All Questions
57
questions
7
votes
1
answer
191
views
Factorization and irreducibilty for $x^n-2x^m+1$ trinomials.
I have encountered a weird phenomenon while trying to solve a problem on Reddit. Here is the phenomenon.
Let $a>b \in \mathbb{N}$ and $p_{(a,b)} = x^a - 2x^b + 1$
It seems that if $gcd(a,b,c,d) = 1,...
- 309
-1
votes
2
answers
71
views
Is a polynomial monotone when the first derivative has only imaginary roots? [closed]
I have a polynomial over a specific the range. The first derivative has only two imaginary roots and no real roots. The first derivative is positive in the lower bound and upper bound. Does that mean ...
- 9
5
votes
1
answer
113
views
Irreducible $p(x) \in \mathbb{Q}[X]$ with roots $r, s$ such that $rs = 1$.
If $p(x) \in \mathbb{Q}[X]$ is irreducible and has two roots $r,s$ such that $rs = 1$, then $p(x)$ is of even degree.
I'm not sure how to solve this problem. My initial idea was to consider the field ...
- 464
0
votes
2
answers
79
views
Number of real roots of $f(x) = (x^6) + 2(x^4) + (x^2) - 2(x) + 1$
The question is to find the number of real roots of the polynomial -
$f(x) = (x^6) + 2(x^4) + (x^2) - 2(x) + 1$
I used the Descartes rule of signs. Using it, it is clear that there are maximum number ...
1
vote
0
answers
41
views
Minimal polynomial of a complex number $z$ that is also an algebraic integer does not have $z$ as a repeated root [duplicate]
Let $z\in \mathbb{C}$ be an algebraic integer, and let $P\in \mathbb{Z}[X]$ be its minimal polynomial over $\mathbb{Z}$. Prove that $(X-z)^2 \nmid P(X)$.
My attempt:
My first idea was to prove that $P'...
- 1,144
2
votes
1
answer
64
views
Is there a 3rd degree irreducible polynomial over Q[x], such that two of it's roots' (over C[x]) product equals the third root?
So we have a polynomial in the form: $ax^3+bx^2+cx+d$, where $a,b,c,d\in\mathbb{Q}$, $a\neq 0$. And this is irreducible over $\mathbb{Q}[x]$, but is of course reducible over $\mathbb{C}[x]$.
We have ...
- 29
0
votes
2
answers
169
views
Let $\alpha = 2\cos(\frac{2\pi}{7})$. What is the minimal polynomial of $\alpha$ over $\mathbb{Q}$?
Let $\alpha = 2\cos(\frac{2\pi}{7})$. What is the minimal polynomial of $\alpha$ over $\mathbb{Q}$?
After a lot of calculation, I found a polynomial such that $\alpha $ is a root: $f(x) = x^6 + 2x^5 -...
- 113
7
votes
3
answers
400
views
Reducing $ax^6-x^5+x^4+x^3-2x^2+1=0$ to a cubic equation using algebraic substitutions
Use algebraic substitutions and reduce the sextic equation to the cubic equation, where $a$ is a real number:
$$ax^6-x^5+x^4+x^3-2x^2+1=0$$
My attempts.
First, I tried to use the Rational root ...
- 652
7
votes
3
answers
362
views
A precalculus solution for $x^6 - 3 x^4 + 2 x^3 + 3 x^2 - 3 x + 1 =0$
Using algebra (precalculus) and suggest the solution method for the polynomial $$x^6 - 3 x^4 + 2 x^3 + 3 x^2 - 3 x + 1 =0$$
I'm solving problems on polynomials. I'm stuck here.
My attempts.
First, I ...
- 652
6
votes
4
answers
437
views
Solve in exact form: $x^6 - x^5 + 4 x^4 - 4 x^3 + 4 x^2 - x + 1=0$ ( WolframAlpha failed)
Solve the polynomial in closed form:
$$x^6 - x^5 + 4 x^4 - 4 x^3 + 4 x^2 - x + 1=0$$
WolframAlpha obviously failed.
I tried several ways:
I tried the Rational Root Thereom, but there is no rational ...
- 652
1
vote
0
answers
40
views
Irreducibilty (reducibility) of a family of polynomials
Let $k\geq 3$ and $p\geq 1$ be integers. Set $f_{k,p}(x):=x^k-px^{k-1}-(p+1)x-1$. I need to prove for which parameters $k$ and $p$, this polynomial is irreducible (over $\mathbb{Z}$). For instance, $...
- 629
2
votes
1
answer
54
views
Equation with polynomial with integer coeficients.
Let $p>3$ be a prime number.
Prove that there doesn't exist a pair of polynomials $(f,g)\in{\mathbb{Z}[X]\times\mathbb{Z}[X]}$ such that:
$X^{2p}+pX^{p+1}-1=[(X+1)^p+p\cdot f(X)]\cdot[(X-1)^p+p\...
- 1,144
0
votes
1
answer
123
views
Monic polynomial of degree 2 or 3 in arbitrary integral domain
Let $R$ be an integral domain and $f\in R[x]$ be a monic polynomial of degree $2$ or $3$. Let $K$ be the field of fractions of $R$. Are the following statements true?
$f$ is irreducible in $K[x]$ $\...
- 349
0
votes
0
answers
86
views
How can you find the roots of a polynomial if all the roots are imaginary?
I am trying to find the roots of a 4th-degree polynomial by hand which is actually coming from the derivative of the open-loop transfer function of a control system to find the breakaway points for ...
- 123
0
votes
1
answer
406
views
$F$ characteristic $0$ and $f(X) \in F[X]$ irreducible over $F$ imply $f(X)$ has distinct roots...why must $F$ be of characteristic $0$?
I have a question on this simple corollary from Abstract Algebra by Saracino:
Corollary 24.11 If $F$ is of characteristic $0$ and $f(X) \in F[X]$ is irreducible over $F$, then $f(X)$ has distinct ...
- 1,054