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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2
votes
3answers
60 views

Polynomial interpolation

I need to find the polynomial of degree $3$ with respect to these conditions: $$\begin{cases} p(0) = 1\\ p(1) = -1\\ p'(0) = 1\\ p''(0) = 0 \end{cases}$$ How do I deal with the condition on ...
1
vote
1answer
14 views

Bound for complex roots of polynomial

I am trying to prove that if $p(z)=z^n+a_{n-1}z^{n-1}+\dots+a_0$ then all the zeros lie in a circle of radius $R= \max\{1,|a_0|+|a_1|+|a_2|+\dots+|a_{n-1}|\}$ I'm trying to use induction and perhaps ...
0
votes
1answer
14 views

For $f(x)=x^4$, find its projection $f(x)^*\in P^2(-1,1)$ onto $W$

Consider the vector space $V=C[-1,1]$ and $W=P^2[-1,1]$. $V$ is an inner product space withe inner product $\langle f, g\rangle=\int_{-1}^1f(x)g(x)dx$. Consider a function $f(x)=x^4$ whcih is in ...
0
votes
0answers
28 views

method of undetermined coefficients and come up with a new quadrature.

I'm trying to solve some problems which is related method of undetermined coefficients to determine some weights and to come up with a new quadrature. the interval x∈[0,1]. given values of a function ...
-2
votes
0answers
19 views

What is the dimension of $\ker f =\{(x^3-x)Q(x):Q \in\mathbb{R}_{n-3}[X]\}$?

I have $$\ker f =\{(x^3-x)Q(x):Q \in\mathbb{R}_{n-3}[X]\}.$$ Here $f$ is the following endomorphism $$f(P) = (x^2-x+1)P(-1)+(x^3-x)P(0)+(x^3+x^2+1)P(1),$$ where $P\in\mathbb{R}_{n}[x]$. My ...
1
vote
0answers
29 views

Polynomial roots in the ring extension

Let $R$ be a ring with identity (not necessarily commutative) and $R[x]$ be a ring of polynomials over $R$. We say that a ring $S$ is an extension of $R$ if there is a subring $\tilde{R}$ in $S$ ...
0
votes
0answers
15 views

Counting the number of roots of multivariate polynomials?

The equation of a circle is well known $$(x-x_0)^2+(y-y_0)^2 - r^2 = 0$$ It has a solution all along the circle with midpoint $(x,y) = (x_0,y_0)$. We also know that $ab = 0$ whenever any of $a$ and/or ...
2
votes
0answers
15 views

Rational function between a constant and a third root

Is there a rational function $f(x)\in{\mathbb Q}(x)$ such that $\sqrt{2} \leq f(x) \leq \sqrt[3]{2x}$ for all $x\geq\sqrt{2}$ ? My thoughts : it is easy to find such an $f$ if we relax the ...
13
votes
3answers
679 views

Can second degree polynomials generate as many as we wish prime numbers in the way described?

While I was getting in my pyjamas, a few minutes ago, the Euler polynomial $n^2+n+41$ came into my mind. As you know, this polynomial is famous because the set $\{f(0),f(1),...f(39)\}$ consists of ...
0
votes
1answer
24 views

solving a pair of simultaneous equations

I have a rather messy pair of simultaneous equations, which I need to solve for x: ...
0
votes
1answer
19 views

Random Walk with overshoot, step sizes $+1, -2$. Solve the polynomial in $e^λ$ [on hold]

If the moment generating function is $$mS(θ) = E(e^{θS}) = pe^θ + qe^{−2θ} = 1$$ Show that setting $$mS(\lambda) = 1$$ yields the unique positive solution: $$ \lambda = \log { \frac{q + \sqrt{4pq + ...
5
votes
2answers
652 views

Solving a 6th degree polynomial equation

I have a polynomial equation that arose from a problem I was solving. The equation is as follows: $$-x^6+x^5+2x^4-2x^3+x^2+2x-1=0 .$$ I need to find $x$, and specifically there should be a real ...
2
votes
1answer
52 views

Sum of a polynomial with all its derivative [duplicate]

Let $$p(x)=x^n+a_1x^{n-1}+...+a_{n-1}x+a_n,$$ with $n$ is even and $p(x)>0$ for all $x\in\mathbb{R}$. Let $$q(x)=p(x)+p'(x)+..+p^{(n-1)}(x)+p^{(n)}(x).$$ Show that $q(x)>0$ for all ...
1
vote
0answers
26 views

Quadrant in which the zeros of a polynomial lies

Consider a polynomial $$p(z) = z^6 + 9z^4 + z^3 + 2z + 4 $$ I need to find which quadrant of the complex plane contains how many zeros that lie in unit circle. Also, I need to find which quadrant ...
1
vote
3answers
42 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 ...
0
votes
2answers
49 views

Quick Question - Complex Roots of Polynomials?

I'm asked to solve for Z where $$\frac{z+i}{2z-i} = \frac{-1}{2} + i\frac{\sqrt 3}{2}$$ As a result i got $$2z = \sqrt{3}zi + \frac{i}{2} - i^2\frac{\sqrt 3}{2} - i$$ The answer is supposed to be ...
-1
votes
1answer
30 views

Factoring polynomials in $\Bbb Z_n$

a). Factor $f(x) = x^3 + 4x^2 + 5x + 2$ completely over $\Bbb Z_7$. b). Give two different factorizations of $x^2 + x + 8$ in $\Bbb Z_{10}[x]$. I have found the zeros of both of these but I am ...
0
votes
1answer
55 views

Find all the zeros of $f(x) = x^3 + 3x + 5$ in $\Bbb Z_7$

Find all the zeros of $f(x) = x^3 + 3x + 5$ in $\Bbb Z_7$. I've tried factoring this into multiple forms but I can't seem to find an easy way to find the $x'$s for $x^3 + 3x + 5 = 0$. Any hints or ...
0
votes
1answer
21 views

Two questions regarding polynomial rings.

Give an example of a natural number $n > 1$ and a polynomial $f(x) ∈ \Bbb Z_n[x]$ of degree $> 0$ that is a unit in $\Bbb Z_n[x]$. For this is set $n=2$. So then $f(x) = x \in \Bbb Z_2[x] $. ...
0
votes
2answers
45 views

General questions about Polynomial Rings [on hold]

I'm learning about polynomial rings in my class. My instructor and book are both spectacularly unhelpful and didn't even bother to define most of the terms in my homework. So I have some general ...
1
vote
1answer
24 views

$f(x) \in\Bbb Q[x]$. Prove that if $f(a + b\sqrt c) = 0$, where $a, b \in\Bbb Q$ and $\sqrt c \in\Bbb Q$ then $f(a − b\sqrt c) = 0$. [on hold]

Let $f(x)\in\Bbb Q[x]$. Prove that if $f(a + b\sqrt c) = 0$, where $a, b \in\Bbb Q$ and $\sqrt c \not\in \Bbb Q$ then $f(a − b\sqrt c) = 0$. I don't really have any idea of where to start on this. ...
0
votes
2answers
30 views

$f(x) \in \mathbb{R}[x]$. Prove that if $z = a + bi$ is a zero of $f(x)$ then $z = a − bi$ is also a zero of $f(x)$.

Let $f(x) \in \mathbb{R}[x]$. Prove that if $z = a + bi$ is a zero of $f(x)$ then $z = a − bi$ is also a zero of $f(x)$. I'm learning about polynomial rings but my book and my instructor never ...
2
votes
2answers
171 views

How do I find out if a polynomial is irreducible?

I have this polynomial: $f(x)=x^4+x^3-4x^2-5x-5$. How can I find out if this polynomial is irreducible over the field $Q$ of rational numbers? I know about mod p irreducibility test but it fails in ...
0
votes
3answers
69 views

Number of real root of the equation $8x^3-6x+1$ lying between -1 and 1 is

Number of real root of the equation $8x^3-6x+1$ lying between -1 and 1 is: I am lagging in solving the inequality portion. Let the roots be $m_1,m_2,m_3$ then $m_1m_2m_3=-\frac{1}{8}$ which means ...
1
vote
0answers
14 views

Primitive polynomials from GF(q) to GF(q^n)

Suppose that over some finite field $GF(q)$, we have two monic primitive polynomials of orders $n$ and $mn$. -From these polynomials, is there always a 'natural' monic primitive polynomial over ...
-2
votes
2answers
35 views

Find $P(n+1)$ using the given information

Suppose that $P(x)$ is a polynomial of degree $n$ such that $P(k)$ = $\frac{k}{k+1}$ for $k=0,1,2,.....n$. Find the value of $P(n+1)$. I have absolutely no idea to start.
1
vote
1answer
255 views

approximating Lipschitz functions by polynomials?

If $f:[0,1]\rightarrow\mathbb{R}$ satisfies $|f(x)-f(y)|<|x-y|\ $ for all $x$, $y\in[0,1]$ and $\epsilon>0$ is fixed, why must there be a polynomial $p$ of degree less than ...
0
votes
1answer
33 views

How to find out if some eigenvalues of a matrix are the same?

I know that in order for a matrix to have two equal eigenvalues, one term in the characteristic polynomial must be in the power of two. Is there any way to tell if two eigenvalues are the same? I have ...
0
votes
1answer
29 views

The relation between $GF(2)$ and $GF(2^3)$

Both $GF(2)$ and $GF(2^3)$ are finite fields of characteristic $2$. Is $GL(2^3)$ an extension of $GF(2)$? Can someone point some links that details something about this, please?
2
votes
1answer
27 views

Quick Question - Complex roots of polynomials?

I was asked to find solutions to $z^3 = 1$ and give my answer in Cartesian form. I got $1, -1/2 \pm i\sqrt{3}/2$ (b) Hence solve the equation $(z+i)^3 = (2z-i)^3$ Little help on this one? Any help ...
-1
votes
0answers
38 views

$A,B\in\mathbb Q[x]$ with $A,B$ monic, and $ AB\in\mathbb Z[x]$, prove $A,B\in\mathbb Z[x]$

It is part of cyclotomic polynomials. But I don't know how to deal with it and what to do next. I have prove $n$-th root is related to Euler's totient fuction. But I don't know how to use it. Thank ...
1
vote
2answers
28 views

How to show that $X^p-t\in\mathbb{F}_p(t)[x]$ is irreducible? [duplicate]

This question is previously asked here, but there is no complete solution of it. I understand that the root $\alpha$ exist in the algebraic closure of $\mathbb{F}_p(t)[x]$, and it is the only root ...
0
votes
2answers
359 views

Why this polynomial is irreducible? [on hold]

Let $K=\mathbb{Z}_p(t)$. How to prove $f(x)=x^p-t$ is irreducible in $K[x]$?
2
votes
0answers
36 views

Zero divisor polynomial [duplicate]

Let $f(x)\in R[x]$ be a zero divisor. How to prove that there is an element $0\neq a\in R$ such that $af(x) = 0$? If $R$ has no nilpotent elements, it is easy. What about the general case? Can ...
1
vote
1answer
75 views

How to construct a polynomial with minimum deviation from zero on the complex region?

I need to compute the analog of Chebyshev polynomials (which give the minimum deviation from zero on [-1,1]) on the given region $\Omega\subset \mathbb C$. More precisely: find $P_n$ such that ...
1
vote
0answers
9 views

Finding the roots with the largest magnitude

Given a non-constant polynomial $p\in\mathbb{Z}[x]=\alpha\prod_{k=1}^nx-\alpha_k$ how can I find the roots $R=\{\beta_1,\ldots,\beta_t\}\subseteq\{\alpha_1,\ldots,\alpha_n\}\subseteq\mathbb{C}$ with ...
0
votes
2answers
25 views

Why is $\varphi(X_i) = X_i + b_i$ an automorphism of $K[X_1,\dots,X_n]$?

I'm trying to justify to myself the assertion (used here) that given a field $K$ and elements $b_1,\dots,b_n\in K$, the map $\varphi(X_i) = X_i + b_i$ is a $K$-automorphism of $K[X_1,\dots,X_n]$. ...
1
vote
0answers
14 views

I have an problem with the function to optimize with lagrange multipliers

I need help with the restriction of the problem, because i cannot find the function to optimize. The problem: Find the maximum and minimum distances from the origin to the curve ...
0
votes
0answers
13 views

Is there an algebraic solution for this rootfinding problem?

I would like to solve for the roots of $f(x)=a_0 + a_1x^\gamma + a_2x^{\gamma+1}$, where $a_0,a_1 \in \mathbb{R}$ and $\gamma \in \mathbb{R}_{\geq 0}$ are arbitrary coefficients. This is possible ...
1
vote
1answer
39 views

Dude with taylor polynomial

Good night, i'm working with an problem of polynomial taylor, but i have a problem with the residue. Get a quadratic approximation $f\left(x,y\right)=\sin\left(x\right)\sin\left(y\right)$ near the ...
1
vote
1answer
46 views

I get a wrong answer for the gcd of two polynomials

Hello first post here, I am trying to get the gcd of the two polynomials using the euclidean algorithm, but as result I get a fraction with huge numbers, instead of 1, which I get as result after ...
0
votes
1answer
25 views

Irreducibility of a Polynomial after a substitution

I am trying to determine whether the polynomial $f(x) = x^6 + 34x^4 + 4x^2 + 89 \in \mathbb{Z}[x]$ is irreducible over $\mathbb{Z}$. Eisenstein's criterion doesn't help and I suppose I could determine ...
1
vote
0answers
13 views

Embedding of a global field to local field of characteristic $p$/

Let $F=\mathbb{F}_q(t)$ and consider its completion $F_P$ with respect to an irreducible polynomial $P(t)$, namely the local field associated to the place $P$ (I understand this could be technically ...
1
vote
5answers
81 views

How to factorize a 4th degree polynomial?

I need help to factorise the following polynomial: $x^4 - 2x^3 + 8x^2 - 14x + 7$ The solution I need to reach is $(x-1)(x^3 - x^2 + 7x - 7)$. I need to factorize to this exactly as it is for ...
11
votes
6answers
11k views

How to prove that a polynomial of degree $n$ has at most $n$ roots?

How can I prove, that a polynomial function $$f(x) = \sum_{0\le k \le n}a_k x^k\qquad n\in\mathbb N,\ a_k\in\mathbb C$$ is zero for at most $n$ different values of $x$? (Except $n=0$ where $f(x)$ is ...
1
vote
1answer
50 views

Is the following described $p(x), q(x)$ the same interpolation polynomial?

Suppose we are given an odd number of data points $x_i$ and the corresponding values $f_i=f(x_i),i=1,...,n+1$($n$ is even), which are symmetric about the origin, i.e for each $x_i$ there is a $j$ such ...
2
votes
1answer
200 views

Find polynomials $q(x)$ and $r(x)$ such that $f(x)=g(x)q(x)+r(x)$ where $r(x)=0$ or $\deg r(x)<\deg g(x)$

Find polynomials $q(x)$ and $r(x)$ such that $f(x)=g(x)q(x)+r(x)$ where $r(x)=0$ or $\deg r(x)<\deg g(x)$ provided that $f(x)=2x^4+x^2-x+1$ and $g(x)=3x^2+2$ in $\Bbb Z[x]$. The problem I'm ...
0
votes
0answers
28 views

Number of roots of a multi-variate polynomial in an integral domain D. [closed]

Let D be an integral domain, and let $ g\in D[X_1,...,X_n], $ with Deg($g$)= $k\ge 0$. Let $S$ be a finite, non-empty subset of D. Show that the number of elements $\left( x_1,...,x_n\right)\in ...
0
votes
0answers
20 views

Computationally check for roots/positiveness of a big polynomial in a given interval

For a proof, I need to check that given a little interval $(0, 0.28)$ some concrete polynomials $\in \mathbb{Q}[w]$ (polynomials in one variable ranging over the real numbers, with degrees around 50) ...
2
votes
1answer
80 views

Solution of $x^2e^x = y$

The other day, I came across the problem (or something that reduced to the problem): Solve for $x$ in terms of $y$ and $e$: $$x^2e^x=y$$ I tried for a while to solve it with logarithms, roots, and ...