All Questions
42
questions
0
votes
1
answer
53
views
Prove that $f(x)$ is irreducible in $\mathbb{Z}$ with $f(b)$ a prime, $f(b-1) \neq 0$ and $\Re(\alpha_i) < b -1/2$
I need some help with a lemma I need to prove.
First I will provide some background with previous lemmas that I already have been able to prove. Maybe these lemmas are needed to proof the last lemma
...
0
votes
0
answers
58
views
Missing and alternating coefficients of polynomials
I will start my question by providing some necessary context:
Let $g(x) = c_lx^l + ... + a_1x + a$ be a polynomial of degree $l$.
$g$ is said to have no missing coefficients if $c_i \neq 0$ for all $...
1
vote
0
answers
80
views
Reduce Polynomial Over Real Numbers
I was given the question $x^8 + 16$ and told to reduce it as much as able over the real numbers.
Here is what I tried
$x^8 + 16$
$(x^4+4)^2-8x^4$
$(x^4+4-2^{3/2}x^2)(x^4+4+2^{3/2}x^2)$
I can not ...
- 203
1
vote
3
answers
197
views
why is $x^6-x^5 + x^3 - x^2 + 1$ irreducible over $\mathbb{Z}[x]$?
Why is $x^6-x^5 + x^3 - x^2 + 1$ irreducible over $\mathbb{Z}[x]$?
It clearly has no integer roots, and in fact no real roots. Every polynomial with real coefficients can be written as a product of ...
- 2,031
2
votes
2
answers
125
views
Can all the roots of $ax^5+bx^2+c=0$, with real coefficients and $a,c\neq0$, be real numbers?
Let $ax^5+bx^2+c=0$ and $a,c\neq 0$ and $a,b,c$ are real numbers.
Can all the roots of this quintic equation be real numbers?
I divided each side by $a$ and I got
$$x^5+\frac bax^2+\frac ca=0$$
Using ...
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9
votes
5
answers
395
views
Find the sum of $\sum_{i=1}^{5}x^5_i+\sum_{i=1}^{5}\frac{1}{x^5_i}$
Suppose $x^5+5x^3+1=0$ and $x_i$ denotes all the complex roots. Find the sum of
$$\sum_{i=1}^{5}x^5_i+\sum_{i=1}^{5}\frac{1}{x^5_i}$$
This polynomial is irreducible over $\Bbb Q[x]$.
I used Vieta's ...
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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
15
votes
3
answers
642
views
Under the which condition, factorisation of $a_1^n+a_2^n+\cdots+a_n^n-na_1a_2a_3...a_n ?$ is possible?
Under the which condition, factorisation of the polynomial
$$a_1^n+a_2^n+\cdots+a_n^n-na_1a_2a_3...a_n ?$$
is possible?
I know possible cases:
$$a^2+b^2-2ab=(a-b)^2$$
and
$$a^3+b^3+c^3-3abc=(a+b+c)(a^...
- 1,659
6
votes
2
answers
401
views
Find the real root of the almost symmetric polynomial $x^7+7x^5+14x^3+7x-1$
Find the real root of following almost symmetric polynomial by radicals $$p(x)=x^7+7x^5+14x^3+7x-1$$
Here are my attempts.
The coefficients of $p(x)$ are : $1,7,14,7,-1$.
I wanted to try possible ...
- 652
3
votes
1
answer
293
views
Why can't wolfram alpha solve this simple quintic?
So I found out that some transformations break wolfram alpha's ability to solve polynomials. The simplest case I could find is the polynomial $$2x^{5}+5x^{4}+5x^{2}+1=0$$ for which the solution is $$x=...
- 193
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 ...
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-2
votes
2
answers
99
views
Show that the polynomial $P(x)=x^4-x^2-x+2$ has no real roots [closed]
Using clever algebra show that the polynomial
$$P(x)=x^4-x^2-x+2$$
has no real roots.
Obviously, we can not use the derivative.
Using the general quartic formula is terrible.
I tried
$$(x^2+1)^2-3x^...
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0
votes
1
answer
80
views
Reduce the degree and solve the polynomial equation $x^6+ax^4-2x^3+1=0$ by algebraic tricks
Reduce the degree of the polynomial and solve by algebraic tricks: $$x^6+ax^4-2x^3+1=0$$ where $a\in\mathbb R$.
$a=0$ is obviously trivial. I tried all possible algebraic variations.
$$\frac {P(x)}{x^...
- 652
3
votes
3
answers
174
views
Find the all real roots of the polynomial $x^6+3 x^5+3 x-1=0$ in closed form
Find the all real roots of the polynomial
$$x^6+3 x^5+3 x-1=0$$
in exact form.
WolframAlpha gives only numerical results. I've asked a few similar questions before. The source of the problem comes ...
- 652
2
votes
1
answer
116
views
Solve the polynomial $x^6+x^5+x^4-2x^3-x^2+1=0$ in exact form
Try to reduce the degree of the polynomial $$P(x)=x^6+x^5+x^4-2x^3-x^2+1$$ by algebraic ways and find the possible solution method to $P(x)=0$.
The source of the problem comes from a non-english ...
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