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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3answers
22 views

Factoring a 4th degree trinomial

I am trying to factor $3x^4-8x^3+16$, but I have no idea how to even start. I put into Wolfram Alpha, and it said that the answer was $(x-2)^2 (3 x^2+4 x+4)$. How would you factor something like this ...
1
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0answers
8 views

Show that every polynomial of degree $1,2,$ or $4$ in $Z_2[x]$ has a root in $Z_2[x]/(x^4+x+1)$.

The problem: Show that every polynomial of degree $1,2,$ or $4$ in $\mathbb{Z}_2[x]$ has a root in $\mathbb{Z}_2[x]/(x^4+x+1)$. My attempt: I know that the polynomials $x$ and $x+1$ have ...
1
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2answers
63 views

gcd of $x$ and $2$ in $\mathbb Z[x]$

In $\mathbb Z[x]$, $x$ and $2$ have gcd $1$. But they cannot be expressed as the linear combination of two polynomials. Then assuming that $1=2f(x)+xg(x)$ we are supposed to arrive ...
5
votes
2answers
242 views

Number Theoretic Transform (NTT) to speed up multiplications

I recently heard that the Number Theoretic Transform (NTT), which is a specialization of Fast Fourier Transformation (FFT) over the ring modulo an integer, can be used to speed up certain ...
1
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0answers
10 views

Spanning cyclic set and the characteristic polynomial

Let $V$ be an $n$-dimensional $\mathbb{R}$-space and $\alpha$ and endomorphism of $V$. I am trying to show that if $\{\mathbf{v},\alpha(\mathbf{v}),\dots ,\alpha^{n-1}(\mathbf{v})\}$ spans $V$ then ...
3
votes
1answer
35 views

Why is $R((X))$ defined as follows?

Let $R$ be a commutative ring. Then $R((X))$ is defined as the set of all $\sum_{n\geq N} a_n X_n$ where $N\in\mathbb{Z}$ and is called "The Formal Laurent series". But why? Why don't we consider ...
0
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2answers
18 views

I need to prove a simple proposition about polynomials

I am working on systems theory and I have reduced one problem that I have into a simple polynomial proposition. All (non-constant) polynomials (with integer degrees) in which the sum of positive ...
2
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0answers
26 views

Bezout relation with integral coefficients

Suppose I have two monic polynomials $f$ and $g$ with coefficients in $\mathbb{Z}$. I also suppose that $f$ and $g$ are coprime as polynomials over $\mathbb{Q}$. In particular, there exists a Bezout ...
2
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0answers
22 views

Similar Triangle dissections

Andrzej Zak found that a triangle with sides based on powers of the root $d^6-d^2-1=0$ ($d=1.15096...$) that can replicate itself with 6 differently sized copies. The numbers are powers of $d$. The ...
0
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0answers
17 views

Find all polynomials $p $ satisfying $p(x+1)=p(x)+2x+1. $

Find all polynomials $p $ satisfying $p(x+1)=p(x)+2x+1. $ I found this on a local question paper, and I am unable to solve it. Any help will be appreciated.
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0answers
22 views

Conjugate roots of a polynomial

If $\sqrt 2 - i$ is a root of $x^5-x^4-2x^3+mx^2+9x+m-11=0$, $m \in \Bbb Q$ find m and the other roots. My question is what other roots can i deduce from what is given? Is $\sqrt 2 + i$ the only one ...
1
vote
1answer
20 views

Isomorphism between the group $(\mathbb Z[x], +)$ and $(\mathbb Q_{>0}, .)$ [duplicate]

In one of my assignment, I was told that $(\mathbb Z[x], +)$ and $(\mathbb Q_{>0}, .)$ are isomorphic, with $$\phi (\sum_{k=0}^n a_k x^k) = \prod_{k=0}^n p_k^{a_k}$$ is a one-to-one surjective ...
1
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0answers
10 views

$3$ intersection points for a quartic polynomial implies 4 intersection points or a local extrema at one of the intersection points

Q: It is given that the graph of $y = x^4+ax^3+bx^2+cx+d$ (where $a,b,c,d$ are real) has at least $3$ points of intersection with the $x$-axis. Prove that either there are exactly $4$ distinct points ...
5
votes
1answer
46 views

Expressing the roots of a cubic as polynomials in one root

All roots of $8x^3-6x+1$ are real. (*) The discriminant of $8x^3-6x+1$ is $5184=72^2$ and so the splitting field of $8x^3-6x+1$ has degree $3$. Therefore, all three roots can be expressed as ...
0
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0answers
9 views

Why D≥0 while finding the range of rational functions

To find the range of a rational expression $f(x)=y$, a) We first make a quadratic in $x$ in terms of $y$. b) Make the discriminant $\Delta ≥ 0$ c) Solve the resulting inequality to ...
0
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1answer
21 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
1answer
21 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 ...
0
votes
1answer
13 views

Problem on field extension related to irreducible polynomial

Suppose $\gamma,\gamma'\in\Bbb C$ are distinct roots of the same irreducible polynomial $p\in\Bbb Q[x]$. Suppose $x^2-5$ is irreducible in $\Bbb Q(\gamma)[x] $. Show that it is also irreducible in ...
6
votes
1answer
65 views

$a,b,c,d\ne 0$ are roots (of $x$) to the equation $ x^4 + ax^3 + bx^2 + cx + d = 0 $

Find all quadruplet(s) of non-zero real numbers $ (a,b,c,d) $ such that $ a,b,c$ and $ d$ are roots (of $x$) to the equation $ x^4 + ax^3 + bx^2 + cx + d = 0 $. My friend found a set of rational ...
1
vote
1answer
22 views

Multiplicative inverse of $x+f(x)$ in $\Bbb Q[x]/(f(x))$

So I have $f(x) = x^3-2$ and I have to find the multiplicative inverse of $x + f(x)$ in $\mathbb{Q}[x]/(f(x))$. I'm slightly confused as to how to represent $x + (f(x))$ in $\mathbb{Q}[x]/(f(x))$. ...
0
votes
1answer
49 views

Can I express some power of $\cos(\frac {2\pi}{5})$ as a rational number without using complex numbers?

I have been trying to express a power of $\cos(\frac {2\pi}{5})$ as a "rational number", or trying to find a "rational number" that results from a linear combination of powers of $\cos(\frac ...
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votes
0answers
26 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 ...
2
votes
4answers
50 views

Is $\Bbb Q[x]/(x^2+x)$ isomorphic to $\Bbb Q[x]/(x^2-x)$?

It seems the statement is true, but I have no idea how to prove it I try to let $f=(x^2+x)Q(x)+\bar f=(x^2-2)P(x)+\bar f'$ Then I construct a function $\phi:\Bbb Q[x]/(x^2+x) \to \Bbb ...
-1
votes
0answers
20 views

Let $F$ be a field and let $F[x]$ be a vector space of polynomials $p(x) = \sum a_ix^i$ in the variable $x$ with coefficients $a_i \in F$ [on hold]

Let $F$ be a field and let $F[x]$ be a vector space of polynomials $p(x) = \sum a_ix^i$ in the variable x with coefficients $a_i \in F$. For $k\geq 1$ define $E_k(p(x)) = a_k$. Show that $E_k$ is a ...
0
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2answers
33 views

Dimension of a vector space of polynomials in 3 variables of degree $d$ over $\mathbb{R}$ [duplicate]

Let $V$ be a vector space of homogeneous polynomials in 3 variables $x_1, x_2$ and $x_3$ over $\mathbb{R}$. What is $\dim V$? I think it will be some expression in terms of $d$ but I am not ...
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votes
3answers
3k views

Why are Vandermonde matrices invertible?

A Vandermonde-matrix is a matrix of this form: $$\begin{pmatrix} x_0^0 & \cdots & x_0^n \\ \vdots & \ddots & \vdots \\ x_n^0 & \cdots & x_n^n \end{pmatrix} \in ...
1
vote
0answers
19 views

A property of ratio of two polynomials

I am given a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(x)= \frac{p(x)}{q(x)}$ for all x, and $p(x),q(x)$ are two polynomials with integer coefficients. I have two questions related to ...
3
votes
1answer
95 views

Proving Extended Eisenstein Criterion

I need to prove this extended Eisenstein criterion. Let $f(x)=a_n x^n + \cdots + a_mx^m +\cdots+ a_1x + a_0\in\mathbb Z[x]$ be given. If for $p$ prime, $p$ does not divide $a_m$, $p$ divides $a_i$ ...
1
vote
1answer
37 views

Let $f(x)=x^5+x^2+1$ with $x_1,x_2,x_3,x_4,x_5$ as zeros and …

Let $f(x)=x^5+x^2+1$ with $x_1,x_2,x_3,x_4,x_5$ as zeros and $g(x)=x^2-2.$ Show that $$g (x_1)g (x_2)g (x_3)g (x_4)g (x_5)-30g(x_1x_2x_3x_4x_5)=7$$. I found this question in a local question paper. ...
3
votes
1answer
27 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 ...
2
votes
3answers
63 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
20 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
0answers
30 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 ...
1
vote
0answers
34 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 $S$ ...
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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 ...
13
votes
3answers
699 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
26 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
20 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
667 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
54 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
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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
50 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
31 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
56 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
46 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
25 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
31 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 ...