s.t. it has zeroes -1, 2/3 and -2 w/ resp multiplicities 2, 8 and 4 and $r(1) = -1$. What I tried: ...
0
votes
0answers
18 views

Polynomial roots conditions vary with coefficients

Polynomial equation $\sum_{i=0}^4 p_i x^i=0$ have the following root conditions: 1) $a_0 \pm b_0 i, a_1 \pm b_1i$ 2) $a_0 \pm b_0 i, a_1, a_2$ 3) $a_0, a_1, a_2 \pm b_2i$ 4) $a_0, a_1, a_2, a_3$ I'm ...
1
vote
0answers
21 views

Finance Algeabra: Converting a Discount Polynomial Function to an Interest Rate Polynomial Function

I have a finance problem that is 99% mathematical. In finance, the price of a bond could be modelled as the discounted value of its future cash flows, so something like: ...
0
votes
0answers
23 views

Bounding the integral of the reciprocal of a complex polynomial

So, I would like to bound $\int_{C_R} \frac{1}{P(z)} dz$ where $C_R$ is the circle radius R centred at the origin, and $P(z)$ is a polynomial of degree $N=0,1,...,n$ i also want to deduce for what ...
0
votes
0answers
21 views

Using Euclid's algorithm, how do I find a polynomial $f_p(x)$ such that $f \cdot f_p \equiv 1 \pmod p$?

Suppose that we have a polynomial $f(x)$ with coefficients in $\mathbb{Z_3}$ and maximum degree $N-1$, where $N$ is prime. (In fact, we consider $f$ as a class of ...
-1
votes
1answer
35 views

$A\subset B$ if $A\cdot R[X] \subset B \cdot R[X]$? [closed]

Can we conclude $A\subset B$ if $A\cdot R[X] \subset B \cdot R[X]$ for ideals $A,B$ in $R$, where R is a commutative ring with unity and $A \cdot R[X]$ the ideal generated by the products $af$, ...
1
vote
3answers
74 views

Prove that $(x^2+1)\mathbb Z[x]$ is a prime ideal of $\mathbb Z[x]$, but not maximal

Prove that $(x^2+1)\mathbb Z[x]$ is a prime ideal of $\mathbb Z[x]$, but not maximal. I'm supposed to show this for my homework. My first thought is to show that $\mathbb Z[x]/(x^2+1)\mathbb ...
0
votes
0answers
40 views

Irreducible over $\mathbb Q[x]$ but reducible over $\mathbb F_p$ for all $p$ [duplicate]

To solve this question I have almost finished the proof but I need a little detail to be rigorous. Let $K$ be the prime fields $\mathbb Q$ or $\mathbb F_p$. Prove that $$f(x)=x^4+1\in ...
2
votes
3answers
34 views

Polynomials equations

I am trying to find the common solutions of this equations: $$ 2x^4+x^3-5x^2+2x=0 $$ $$ 2(2x-1)^{1998}+(4x-1)^{1997}=4x+1 $$. My idea is to solve the first one. I find the solutions:$ x=1$ , $x=0$ , ...
2
votes
2answers
20 views

What does $\mathbb{Z}[X]$ for a polynom mean?

I have a proof saying that a Polynom $p \in \mathbb{Z}[X_1,...,X_m]$ I'm a bit confused of this notation because neither X nor the m is explains somewhere. Does somebody of you know the notation?
0
votes
1answer
26 views

Find the value of $k$ in the equation [closed]

Find the value of $k$ for which the equation $$kx^2-2015x+(k-2015)=0$$ has one positive and one negative root.
0
votes
1answer
54 views

Find all the Zeros and their multiplicities of $f(x)=x^5 +4x^4 +4x^3 -x^2-4x +1$ over $\Bbb Z_5$.

Find all the Zeros and their multiplicities of $f(x)=x^5 +4x^4 +4x^3 -x^2-4x +1$ over $\Bbb Z_5$. Firstly,I've found the zeros of $f(x)$,just by simply substituting the elements of $\Bbb ...
2
votes
2answers
46 views

Proving $f(x)$ is not a square in $k[x]$

Let the field $k$ be algebraically closed, let $f(X) \in k[X]$ be a separable polynomial of degree at least $2$, let $$ B = \frac{k[Y,X]}{(Y^2 - f(X))} $$ and write $y,x$ for the images in $B$ of $Y$ ...
5
votes
2answers
59 views

If a polynomial ring is a free module over some subring, is that subring itself a polynomial ring?

Suppose I have a graded polynomial ring $k[x_1,\ldots,x_n]$ on homogeneous generators, where $k$ is a field and the $x_i$ indeterminates, and further that I have a homogeneous graded subring $A$ such ...
0
votes
0answers
17 views

Two different random processes for generating polynomials

Consider two processes for generating random complex polynomials: choosing the roots uniformly and independently throughout the unit disc, and choosing the coefficients uniformly and independently ...
0
votes
0answers
26 views

how can i convert a polynomial to a symmetric matrix

is there any way i can convert a polynomial $P(x)$ into a symmetric matrix $A$ ? so that $det(A-I\lambda )=P(\lambda)$
0
votes
1answer
36 views

What is the likelihood that a degree 3 monic polynomial will have 3 real roots.

The discriminant of a monic cubic polynomial $x^3 + bx^2 + cx + d $ is given by the expression $b^2 c^2 - 4 c^3 - 4 b^3d + 18 bcd - 27 d^2$. If this expression is positive then all the roots are real. ...
5
votes
0answers
57 views

The probability that a random (real) cubic has three real roots

We can formalize the notion of the probability that a randomly selected quadratic real polynomial has real roots as follows: Suppose $R > 0$, and suppose the random variables $a, b, c$ are ...
1
vote
2answers
129 views

Has the polynomial distinct roots? How can I prove it?

I want to prove that the polynomial $$ f_p(x) = x^{2p+2} - cx^{2p} - dx^p - 1 $$ ,where $c>0$ and $d>0$ are real numbers, has distinct roots. Also $p>0$ is an even integer. How can I prove ...
1
vote
1answer
35 views

How to prove $\sum_{i=1}^n |a_i|^r\leq (\sum_{i=1}^n|a_i|)^r$

I want to establish $\sum_{i=1}^n |a_i|^r\leq (\sum_{i=1}^n|a_i|)^r$, where $a_i,r \in R$ and $|.|$ is the absolute value. Is the condition $r>0$ correct? How to prove this inequality?
1
vote
1answer
37 views

are these the only answers of $x^4+y^4+z^4+1=4xyz$?

Given an equation $$x^4+y^4+z^4+1=4xyz$$Find out the number of possible ordered tuple $(x,y,z)\mid x,y,z\in\Bbb{R}$. I am getting it as $(1,1,1),(-1,-1,1),(1,-1,-1),(-1,1,-1)$ so $\boxed{4}$ ...
-1
votes
2answers
32 views

Construct a degree 3 monic polynomial with integer coefficients that has 3 irrational roots.

The polynomial $x^3 - 3x + 1$ is monic, degree $3$, has integer coefficients and all its roots are irrational. I found this polynomial using Mathematica to generate random polynomials and then ...
0
votes
2answers
46 views

Irreducible implies Separable in a Finite Field

Proposition 37 on page 549 of Abstract Algebra, 3rd Ed. by Dummit and Foote claims that irreducible implies separable over a finite field. Suppose $p(x)$ is irreducible over a finite field of ...
4
votes
2answers
224 views

Sum of cube roots of a quadratic

If $a$ and $b$ are the roots of $x^2 -5x + 8 = 0$. How do I find $\sqrt[3]{a} + \sqrt[3]{b}$ without finding the roots? I know how to evaluate $\sqrt[2]{a} + \sqrt[2]{b}$ by squaring and subbing for ...
0
votes
1answer
35 views

Solving a “simple” quadratic/quartic equation

Despite having solved quadratic quations for years I can't seem to be able to get the same result than maple on this one (not as simplified as Maple's), so I wonder if someone could not explain: I'm ...
2
votes
1answer
41 views

Integrating Lagrange polynomials

Could you suggest some efficient way to numerically compute $\int\limits_0^{t_i}l^N_j(t)dt$, where $l^N_j(t)$ is the $j$th N-point Lagrange polynomial and $t_i$ is the $i$th interpolation point?
-1
votes
1answer
20 views

Proving that f(X) factors in $\mathbb Z_3[X]$ as a product of three linear factors [closed]

I have no idea how to answer this question. Any help would be appreciated! Let $f(X) = X^3 - X + 1$ be a polynomial in $\mathbb Z_3[X]$. How to prove that $f$ factors in $\mathbb Z_3[X]$ as a ...
0
votes
1answer
45 views

$I=(f_1, \ldots, f_n)\subset k[x_1, \ldots, x_n]$ with $f_i\in k[x_i]$ irreducible polynomials

Let $A=k[x_1,\ldots, x_n]$ and $I=(f_1, \ldots, f_n)\subset A$ with $f_i\in k[x_i]$ irreducible polynomials. Is it true that $I$ is a maximal ideal in $A$? $I$ is a maximal ideal $\iff$ $1\in ...
-9
votes
2answers
37 views

Find the perfect square trinomial! [closed]

Fill in the blank to complete the perfect square trinomial: $$ z^8+\_\_\_+144 $$
2
votes
3answers
118 views

Finding the range of a $y=-x^2(x+5)(x-3)$ without calculus?

I was helping a precalculus student with this question. The graph wasn't given. My only idea was to find the inverse and try to find its domain. When trying to find the inverse, I arrived at ...
12
votes
3answers
136 views

Why $|x|$ is not rational expression?

I'm 9th grade student, and my teacher said that $|x|$ is not rational expression ( expression like $\frac{p(x)}{q(x)}$ s.t $p(x)$ and $q(x)\neq 0$ are polynomial) but he didn't have convincing reason. ...
0
votes
1answer
28 views

Characterization of the elements of a quotient ring

I'm in trouble with the following exercise: Consider the ideal $ I = (X^2-Y^3,Y^2-Z^3) $ in the polynomial ring $ k[X,Y,Z] $, where $k$ is any algebraically closed field. Show that every element of $ ...
0
votes
1answer
10 views

Factors/divisibility of monotonically-increasing integer polynomial

For positive integers $n$ and $x$, let $f_n(x)$ be a polynomial in $x$ of degree $n-1$, such that $f_n(x)$ is monotonically increasing for increasing $x \ge 1$. Now assume that there exist positive ...
0
votes
0answers
45 views

Why $f_1,f_2,f_3$ don't have a common factor?

Let $p(x,y)$ and $q(x,y)$ are polynomials in $x,y\in \mathbb{R}$. ($p,q\ne0$) $p$ and $q$ are coprime to each other ,($\frac{{\partial p}}{{\partial x}}=p_x$,$\frac{{\partial p}}{{\partial y}}=p_y$) ...
1
vote
1answer
25 views

How to draw cubic plane curve?

In Python, using MatPlotLib, given [vector] parameters $a$ and $b$ and [scalar] parameter $c$, I want to draw a general cubic plane curve in 2-dimensional space (regular plane with $x$ and $y$ axes): ...
1
vote
1answer
27 views

What is the family of parabolas with the following characteristics? [closed]

How does one describe the family of parabolas that contain the point $(0,0)$ and are tangent to the parabola $x = y^2 + 1$?
0
votes
0answers
37 views

Exhibit a reducible polynomial of the form $x^p -x-c$ having no roots in a field of characteristic 0

Is it possible for a polynomial, $x^p -x-c$ where $p$ is prime, to be reducible in a field of characteristic $0$, yet have roots in that field? I know for a fact that the general form is true, ...
1
vote
1answer
15 views

Do the integer roots of a polynomial $P(x) \in \Bbb Z[x]$ have to divide the constant coefficient?

By Gauss Lemma, the roots of a polynomial $P(x) = a_nx^n + \cdots + a_1x + a_0 \in \Bbb Z[x]$ are either integer, irrational or complex. Vietà's formulas imply that the product of all roots equals ...
1
vote
1answer
71 views

Logarithm as limiting case of $n$th root

Let $f_n(x) = x^{1/n}$ where $n \in \mathbb N$, and let $g(x) = \log(x)$. We can compute $f_n'(x) = \frac{1}{n}x^{-1 + \frac{1}{n}}$ and $g'(x) = x^{-1}$. Let's define $f_\infty(x) = \lim_{n ...
2
votes
1answer
40 views

Denominator is product of irreducibles with cyclic Galois group

Short version of the question: Guess the next terms in the sequence : $D_{17},D_{19},D_{23}$ etc where $$ \begin{array}{lcl} D_3 &=& (a\pm 1) \\ D_5 &=& (a\pm 1) (a^2-1 \pm 11a) \\ ...
0
votes
1answer
32 views

solution of system of polynomials

I have 3 equations as following: $$ \left\{ \begin{array}{c} (\Delta_{11}*y^2 + \Delta_{12}*y + \Delta_{13})x^2 + (\Delta_{21}*y^2 + \Delta_{22}*y + \Delta_{23})x + \Delta_{31}*y^2 + \Delta_{32}*y + ...
0
votes
1answer
16 views

Jacobian of a system of equations

I'm asked to compute the Jacobian of a system of equations $x_1^4+x_2^4-1=0$ $x_2-\sin(5x_1)=0$ $x_1-x_3^2=0$ What's the Jacobian of a system of equations? Do I perhaps need to infer the individual ...
1
vote
2answers
36 views

Find a polynomial f(x) of degree 5 such that 2 properties hold.

I have been trying to find a polynomial $f(x)$ such that these $2$ properties hold: $f(x)-1$ is divisible by $(x-1)^3$ $f(x)$ is divisible by $x^3$ To start, I set $f(x) =ax^5 + bx^4 + cx^3 + dx^2 ...
2
votes
1answer
27 views

Show that this sum of polynomials has no zeros with positive real part

Let $0 < \lambda_1 \leq \ldots \leq \lambda_n $ and $k_1, \ldots, k_n> 0$. Let further $$ \begin{align} P(x)&:=\prod_{i=1}^n (x+\lambda_i) = (x+\lambda_1)\cdot \ldots \cdot (x+\lambda_n) ...